Xuyuan Li♠†⋆ Zengqiang Shang♢♠⋆ Peiyang Shi♠ Hua Hua♠† Ta Li♠† Pengyuan
Zhang♢♠†

# Expressive paragraph text-to-speech synthesis with multi-step variational
autoencoder

###### Abstract

Neural networks have been able to generate high-quality single-sentence
speech. However, it remains a challenge concerning audio-book speech synthesis
due to the intra-paragraph correlation of semantic and acoustic features as
well as variable styles. In this paper, we propose a highly expressive
paragraph speech synthesis system with a multi-step variational autoencoder,
called EP-MSTTS. EP-MSTTS is the first VITS-based paragraph speech synthesis
model and models the variable style of paragraph speech at five levels: frame,
phoneme, word, sentence, and paragraph. We also propose a series of
improvements to enhance the performance of this hierarchical model. In
addition, we directly train EP-MSTTS on speech sliced by paragraph rather than
sentence. Experiment results on the single-speaker French audiobook corpus
released at Blizzard Challenge 2023 show EP-MSTTS obtains better performance
than baseline models.

###### keywords:

Paragraph speech synthesis, Expressive speech synthesis, Text-to-speech,
Multi-step VAE

††footnotetext: ⋆ Equal contribution.††footnotetext: ♢ Corresponding author.

##  1 Introduction

Thanks to the development of neural networks, current text-to-speech (TTS)
models [1, 2, 3, 4] have achieved near-human synthesis results on customized
datasets recording independent sentences. However, it remains a challenge for
synthesizing audiobooks. Different from the customized datasets, audiobooks
have two special characteristics: 1) The speech of audiobooks is usually given
in the form of paragraphs because the semantic and acoustic features
(unrelated to semantics such as acoustic environment, hoarse voice, and
nasality) of different sentences in the same paragraph are strongly
correlated. 2) There are dramatic shifts in style within a single paragraph,
as the reader needs to switch between styles to play the role of narrator and
different characters. Recent research [5] points out that TTS methods still
have obvious drawbacks when used to synthesize long stories in audiobooks
compared with humans.

There are many works have been devoted to capturing semantic dependence. [6,
7, 8] used the embedding of the context sentences extracted by pre-trained
language model [9, 10] to predict the global style of the current sentence. In
addition to the global style, [11] additionally models local styles based on
context semantics. StyleSpeech [12] followed this two-stage style modeling and
used unpaired text and audio data to pre-train the context encoder and
reference encoder to improve the generalization. However, these works are
trained at the sentence level and ignore modeling the acoustic features which
are independent of context semantics. These acoustic features are generally
consistent across paragraph-level speech but change throughout the dataset due
to the change of record location and the physiological state of speakers.
Thus, given different sentences in a paragraph, these sentence-level models
could generate speech with very different acoustic features. To generate
speech with consistent acoustic features [13, 14, 15] considered previously
synthesized speech when synthesizing the current sentence. Another solution is
training the model directly on the whole paragraph speech. ParaTTS [16] first
performed this paragraph-level training on a modified Tacotron2 [17]. [18]
applied it on Fastspeech2 [1] and found paragraph-level training has a huge
advantage over single-sentence training in the following two cases: 1)
Paragraphs containing many very short sentences. 2) Paragraphs with dramatic
shifts in style. Another recent work [19] found that increasing the duration
limit of this long-form training can further enhance the performance of
paragraph speech synthesis model.

However, these models are based on FastSpeech or Tacotron, which performs not
as well as VITS [3]. In addition, they model the style of paragraph speech at
no more than two levels of hierarchy, leading to average rhythms. In this
paper, we work on adapting the VITS to generate paragraph speech with highly
expressive. To achieve this, we expand the single-layer variational
autoencoder (VAE) of the original VITS model into five layers, inspired by
HierTTS [20]. We describe how it captures the variable style of paragraph
speech at five levels: frame, phoneme, word, sentence, and paragraph in
Section 2. After that, in Section 3, we propose a novel training strategy to
alleviate the posterior collapse problem of deep VAE. Finally, we apply
paragraph-level training to our proposed model to generate paragraph speech
with smooth transitions in style and acoustic features. The subjective and
objective evaluation results on a French audiobook dataset show that our
proposed model outperforms all baseline models. The results of ablation
studies also demonstrate the effectiveness of our proposed module and training
strategy. Audio can be found at https://lixuyuan102.github.io/EP-MSTTS/.

Figure 1: EP-MSTTS architecture including MSAE, MSTE, and MSD. The DS and US
denote "Downsampling" and ”Upsampling" operations. The PoEnc and PriEnc denote
"Posterior Encoder" and "Prior Encoder". "$\oplus$" represents concatenating
two tensors.

##  2 Model Structure

For convenience, we refer to our proposed model as EP-MSTTS. The overall
architecture of EP-MSTTS can be divided into three modules: Multi-step Audio
Encoder (MSAE), Multi-step Text Encoder (MSTE), and Multi-step Decoder (MSD),
as illustrated in Fig. 1. All modules have five layers including $L_{1}$,
$L_{2}$, $L_{3}$, $L_{4}$, and $L_{5}$, which represent the frame, phoneme,
word, sentence, and paragraph layer, from fine to coarse.

###  2.1 Multi-step Audio Encoder

Within a paragraph of audiobooks, different characters usually have different
styles of sentences. The speaker usually expresses these styles by regularly
changing the pitch and volume of words. To capture the variable style and
express it well, the MSAE breaks down a paragraph speech at five grammatical
levels. Specifically, as shown in the upper left (yellow) part of Fig. 1, we
downsample the hidden state of paragraph linear-spectrogram step by step
according to the pre-given alignment. The acoustic hidden state
$\bm{h_{i}^{au}}$ at each step is encoded into a posterior latent variable by
posterior encoder $\mathrm{G_{post}}$:

$\bm{u_{i}},\bm{\sigma_{i}}=\mathrm{G_{post}}(\bm{h_{i}^{au}})$ (1)

where $\bm{u_{i}}$ and $\bm{\sigma_{i}}$ represent the means and variances of
the posterior latent variables. Since the information in the initial step is
much larger than in the subsequent steps. We only employ the Posterior Encoder
mentioned in VITS [3] as the audio encoder of the frame layer. For the other
levels, we follow the HierTTS [20], which uses a bidirectional GRU with
attention-pooling as the downsample operation to extract coarser-grained
information from the hidden state of the upper layer. All the posterior
encoders are composed of a convolution layer whose output is activated by
Softplus. We emphasize that the MSAE is different from the Hierarchical Audio
Encoder (HAE) in HierTTS, which uses text information as an additional input.
To avoid posterior latent variables learning useless information directly from
the text, our MSAE only learns information from audio.

###  2.2 Multi-step Text Encoder

Coarse-to-fine iterative generation can produce better results than single-
step generation[21]. Based on this principle, we followed the structure of the
Hierarchical Audio Decoder from [20] and modified its layer depth to fit the
grammatical hierarchy of French paragraphs, as shown in the lower left (green)
part of Fig. 1. Starting from the paragraph level, the MSTE encodes semantic
information to the prior latent variables. The specific operation of each step
is written as follows:

$\bm{s_{i}},\bm{u_{i}},\bm{\sigma_{i}}=\mathrm{G_{pri}}(\mathrm{US}(\bm{s_{i-1}}+\bm{z_{i-1})},\bm{h_{i}^{txt}})$
(2)

where prior encoder $\mathrm{G_{pri}}$ predicts the means $\bm{u_{i}}$ and
variances $\bm{\sigma_{i}}$ of prior latent variables from semantic
information $\bm{h_{i}^{txt}}$, posterior latent variables $\bm{z_{i-1}}$ from
the previous step, and the hidden state of prior predictor $\bm{s_{i-1}}$ from
the previous step. $\mathrm{US}$ means Upsampling operation, which will be
described in Sec. 2.3. We use a lookup table to obtain the hidden state of
paragraph phonemes, two convolution layers with a kernel size of 3, and a
Downsampling module to extract the hidden state of words from the Bert subword
embedding. The Downsampling operation is implemented in the same way as in
MSAE. Because semantic information cannot be split to the frame level, we do
not set semantic information input for the frame layer. All prior encoders
consist of $N$ feed-forward transformer blocks [22] and a linear layer.
Following the order from the frame layer to the paragraph layer, we set $N$ to
[4,4,3,3,2] respectively.

###  2.3 Multi-step Decoder

We hypothesize that generating speech waveforms using only $\bm{z}$ from the
frame layer leads to redundancy of information between layers and accumulates
inference errors across layers, since MSTE inference $\bm{z}$ in a recursive
manner. To address this, we upsample the hidden states $\bm{h}$ of the upper
step and concatenate it with $\bm{z}$ at each step, as illustrated in the
right (purple) part of Fig. 1. The Upsampling operation is implemented by the
Length Regulator in [22]. The hidden states at each step are given by equation
(3).

$\bm{h_{i}}=\mathrm{US}(\bm{h_{i-1})}+\bm{z_{i}}$ (3)

We employ the Univnet [23] to generate speech waveforms from the hidden states
in the last step. Since the $\bm{z}$ of each layer can contribute directly to
the reconstruction of the speech waveform. Encoders tend to encode independent
information into different layers. For vocoder, this decoding method also
mitigates the mismatch between training and inference caused by multi-layer
cumulative errors. In addition, we employ a linear layer as the parallel
structure of the vocoder to predict the linear-spectrogram of the paragraph
speech. The specific effect of this parallel structure is described in section
3.

##  3 Training Strategy

We employ KL loss to make the output of MSTE consistent with that of MSAE. The
KL loss $\mathcal{L}_{kl}$ of all grammatical layers can be expressed as:

$\mathcal{L}_{kl}=\alpha_{1}\mathcal{L}_{1}+\alpha_{2}\mathcal{L}_{2}+\alpha_{3}\mathcal{L}_{3}+\alpha_{4}\mathcal{L}_{4}+\alpha_{5}\mathcal{L}_{5}$
(4)

We set the $\alpha_{1}$, $\alpha_{2}$, $\alpha_{3}$, $\alpha_{4}$, and
$\alpha_{5}$ as a decreasing sequence: [1, 0.25, 0.07, 0.01, 0.005] because
the information from the finer-grained layer contributes more to speech
reconstruction. For MSD, we apply the adversarial training on vocoder with
Multi-Period Discriminator [2] and Multi-Resolution Discriminator [23]. The
adversarial loss is denoted by $\mathcal{L}_{g}$ . We apply the L1 loss of
mel-spectrogram $\mathcal{L}_{m}$ and multi-resolution STFT loss [24]
$\mathcal{L}_{s1}$ on the output of the vocoder. In addition, another multi-
resolution STFT loss $\mathcal{L}_{s2}$ is used to constrain the output of the
linear layer parallel to the vocoder. Finally, we use the L2 loss
$\mathcal{L}_{d}$ to train the duration predictor.

We observed that when directly training the total model, Univnet tends to
reconstruct waveform only from the $\bm{z}$ at low levels due to its
overpowering reconstruction capability. This phenomenon, known as the
posterior collapse, underscores the necessity for a less potent vocoder to
force the encoder to supply more useful information. However, restructuring
the waveform is difficult, and we cannot risk compromising the quality of the
speech reconstruction by choosing a simple vocoder structure. Inspired by
MSMC-TTS [25], we designed another task: predicting the linear-spectrogram
with a simple linear layer parallel to the vocoder. We also use the weight
annealing [26] on the weight of $\mathcal{L}_{kl}$ to ensure that our model
has enough time to extract information from audio without text constraints.
Specifically, our training strategy is divided into the following three
stages:

  1. 1.

First, only the MSAE and MSD are trained with the goal of reconstructing the
linear spectrogram features. The weight $\lambda_{kl}$ of $\mathcal{L}_{kl}$
is set to 0.00001. As a result, the total loss can be given as:

$\mathcal{L}_{tot}=2.5\mathcal{L}_{s2}+5\mathcal{L}_{d}+\lambda_{kl}\mathcal{L}_{kl}$
(5)

  2. 2.

Then the $\lambda_{kl}$ starts to increase slowly, which means the latent
variable of each layer begins to adjust the style information it contains with
the constraints of the MSTE. We change the $\lambda_{kl}$ as following
equation:

$\lambda_{kl}=1e^{-5}(Step_{g}-Step_{w})$ (6)

$Step_{g}$ is the global training iterations. $Step_{w}$ is the total number
of iterations for the first stage.

  3. 3.

Finally, the network gives up reconstructing the linear spectrogram features
and instead reconstructs the speech waveform. Meanwhile, $\lambda_{kl}$ keeps
increasing until 1. As a result, the total loss is given as:

$\mathcal{L}_{tot}=\mathcal{L}_{g}+1.5\mathcal{L}_{s1}+2.5\mathcal{L}_{m}+\lambda_{kl}\mathcal{L}_{kl}+\mathcal{L}_{d}$
(7)

The first and second stages were set to 10,000 and 30,000 iterations,
respectively.

##  4 Experiments

###  4.1 Dataset

In this work, we used a portion of the French audiobook dataset [27]. This
part consists of 289 chapters of 5 audiobooks from Librivox read by a female
native French speaker. After slicing, we compiled 18,155 paragraphs 111We cut
26 paragraphs that exceeded 218 seconds into smaller segments that did not
exceed 218 seconds. totaling 51.6 hours. The length distribution of these
paragraph speech is shown in Fig 2. All the audios are encoded in the PCM
format at a sampling rate of 22.05 kHz with transcribed text and partial
aligned phoneme sequences. We used a transformer-based sequence-to-sequence
model to generate the remaining phonemes and used the MFA [28] to align these
phonemes and audios. In our experiments, 150 paragraphs were selected for
testing, 150 paragraphs for validation, and the remaining paragraphs for
training. We extracted the 512-dimensional linear-spectrograms with a hop
length of 300 points and a window length of 800 points. A pre-trained Bert
released on Huggingface222https://github.com/huggingface/transformers was used
to extract the subword embedding for paragraph text.

Figure 2: Probability distribution of paragraph duration length.

###  4.2 Experiment settings

Four models were implemented as baseline models. 1) VITS-U: To demonstrate
that our VITS-based model is more suitable than the original VITS for
synthesizing paragraph speech in audiobooks. We cloned the official
implementation333https://github.com/jaywalnut310/vits of VITS as one of the
baseline models. To eliminate the bias introduced by the vocoder, we replaced
the vocoder structure in VITS with
Univnet444https://github.com/rishikksh20/UnivNet-pytorch. 2) VITS-U-S: We
split the audios in the dataset by sentence and trained the VITS-U on them, to
compare the performance of sentence-level training with paragraph-level
training on VITS. 3) VITS-U-B: We add a style encoder for VITS to represent
models that utilize semantic information to predict the style of paragraph
speech at a limited number of hierarchical levels. The style encoder first
upsample the Bert subword embedding to the word level, as described in Sec
2.2, then downsample it to the phoneme level, and finally converts it to a
style embedding utilizing the same structure as the text encoder in VITS.
VITS-U-B concatenates this style embedding and text embedding together to
predict the prior latent variables. 4) HierTTS: To demonstrate that our
proposed improvements enhance the hierarchical modeling. We cloned the
official implementation555https://github.com/shang0712/HierTTS of HierTTS and
modified it to the same five-layer structure as EP-MSTTS. All models except
VITS-U-S are trained at the paragraph level using a dynamic batch with a total
duration of no more than 218 seconds.

Figure 3: Speech of three continuous sentences generated by VITS-U-S (top) and
VITS-U (bottom).

###  4.3 Evaluation

####  4.3.1 Training Evaluation

We visualized the speech of three continuous sentences generated by VITS-U-S
and VITS-U. The speech of VITS-U-S is generated sentence by sentence and is
concatenated together, while the speech of VITS-U is generated in a single
inference. As shown in Fig. 3, besides its inability to predict the pauses
between sentences, VITS-U-S generates speech with discontinuities in loudness
(the second sentence) and high frequency (the third sentence). On the
contrary, the acoustic features of the speech synthesized by VITS-U are
consistent. In addition, we further computed the mel cepstral distortion (MCD)
and Log-F0 root mean square error (log-F0 RMSE) of generated speech with the
scripts in
ESPnet666https://github.com/espnet/espnet/tree/master/egs2/TEMPLATE/tts1. As
shown in Tab. 1, compared to VITS-U-S, VITS-U decreased by 0.034 on MCD and
remained unchanged on log-F0 RMSE. These results illustrate that paragraph-
level training leads to better paragraph audiobook synthesis on VITS, which is
consistent with the findings of previous experiments on FastSpeech-based and
Tacotron-based models.

Table 1: Subjective and objective comparison on test set

Metric MOS (±95%CI) $\uparrow$ MCD $\downarrow$ log-F0 RMSE $\downarrow$ GT
4.21±0.07 N/A N/A VITS-U-S - 5.692 0.266 VITS-U 3.35±0.09 5.658 0.266 VITS-U-B
3.52±0.07 5.574 0.246 HierTTS 3.80±0.08 5.458 0.250 EP-MSTTS 3.98±0.07 4.876
0.215

####  4.3.2 Model Evaluation

We perform subjective and objective evaluations on models that are trained at
the paragraph level. 25 native French speakers were invited to score as the
role of the 5-scale mean opinion score (MOS) in terms of sound quality,
naturalness, and expressiveness. Each listener was randomly assigned 20
paragraphs from the test set. As Tab. 1 shown, VITS-U-B outperforms VITS-U on
both subjective and objective metrics, which demonstrates the importance of
contextual semantic information for capturing the style of paragraph speech.
In addition, HierTTS, which models the style with five grammatical levels,
outperforms the single-level VITS-U-B on MOS and MCD, but its Log-F0 RMSE
rises by a relative 0.004. This is reasonable because, as described in its
original paper [20], the F0 of speech generated by HierTTS fluctuates over a
wide range. We believe this is mainly due to the recursive five-layer decoder
accumulating inference errors at each layer. Finally, our proposed model
obtains the highest MOS and has significant reductions over HierTTS on
MCD/log-F0 with 0.582/0.035. These results illustrate that our proposed
modifications significantly improve the performance of the VITS-based s
hierarchical model on audiobook paragraph speech synthesis.

Table 2: CMOS results on ablation studies. Models | CMOS
---|---
EP-MSTTS | 0
+Text in MSAE | -0.12
-MSD | -0.26
-Training Strategy | -0.17

####  4.3.3 Ablation Study

EP-MSTTS differs from previous Hierarchical VAE models mainly in MSAE, MSD,
and the training strategy. We conducted the comparative MOS (CMOS) test (on a
7-point scale) to demonstrate the effectiveness of these proposed
modifications. As shown in Tab. 2: 1) After introducing the text input in MASE
like HierTTS, the CMOS is reduced by 0.12. We also observe that the KL loss
converges faster at the word, sentence, and paragraph levels, which means that
less information is compressed into these levels. 2) Removing the MSD and
using only the $\bm{z}$ from the frame layer reduces CMOS by 0.26. This most
significant decrease proves our hypothesis in Sec. 2.3. 3) Finally, replacing
our training strategy with the staged KL weighted annealing in HierTTS reduces
the CMOS by 0.17. It is worth noting that, unlike the staged KL weighted
annealing, our training strategy does not require extensive experience in
deciding when to increase the KL weights of a particular level.

##  5 Conclusion

In this paper, we introduce multi-step VAE into the VITS to enhance its
performance on audiobook paragraph synthesis. In addition, we propose a series
of modifications to improve the multi-step VAE. Experiment results demonstrate
that the paragraph speech synthesized by EP-MSTTS outperforms all the baseline
models. Ablation studies confirm the effectiveness of our proposed
improvements. In the future, we will further analyze the information that has
been decoupled into different grammatical layers and explore to realize the
smooth transition of synthesized speech between paragraphs, using coarse-
grained information of the paragraph layer.

##  6 Acknowledgements

This work is supported by the National Key Research and Development Program of
China(No. 2021YFC3320102) and Postdoctoral Fellowship Program of CPSF(No.
GZB20230811)

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$\displaystyle\,|\gamma|\big{|}v(\gamma)-v({N^{\downarrow}_{\pm}})\big{|}^{+}\big{[}|{N^{\downarrow}_{\pm}}^{\prime}|-|{N^{\downarrow}_{\pm}}|\big{]}+\mathcal{O}(|\gamma|\cdot|\beta_{j}|\cdot|{N^{\downarrow}_{\pm}}|)$
$\displaystyle\leq$
$\displaystyle\,\mathcal{O}(|\gamma|\cdot|\beta_{j}|\cdot|{N^{\downarrow}_{\pm}}|).$

If $\gamma$ is a weak wave coming from the right and interacting with the
strong wave, we have

$\displaystyle[Q_{2}]=$
$\displaystyle\,|\gamma|\cdot\Big{[}\big{|}v({N^{\downarrow}_{\pm}}^{\prime})-v(\gamma)\big{|}^{+}|{N^{\downarrow}_{\pm}}^{\prime}|+\big{|}v(\beta_{i}^{\prime})-v(\gamma)\big{|}^{+}|\beta_{i}^{\prime}|-\big{|}v({N^{\downarrow}_{\pm}})-v(\gamma)\big{|}^{+}|{N^{\downarrow}_{\pm}}|\Big{]}$
$\displaystyle\leq$
$\displaystyle\,\mathcal{O}(|\gamma|\cdot|\beta_{j}|\cdot|{N^{\downarrow}_{\pm}}|).$

In conclusion, the usual a priori condition $\sum|\gamma|\leq\mathcal{O}(1)$
implies that

$[Q]=[Q_{1}]+[Q_{2}]=-\Big{|}\mathcal{O}\big{(}|\beta_{j}|\cdot|{N^{\downarrow}_{\pm}}|\big{)}\Big{|}.\qed$

######  Lemma 3.21 (Case 7. $\beta_{j}C\rightarrow
C^{\prime}\beta_{j}^{\prime}$ or
$C\beta_{j}\rightarrow\beta_{j}^{\prime}C^{\prime}$).

Consider at some time $t$ a weak wave $\beta_{j}$ belonging to the family
$j\neq i$ and crossing $C={C^{\downarrow}_{\pm}}$ or ${C^{\uparrow}}$. Then if
the perturbation satisfies

$\epsilon(t)\leq\mathcal{O}(1),$ (3.46)

then one has

$[Q]=-\big{|}\mathcal{O}\big{(}|\beta_{j}|\cdot|C|\big{)}\big{|}.$

###### Proof.

Consider first the situation when $C={C^{\downarrow}_{\pm}}$. We have the
following estimates,

$|{C^{\downarrow}_{\pm}}|=|{C^{\downarrow}_{\pm}}^{\prime}|+\mathcal{O}(|\beta_{j}|\cdot|{C^{\downarrow}_{\pm}}|),\qquad
v({C^{\downarrow}_{\pm}})=v({C^{\downarrow}_{\pm}}^{\prime})+\mathcal{O}(|\beta_{j}|\cdot|{C^{\downarrow}_{\pm}}|).$

We observe that

$[Q_{1}]=-\big{|}v(\beta_{j})-v({C^{\downarrow}_{\pm}})\big{|}\cdot|{C^{\downarrow}_{\pm}}|\cdot|\beta_{j}|=-\Big{|}\mathcal{O}\big{(}|\beta_{j}||{C^{\downarrow}_{\pm}}|\big{)}\Big{|}.$

For a weak wave $\gamma$ interacting from the left

$\displaystyle[Q_{2}]=$
$\displaystyle\,|\gamma|\cdot\Big{[}\big{|}v(\gamma)-v({C^{\downarrow}_{\pm}}^{\prime})\big{|}^{+}|{C^{\downarrow}_{\pm}}^{\prime}|-\big{|}v(\gamma)-v({C^{\downarrow}_{\pm}})\big{|}^{+}|{C^{\downarrow}_{\pm}}|\Big{]}$
$\displaystyle=$
$\displaystyle\,|\gamma|\cdot\big{|}v(\gamma)-v({C^{\downarrow}_{\pm}})\big{|}^{+}\big{[}|{C^{\downarrow}_{\pm}}^{\prime}|-|{C^{\downarrow}_{\pm}}|\big{]}+\mathcal{O}(|\gamma|\cdot|\beta_{j}|\cdot|{C^{\downarrow}_{\pm}}|)$
$\displaystyle\leq$
$\displaystyle\,\mathcal{O}(|\gamma|\cdot|\beta_{j}|\cdot|{C^{\downarrow}_{\pm}}|).$

If $\gamma$ is a weak wave coming from the right and interacting with the
nonclassical wave, then the same argument shows that

$\displaystyle[Q_{2}]=|\gamma|\cdot\Big{[}\big{|}v({C^{\downarrow}_{\pm}}^{\prime})-v(\gamma)\big{|}^{+}|{C^{\downarrow}_{\pm}}^{\prime}|-\big{|}v({C^{\downarrow}_{\pm}})-v(\gamma)\big{|}^{+}|{C^{\downarrow}_{\pm}}|\Big{]}\leq\mathcal{O}(|\gamma|\cdot|\beta_{j}|\cdot|{C^{\downarrow}_{\pm}}|).$

The required a priori condition is again of the form
$\epsilon(t)=\sum|\gamma|\leq\mathcal{O}(1)$.

We now study the case when $C={C^{\uparrow}}$ and we begin with the estimates.

$|{C^{\uparrow}}|=|{C^{\uparrow}}^{\prime}|+\mathcal{O}(|\beta_{j}|\cdot|{C^{\uparrow}}|),\qquad({C^{\uparrow}})=v({C^{\uparrow}}^{\prime})+\mathcal{O}(|\beta_{j}|\cdot|{C^{\uparrow}}|).$

During this interaction,

$[Q_{1}]=-\big{|}v(\beta_{j})-v({C^{\uparrow}})\big{|}\cdot|{C^{\uparrow}}|\cdot|\beta_{j}|=-\big{|}\mathcal{O}(|\beta_{j}|\cdot|{C^{\uparrow}}|)\big{|}.$

The same proofs as in the last two lemmas show that when an a priori condition
$\epsilon(t)\leq\mathcal{O}(1)$ holds, then

$[Q]=[Q_{1}]+[Q_{2}]=-\big{|}\mathcal{O}(|\beta_{j}|\cdot|{C^{\uparrow}}|)\big{|}.\qed$

###  3.5 Global estimates

We now possess all the required estimates for interactions within splitting-
merging solutions. We need to find a constant $K$ for which $W+KQ$ is
decreasing during all interactions. We now reinterpret Theorem 3.5 and state
the a priori conditions on the perturbation $\epsilon(t)$ and on the constants
$k_{*}^{\lessgtr},K$ required in the proof.

#### Constraints on the weights in $W$

The key is to choose the constants $K,k_{*}$ and $k_{*}^{\lessgtr}$ in such a
way that waves crossing through the strong shocks lead to decreases in $W+KQ$.
The analysis of the various interactions, to be presented immediately below,
will lead to the following constraints on these constants.

  * (W1)

The constants $k_{L},k_{M}$ and $k_{R}$ must satisfy

$\displaystyle k_{L}>$ $\displaystyle\,(1+C^{\flat\flat})k_{M}.$ (3.47)

  * (W2)

The weights $k_{L}^{<},k_{M}^{<}$ and $k_{R}^{<}$ must satisfy

$k_{L}^{<}<k_{M}^{<}<k_{R}^{<},$ (3.48)

  * (W3)

The weights $k_{L}^{>},k_{M}^{>}$ and $k_{R}^{>}$ must satisfy

$k_{L}^{>}>k_{M}^{>}>k_{R}^{>}.$ (3.49)

The validity of the following lemma and the proposed values for the constants
can be checked by inspection.

######  Lemma 3.22.

Assuming only property (H4) of the kinetic function $\Phi^{\flat}_{i}$, the
conditions on the weights (3.47)-(3.49) can be satisfied independently of the
size $\epsilon(t)$ of the perturbation, of the strength of the nonclassical
shock and of the constant $K$. For each $\zeta>0$, one such a choice is given
by

$\begin{array}[]{ccc}k_{L}=1+C^{\flat\flat}+\zeta,&k_{M}=1,&k_{R}=1,\\\
k_{L}^{<}=1-\zeta,&k_{M}^{<}=1,&k_{R}^{<}=1+\zeta,\\\
k_{L}^{>}=1+\zeta,&k_{M}^{>}=1,&k_{R}^{>}=1-\zeta.\end{array}$ (3.50)

#### Constraints on the perturbation

We present conditions that the perturbation $\epsilon(t)$ must satisfy at all
times $t$:

  * (P1)

Lemmas 3.15-3.18 require the following constraints

$\displaystyle\epsilon(t)\leq$
$\displaystyle\,\mathcal{O}\big{(}|{C^{\downarrow}_{\pm}}|\big{)},$ (3.51)
$\displaystyle\epsilon(t)\leq$
$\displaystyle\,\mathcal{O}\big{(}|{C^{\uparrow}}|\big{)},$ (3.52)
$\displaystyle\epsilon(t)\leq$
$\displaystyle\,\mathcal{O}\big{(}|{N^{\downarrow}_{\pm}}|\big{)}.$ (3.53)

Note that Lemmas 3.15 and 3.17 require two different bounds of the form
(3.51).

  * (P2)

Lemmas 3.19-3.21 require three different bounds of the form

$\epsilon(t)\leq\mathcal{O}\big{(}1\big{)}.$ (3.54)

The functions involved in the $\mathcal{O}$ bounds (3.51)-(3.54) are all the
result of quadratic terms in Glimm’s interaction estimates, cf. Theorem 3.7.

#### Constraints on the weights in $W+KQ$

The following fundamental constraints are required to insure that decreases in
the interaction potential compensate for the introduction of spurious second-
order waves during each interaction. The analysis of the various interactions,
to be presented immediately below, will lead to the following constraints on
these constants.

  * (Q1)

The weight $K$ on the interaction potential must satisfy

$\displaystyle k_{L}^{<}+k_{R}^{>}<$
$\displaystyle\,K\big{|}\mathcal{O}(|{C^{\downarrow}_{\pm}}|)\big{|},$ (3.55)
$\displaystyle k_{M}^{<}+k_{R}^{>}<$
$\displaystyle\,K\big{|}\mathcal{O}(|{C^{\uparrow}}|)\big{|},$ (3.56)
$\displaystyle k_{M}+k_{L}^{<}+k_{M}^{>}<$
$\displaystyle\,K\big{|}\mathcal{O}(|{N^{\downarrow}_{\pm}}|)\big{|},$ (3.57)
$\displaystyle k_{L}^{<}+k_{R}^{>}<$
$\displaystyle\,K\big{|}\mathcal{O}(1)\big{|},$ (3.58) $\displaystyle
k_{M}+k_{L}^{<}+k_{M}^{>}<$ $\displaystyle\,K\big{|}\mathcal{O}(1)\big{|},$
(3.59) $\displaystyle k_{M}^{<}+k_{R}^{>}<$
$\displaystyle\,K\big{|}\mathcal{O}(1)\big{|}.$ (3.60)

We now summarize our earlier estimates by combining, case by case, the
estimates for the total variation, Lemmas 3.8-3.13, with the estimates for the
interaction potential, Lemmas 3.15-3.21. Each time, we will indicate which
assumptions on the weights (W1)-(W3), the perturbation (P1)-(P2) and on the
interaction potential (Q1). are required to show that $[W+KQ]$ is negative. We
have already showed in Lemma 3.22 that the constraints on the weights can be
satisfied. As should be clear, for the moment we have not demonstrated that
$\epsilon(t)$, defined in (3.29), is a priori bounded in time. The estimates
summarized below are only valid at the time $t$ of a fixed interaction, and
show that for the strength $\epsilon(t)$ of the perturbation at that time, the
constants can be adjusted to insure that $W+KQ$ will decrease during that
interaction. We will eventually have to show that the bounds on the
perturbation can be satisfied uniformly in time.

Case 1:
($\alpha_{i}{C^{\downarrow}_{\pm}}\rightarrow{N^{\downarrow}_{\pm}}{C^{\uparrow}}$
or
${C^{\downarrow}_{\pm}}\alpha_{i}\rightarrow{N^{\downarrow}_{\pm}}{C^{\uparrow}}$).
In Lemma 3.8, we showed that $\alpha_{i}$ is faster than
${C^{\downarrow}_{\pm}}$, then

$[W]=|\alpha_{i}|\Big{(}-k_{L}+\big{(}k_{L}^{<}+k_{R}^{>}\big{)}\mathcal{O}({C^{\downarrow}_{\pm}})\Big{)},$

while Lemma 3.15 has that
$[Q]=-|\mathcal{O}(|\alpha_{i}||{C^{\downarrow}_{\pm}}|^{2})|$ when (3.51) is
imposed. Therefore, if the constraint (3.55) is satisfied, then $W+KQ$ is
decreasing. Similarly, Lemma 3.8 showed that when $\alpha_{i}$ is slower than
${C^{\downarrow}_{\pm}}$, then

$[W]=|\alpha_{i}|\Big{(}-k_{R}+\big{(}k_{L}^{<}+k_{R}^{>}\big{)}\mathcal{O}({C^{\downarrow}_{\pm}})\Big{)}$

Assuming that the perturbation satisfies an a priori bound (3.51), then
$[Q]=-|\mathcal{O}(|\alpha_{i}||{C^{\downarrow}_{\pm}}|^{2})|$. For weights
satisfying (3.55), then $[W+KQ]$ is negative.

Case 2:
(${N^{\downarrow}_{\pm}}{C^{\uparrow}}\rightarrow{C^{\downarrow}_{\pm}}$). For
the strong interaction
${N^{\downarrow}_{\pm}}{C^{\uparrow}}\rightarrow{C^{\downarrow}_{\pm}}$ Lemma
3.9 stated that

$\displaystyle[W]=$
$\displaystyle\,\,(k_{L}^{<}+k_{R}^{>})\mathcal{O}(|{N^{\downarrow}_{\pm}}||{C^{\uparrow}}||v({N^{\downarrow}_{\pm}})-v({C^{\uparrow}})|).$

Lemma 3.16 states that if the inequality (3.52) is enforced, then $[Q]$ is
negative but proportional to the same quantity, and therefore

$[W+KQ]=\big{(}k_{L}^{<}+k_{R}^{>}-K|\mathcal{O}(1)|\big{)}\Big{|}\mathcal{O}\big{(}|{N^{\downarrow}_{\pm}}||{C^{\uparrow}}||v({N^{\downarrow}_{\pm}})-v({C^{\uparrow}})|\big{)}\Big{|}.$

is negative if $K$ is chosen to satisfy an a priori bound of the form (3.58).

Case 3: ($\alpha_{i}C\rightarrow{C}^{\prime}$ or
$C\alpha_{i}\rightarrow{C}^{\prime}$). According to Lemma 3.10, we have that
if the weak wave crosses the classical strong shock $C={C^{\downarrow}_{\pm}}$
from the left, then

$\displaystyle[W]=$
$\displaystyle\,\,|\alpha_{i}|\Big{(}-k_{L}+(k_{L}^{<}+k_{R}^{>})\mathcal{O}(|{C^{\downarrow}_{\pm}}|)\Big{)}.$

Lemma 3.17 shows that if (3.51) holds, then
$[Q]=-|\mathcal{O}(|\alpha_{i}||{C^{\downarrow}_{\pm}}|^{2})|$. Therefore, if
the constraint (3.55) is satisfied, then the variation $[W+KQ]\leq-
k_{L}|\alpha_{i}|$ is strictly negative.

If the weak wave $\alpha_{i}$ crosses $C={C^{\downarrow}_{\pm}}$ from the
right, then Lemma 3.10 states that the change in $W$ is

$\displaystyle[W]=$
$\displaystyle\,\,|\alpha_{i}|\Big{(}-k_{R}+(k_{L}^{<}+k_{R}^{>})\mathcal{O}(|{C^{\downarrow}_{\pm}}|)\Big{)}.$

Again,
$[Q]=-\big{|}\mathcal{O}(|\alpha_{i}||{C^{\downarrow}_{\pm}}|^{2})\big{|}$
when (3.51) holds, and so

$[W+KQ](t)\leq-k_{R}|\alpha_{i}|$ (3.61)

when the bound (3.55) holds.

Suppose now that $C={C^{\uparrow}}$ is a classical shock. If $\alpha_{i}$
crosses ${C^{\uparrow}}$ from the left, then lemmas 3.10 and 3.17 state

$\displaystyle[W]=$
$\displaystyle\,|\alpha_{i}|\big{(}-k_{M}+(k_{M}^{<}+k_{R}^{>})\mathcal{O}(|{C^{\uparrow}}|)\big{)},$
$\displaystyle[Q]=$
$\displaystyle\,-\big{|}\mathcal{O}(|\alpha_{i}||{C^{\uparrow}}|^{2})\big{|}.$

If the constraint (3.56) is satisfied a priori, then $[W+KQ]\leq-
k_{M}|\alpha_{i}|$ is negative.

Similarly, if $\alpha_{i}$ comes from the right, then

$[W]=|\alpha_{i}|\big{(}-k_{R}+(k_{M}^{<}+k_{R}^{>})\mathcal{O}(|{C^{\uparrow}}|)\big{)}.$

Using the estimate for $[Q]$ from Lemma 3.17 and the a priori condition
(3.56), we conclude that $[W+KQ]\leq-k_{R}|\alpha_{i}|$ is strictly
decreasing.

Case 4:
($\alpha_{i}{N^{\downarrow}_{\pm}}\rightarrow{{N^{\downarrow}_{\pm}}}^{\prime}\alpha_{i}^{\prime}$).
According to Lemma 3.11 and Lemma 3.18

$\displaystyle[W]=$
$\displaystyle\,|\alpha_{i}|\big{(}-(k_{L}-\operatorname{sign}(\alpha_{i})k_{M})+C^{\flat\flat}k_{M}+(k_{M}+k_{L}^{<}+k_{M}^{>})\mathcal{O}(|{N^{\downarrow}_{\pm}}|)\big{)},$
$\displaystyle[Q]=$
$\displaystyle\,-\big{|}\mathcal{O}(|\alpha_{i}||{N^{\downarrow}_{\pm}}|^{2})\big{|},$

where the last bound required (3.53). We impose the bound (3.47) and the
constraint (3.57) on $K$ and these imply that

$[W+KQ]\leq\big{(}-k_{L}+(1+C_{i})k_{M}\big{)}\,|\alpha_{i}|.$ (3.62)

Case 5:
($\beta_{j}{C^{\downarrow}_{\pm}}\rightarrow{{N^{\downarrow}_{\pm}}}^{\prime}{{C^{\uparrow}}}^{\prime}$
or
${C^{\downarrow}_{\pm}}\beta_{j}\rightarrow{{N^{\downarrow}_{\pm}}}^{\prime}{{C^{\uparrow}}}^{\prime}$).
Lemma 3.12 and Lemma 3.20 states that when a weak wave $\beta_{j}$, with
$j>i$, approaches ${C^{\downarrow}_{\pm}}$ from the left, and if the
perturbation satisfies (3.54), then

$\displaystyle[W]=$
$\displaystyle\,|\beta_{j}|\Big{(}-k_{L}^{>}+k_{R}^{>}+(k_{L}^{<}+k_{R}^{>})\mathcal{O}(|{C^{\downarrow}_{\pm}}|)\Big{)},$
$\displaystyle[Q]=$
$\displaystyle\,-\big{|}\mathcal{O}(|\beta_{j}||{C^{\downarrow}_{\pm}}|)\big{|}.$

The linear terms in $[W]$ must control the possible increase in $[Q]$ and so
we balance the terms as

$\displaystyle[W+KQ]=$
$\displaystyle\,|\beta_{j}|\Big{\\{}-k_{L}^{>}+k_{R}^{>}+\big{(}k_{L}^{<}+k_{R}^{>}-K|\mathcal{O}(1)|\big{)}\mathcal{O}(|{C^{\downarrow}_{\pm}}|)\Big{\\}}.$

We impose the zero-th order bound (3.49) the a priori condition (3.55) so that
$W+KQ$ becomes decreasing.

Similarly, if $\beta_{j}$ is a weak wave approaching ${C^{\downarrow}_{\pm}}$
from the right and the perturbation is small in the sense of (3.54), then

$\displaystyle[W+KQ]=$
$\displaystyle\,|\beta_{j}|\Big{\\{}-k_{R}^{<}+k_{L}^{<}+\big{(}k_{L}^{<}+k_{R}^{>}-K\mathcal{O}(1)\big{)}\mathcal{O}(|{C^{\downarrow}_{\pm}}|)\Big{\\}}.$

The conditions and (3.48) and (3.58) are sufficient to make the decreasing
functional.

Case 6:
($\beta_{j}{N^{\downarrow}_{\pm}}\rightarrow{N^{\downarrow}_{\pm}}^{\prime}\beta_{j}^{\prime}$
or
${N^{\downarrow}_{\pm}}\beta_{j}\rightarrow{N^{\downarrow}_{\pm}}^{\prime}\beta_{j}^{\prime}$).
Consider first the interaction with $\beta_{j}$ initially to the left of
${N^{\downarrow}_{\pm}}$. Lemmas 3.13 and 3.21 state that with the condition
(3.54), then

$\displaystyle[W]=$
$\displaystyle\,|\beta_{j}|\Big{(}-k_{L}^{>}+k_{M}^{>}+(k_{M}+k_{L}^{<}+k_{M}^{>})\mathcal{O}(|{N^{\downarrow}_{\pm}}|)\Big{)},$
$\displaystyle[Q]=$
$\displaystyle\,-\big{|}\mathcal{O}(|\beta_{j}||{N^{\downarrow}_{\pm}}|)\big{|}.$

To balance these terms, we must examine

$\displaystyle[W+KQ]=$
$\displaystyle\,|\beta_{j}|\Big{\\{}-k_{L}^{>}+k_{M}^{>}+\big{(}k_{M}+k_{L}^{<}+k_{M}^{>}-K|\mathcal{O}(1)|\big{)}\mathcal{O}(|{N^{\downarrow}_{\pm}}|)\Big{\\}}.$

This functional is decreasing if we further assume that conditions (3.49) and
(3.59) holds, and the exact decrease is

$[W+KQ]\leq\big{(}-k_{L}^{>}+k_{M}^{>}\big{)}|\beta_{j}|.$ (3.63)

Similarly, when $\beta_{j}$ is initially to the right of
${N^{\downarrow}_{\pm}}$, then we impose the bound (3.54) to deduce

$[W+KQ]=|\beta_{j}|\Big{\\{}-k_{M}^{<}+k_{L}^{<}+(k_{M}+k_{L}^{<}+k_{M}^{>}-K|\mathcal{O}(1)|)\mathcal{O}(|{N^{\downarrow}_{\pm}}|)\Big{\\}}\,$

and then we conclude that this is negative if conditions (3.48) and (3.59)
holds. In fact, the amount of decrease is

$[W+KQ]\leq\big{(}-k_{M}^{<}+k_{L}^{<}\big{)}|\beta_{j}|.$ (3.64)

Case 7: ($\beta_{j}C\rightarrow C^{\prime}\beta_{j}^{\prime}$ or
$C\beta_{j}\rightarrow\beta_{j}^{\prime}C^{\prime}$). When the weak wave
$\beta_{j}$ is incoming from the left and $C={C^{\downarrow}_{\pm}}$, then
using lemmas 3.13 and 3.21 under the assumption (3.54), we find

$\displaystyle[W+KQ]=$
$\displaystyle\,|\beta_{j}|\Big{\\{}-k_{L}^{>}+k_{R}^{>}+\big{(}k_{L}^{<}+k_{R}^{>}-K|\mathcal{O}(1)|\big{)}\mathcal{O}(|{C^{\downarrow}_{\pm}}|)\Big{\\}}\leq-
k_{L}^{>}|\beta_{j}|.$

which requires the inequality (3.49) and the a priori condition (3.58) in
order to imply that it is strictly negative.

Similarly, for right incoming weak waves and $C={C^{\downarrow}_{\pm}}$ and
(3.54), we have

$\displaystyle[W+KQ]=$
$\displaystyle\,|\beta_{j}|\Big{\\{}-k_{R}^{<}+k_{L}^{<}+\big{(}k_{L}^{<}+k_{R}^{>}-K|\mathcal{O}(1)|\big{)}\mathcal{O}(|{C^{\downarrow}_{\pm}}|)\Big{\\}}.$

This quantity is negative if the additional conditions (3.48) and (3.58)
holds, in which case the decrease is

$[W+KQ]\leq(-k_{R}^{<}+k_{M}^{<})|\beta_{j}|.$

The last two interactions covered by Case 7. involve either a left incoming
$\beta_{j}$ or a right incoming $\beta_{j}$ interacting with
$C={C^{\uparrow}}$, in which case the a priori bound (3.54) implies
respectively

$[W+KQ]=|\beta_{j}|\Big{\\{}-k_{M}^{>}+k_{R}^{>}+\big{(}k_{M}^{<}+k_{R}^{>}-K|\mathcal{O}(1)|\big{)}\mathcal{O}(|{C^{\uparrow}}|)\Big{\\}}\leq(-k_{M}^{>}+k_{R}^{>})|\beta_{j}|.$
(3.65)

or

$[W+KQ]=|\beta_{j}|\Big{\\{}-k_{R}^{<}+k_{M}^{<}+\big{(}k_{M}^{<}+k_{R}^{>}-K|\mathcal{O}(1)|\big{)}\mathcal{O}(|{C^{\uparrow}}|)\Big{\\}}\leq(-k_{R}^{<}+k_{M}^{<})|\beta_{j}|$
(3.66)

The functional $W+KQ$ is strictly decreasing, as written above, if the bounds
(3.49), (3.48) and (3.60) are satisfied.

###### Proof of Theorem 3.5.

The remaining steps in the proof of existence of a weak solution using front-
tracking approximations are similar in every published proof [32, 11] and we
will only sketch the final argument. The key step is to demonstrate that the
family of front-tracking approximations, parameterized by $h$, possesses a
convergent subsequence that converges as $h\to 0$. The proof that the limit is
a weak solution is straightforward and will be omitted. Recall that $h$ is a
bound on the strength of rarefaction-shocks, on the errors in the propagation
speeds of those discontinuities, and on the approximation of the initial data.
The bound $\epsilon_{0}$ on the size of the initial perturbation (3.4) is not
to be confused with the parameter $h$.

We begin by choosing the state $u_{*}\in\mathcal{B}({\delta_{1}})$ so that
$\Phi^{\sharp}_{i}(u_{*})$ and $\Phi^{\flat}_{i}(u_{*})$ are also in
$\mathcal{B}({\delta_{1}})$. The euclidean distance in phase-space is
equivalent to the generalized shock strength, hence there exists $\kappa_{1}$
for which the bound

$\|v_{0}\|_{L^{\infty}}\leq\kappa_{1}\operatorname{TV}(v_{0})<\kappa_{1}\epsilon_{0}$

can be used to ensure that all states in $u_{0}=\overline{u}_{0}+v_{0}$ belong
to $\mathcal{B}({\delta_{1}})$. To simplify the notation, the function
$\epsilon(t)$ will denote the size of the perturbation in $\operatorname{TV}$,
that is the size of the splitting-merging solution after subtracting the
strong waves.

The analysis in Section 3.5 has shown that under certain conditions, the
functional $W+KQ$ is decreasing. We need to establish that this functional is
equivalent to the quasi-norm $\operatorname{TV}$ for values of $\epsilon_{0}$
and $K$ that are compatible with the constraints. Clearly, there exists
$\kappa_{2}$ such that

$\epsilon(t)\leq\kappa_{2}W(u(t))\leq\kappa_{2}\big{(}W+KQ\big{)}\big{(}u(t)\big{)},$

but the existence of a $\kappa_{3}$ for which the inverse inequality holds,

$\kappa_{3}\big{(}W+KQ\big{)}\big{(}u(t)\big{)}\leq\epsilon(t),$

requires an a priori bound of the form $KW\leq\kappa_{4}$ for some constant
$\kappa_{4}$. The constant $K$ must satisfy the a priori lower bounds (Q1),
namely (3.55)-(3.60), but since Lemma 3.22 shows that for $\zeta<1/2$ the
lower bounds are all at most $3$, these bounds can be rewritten as

$3\leq
K\min\Big{\\{}\big{\|}\mathcal{O}(1)\big{\|}_{\infty},\big{\|}\mathcal{O}\big{(}|{C^{\downarrow}_{\pm}}|\big{)}\big{\|}_{\infty},\big{\|}\mathcal{O}\big{(}|{N^{\downarrow}_{\pm}}|)\big{\|}_{\infty},\big{\|}\mathcal{O}\big{(}|{C^{\downarrow}}|\big{)}\big{\|}_{\infty}\Big{\\}},$
(3.67)

where each of the uniform norms are taken over $\mathcal{B}({\delta_{1}})$.
Assume that $K$ is sufficiently large to satisfy this constraint and then
impose that $\epsilon_{0}$ is sufficiently small that there exists a constant
$\kappa_{4}$ for which the constraint $KW(0)\leq\kappa_{4}$ is satisfied
initially. On the other hand, constraints (P1) and (P2) also require the total
variation of the perturbation to decrease as the size of the strong waves
decreases,

$\epsilon(t)\leq\min\Big{\\{}\big{\|}\mathcal{O}(1)\big{\|}_{\infty},\big{\|}\mathcal{O}\big{(}|{C^{\downarrow}_{\pm}}|\big{)}\big{\|}_{\infty},\big{\|}\mathcal{O}\big{(}|{N^{\downarrow}_{\pm}}|)\big{\|}_{\infty},\big{\|}\mathcal{O}\big{(}|{C^{\downarrow}}|\big{)}\big{\|}_{\infty}\Big{\\}}.$
(3.68)

This upper bound on $\epsilon(t)$ can be satisfied simultaneously with the
upper bound $W\leq\kappa_{4}/K$. Under these conditions, because $W$ will
remain uniformly bounded above in time, the equivalence between
$\operatorname{TV}$ and $W+KQ$ will hold for all time.

Assume that the state $u_{*}$ is fixed and thus, the strength of the strong
waves are only small perturbations of the initial configuration
$\overline{u}_{0}$. Under these conditions, $(W+KQ)(u(t))$ is monotone
decreasing and therefore bounded until the approximation leads to states
outside $\mathcal{B}({\delta_{1}})$. Since $(W+KQ)(u(t))$ is equivalent to
$\operatorname{TV}(u(t)-\overline{u}_{0})$, at the cost of further restricting
$\operatorname{TV}(v_{0})$, the boundedness of $W+KQ$ implies that the
approximation cannot grow in $L^{\infty}$ outise of
$\mathcal{B}({\delta_{1}})$. Glimm’s functional is therefore bounded for all
time. The front-tracking approximations are thus continuous with respect to
time in $L^{1}(\mathbb{R})$, and uniformly bounded in $\operatorname{TV}$ and
$L^{\infty}$. Helley’s Theorem therefore provides a subsequence
$\\{h_{k}\\}_{k}$ converging to zero for which $u_{h_{k}}$ converges in
$L^{1}_{\text{loc}}\cap\operatorname{BV}$. ∎

###### Proof of Theorem 3.6.

The proof is almost identical to the earlier proof, except in this case we
need to propose an explicit relation between the strength of the perturbation
and the strength of the strong shocks.

Consider now the case where the initial state $u_{*}^{(k)}$ belongs to a
sequence in $\mathcal{B}({\delta_{1}})$ that converges to a single point on
the manifold $\mathcal{M}_{i}$ where the wave speed $\lambda_{i}$ has a
critical point. For each index $k$, we may assume that the states
$\Phi^{\sharp}_{i}(u_{*}^{(k)})$ and $\Phi^{\flat}_{i}(u_{*}^{(k)})$ are also
in $\mathcal{B}({\delta_{1}})$. The uniform $\mathcal{O}(1)$ bounds for $K$
and the total variation continue to hold.

In the proof of Theorem 3.5, we saw that as the strength of the strong shock
decreases, then the strength of the perturbation must decrease while the
weight $K$ must increase. More precisely, suppose that the constant $K$
satisfies exactly the constraint (3.67), that is

$K=\frac{3}{F_{1}\big{(}|{C^{\downarrow}_{\pm}}|,|{N^{\downarrow}_{\pm}}|,|{C^{\downarrow}}|\big{)}},$

for a function $F_{1}$ linear with respect to the wave strengths. We remark
that the strength of these strong waves depend only on the left-hand state of
${C^{\downarrow}_{\pm}}$, or ${N^{\downarrow}_{\pm}}$, and these strengths are
all uniform ratios of each other since $\Phi^{\sharp}_{i}$ and
$\Phi^{\flat}_{i}$ are Lipschitz. On the other hand, the condition (3.68)
requires that

$\epsilon(t)\leq
F_{2}\big{(}|{C^{\downarrow}_{\pm}}|,|{N^{\downarrow}_{\pm}}|,|{C^{\downarrow}}|\big{)},$

for some function $F_{2}$ linear in the wave strengths. Taking (3.67) and
(3.68) together, we have

$KW\leq
K\cdot\frac{\epsilon(t)}{\kappa_{3}}=\frac{3}{\kappa_{3}}\cdot\frac{F_{2}(|{C^{\downarrow}_{\pm}}|,|{N^{\downarrow}_{\pm}}|,|{C^{\downarrow}}|)}{F_{1}(|{C^{\downarrow}_{\pm}}|,|{N^{\downarrow}_{\pm}}|,|{C^{\downarrow}}|)}.$

By uniform continuity over $\mathcal{B}({\delta_{1}})$, the ratio
$F_{2}/F_{1}$ is uniformly bounded and hence there exists a constant
$\kappa_{4}$ for which $KW\leq\kappa_{4}$. The equivalence between
$\operatorname{TV}$ and $W+KQ$ is therefore uniform with respect to the
vanishing strengths of the strong waves.

Constraints (P1) are summarized by the condition (3.68) which, because the
strengths of the strong waves are proportional to each other, can be replaced
by a simpler inequality

$\epsilon_{0}\leq\kappa_{*}|{N^{\downarrow}_{\pm}}|.$

The rest of the proof is the same as the one presented for Theorem 3.5. ∎

##  4 Splitting-merging solutions with nucleation

###  4.1 Preliminaries

The objective of this section is to show that splitting-merging solutions can
only undergo a finite number of splitting-merging cycles when the creation of
nonclassical waves requires the more demanding nucleation condition; see
Definition 2.16. This is a generalization of an earlier result by LeFloch and
Shearer in the scalar case [30].

The proof does not follow the techniques of the earlier work [30], which
nonetheless inspired preliminary work of two of the authors on the total
variation for scalar laws [25], and hence the current definition of wave
strength. The key step of the proof presented here is an extension of Lemma
5.2 from [25] which translates the nucleation condition into a lower bound on
the signed variation of the waves crossing ${N^{\downarrow}_{\pm}}$ and
${C^{\uparrow}}$ during a single splitting-merging cycle. The proof therefore
requires some careful accounting of all the waves which interact with, and
ultimately remove, the nonclassical wave.

This section begins with a description of the geometry and processes involved
in a single splitting-merging cycle. The extension of Lemma 5.2 from [30] is
then presented. The signed variation is introduced and related to the previous
bound. We conclude the result by demonstrating that the total variation
decreases during each splitting-merging by an amount proportional to the
strength of the nucleation condition cycle, and therefore, that a splitting-
merging solution for a finite perturbation can only undergo a finite number of
splitting-merging cycles.

###  4.2 Nucleation and the splitting-merging cycle

This section contains a definition of nucleation, a statement of the main
theorem, and a detailed description of the waves and interactions one can
expect to see during a splitting-merging cycle.

The nucleation condition and its associated Riemann solver were defined in
Section 2, and in particular in Lemma 2.6. For the sake of convenience, we
repeat its definition. The nucleation condition assumes that there exists a
$C^{2}$ function
$\mu_{i}^{n}:\mathcal{B}({\delta_{1}})\longrightarrow\mathbb{R}$ satisfying

$\mu_{i}^{\natural}(u)\leq\mu_{i}^{n}(u)\leq\mu_{i}^{\sharp}(u),\quad\text{
for all }u\in\mathcal{B}({\delta_{1}}).$

The nucleation criteria states that $i$-classical shocks are preferred over
nonclassical shocks if

$\big{|}\mu_{i}(u_{-})-\mu_{i}(u_{+})\big{|}<\big{|}\mu_{i}(u_{-})-\mu_{i}^{n}(u_{-})\big{|}.$
(4.1)

Lemma 2.6 describes the resulting Riemann solver. Furthermore, we will assume
that $\mu_{i}^{n}(u_{-})<\mu_{i}^{\sharp}(u_{-})$ uniformly over
$\mathcal{B}({\delta_{1}})$, that is to say that there exists a strictly
positive constant $\eta$ such that

$\eta<\big{|}\mu_{i}^{\sharp}(u)-\mu^{n}_{i}(u)\big{|},$

for all states $u$ that are perturbations of the left-hand state $u_{*}$ in
the splitting-merging solution. We are now in a position to state the main
result of this section.

######  Theorem 4.1 (Finiteness of splitting–merging patterns under
nucleation).

Given any strictly hyperbolic system (1.1) with genuinely nonlinear, linearly
degenerate, or concave-convex characteristic families, together with a
nonclassical Riemann solver satisfying a nucleation condition, there exist
constants $\kappa_{*}$ and $\delta_{*}<\delta_{0}$ depending only upon the
flux function and the constant $C^{\flat\flat}$ arising in (H4), so that the
following property holds: for every family of kinetic mappings
$\Phi^{\flat}_{i}$ associated with concave-convex families (indexed by $i$),
provided they satisfy the conditions (H1)–(H4), any three state pattern
$\big{(}u_{*},\Phi^{\flat}_{i}(u_{*}),\Phi^{\sharp}_{i}(u_{*})\big{)}\in\mathcal{B}(\delta_{*})^{3}$
can only undergo a finite number of splitting-merging cycles, that is, for
every perturbation with total variation at most
$\kappa_{*}|\sigma(u_{*},\Phi^{\flat}_{i}(u_{*}))|$ there exists a
global–in–time solution $u=u(t,x)$ to (1.1) satisfying the initial condition
(3.2) and containing only a finite number of interactions
${N^{\downarrow}_{\pm}}{C^{\uparrow}}\longrightarrow{C^{\downarrow}_{\pm}}$.

Before the nucleation condition can be related to the signed variation of the
strong waves during one splitting-merging cycle, we need to identify all the
waves involved in the interactions. Figure 1 illustrates the waves in a
splitting-merging cycle. Suppose that a classical shock
${C^{\downarrow}_{\pm}}$ interacts with a weak wave at time $t_{0}$ and splits
into a nonclassical shock ${N^{\downarrow}_{\pm}}(t_{0}+)$ and a faster
classical shock ${C^{\uparrow}}(t_{0}+)$. The nucleation condition (4.1) is
satisfied at time $t_{0}$. We assume that the nonclassical shock
${N^{\downarrow}_{\pm}}$ persists until a time $t_{f}$ at which the classical
shock ${C^{\uparrow}}$ crosses the nonclassical shock and results in a single
outgoing classical shock ${C^{\downarrow}_{\pm}}(t_{f}+)$. Suppose that the
left-hand state of ${N^{\downarrow}_{\pm}}(t)$ at some time $t$ is denoted by
$u_{\ell}(t)$ while the right-hand state of ${C^{\uparrow}}(t)$ is written as
$u_{r}(t)$. Note that these are also the left and right-hand states of the
classical shock ${C^{\downarrow}_{\pm}}$ when it is present. Immediately after
the splitting, the intermediate state $u_{m}(t_{0}+)$ is well-defined, just as
$u_{m}(t_{f}-)$ is well-defined immediately prior to the merging.

######  Lemma 4.2.

When nucleation is present, a single splitting-merging cycle from $t=t_{0}$ to
$t_{f}$ satisfies

$\displaystyle\eta<$
$\displaystyle\,\Big{(}\mu_{i}\big{(}\Phi^{\sharp}_{i}\big{(}u_{\ell}(t_{0}+)\big{)}\big{)}-\mu_{i}\big{(}\Phi^{\sharp}_{i}\big{(}u_{\ell}(t_{f}-)\big{)}\big{)}\Big{)}+\Big{(}\mu_{i}\big{(}u_{r}(t_{f}-)\big{)}-\mu_{i}\big{(}u_{r}(t_{0}+)\big{)}\Big{)}.$
(4.2)

###### Proof.

The nucleation condition states that

$0<\eta<\mu_{i}\big{(}\Phi^{\sharp}_{i}\big{(}u_{\ell}(t_{0}+)\big{)}\big{)}-\mu_{i}\big{(}u_{r}(t_{0}+)\big{)},$
(4.3)

while the subsequent merging requires that the speed of
${N^{\downarrow}_{\pm}}(t_{f}-)$ be greater than that of
${C^{\uparrow}}(t_{f}-)$, or more specifically

$a\big{(}u_{r}(t_{f}-),u_{m}(t_{f}-)\big{)}<a\big{(}u_{\ell}(t_{f}-),\Phi^{\flat}_{i}(u_{\ell}(t_{f}-))\big{)}=a\big{(}\Phi^{\sharp}_{i}(u_{\ell}(t_{f}-)),\Phi^{\flat}_{i}(u_{\ell}(t_{f}-))\big{)}.$
(4.4)

Since $u_{m}(t_{f}-)=\Phi^{\flat}_{i}\big{(}u_{\ell}(t_{f}-)\big{)}$, this
inequality on the speeds is equivalent to

$\mu_{i}\Big{(}\Phi^{\sharp}_{i}\big{(}u_{\ell}(t_{f}-)\big{)}\Big{)}<\mu_{i}\big{(}u_{r}(t_{f}-)\big{)}.$
(4.5)

Taking (4.3) and (4.5) together, we obtain that each splitting-merging cycle
with nucleation must satisfy (4.2). ∎

Analysis of the two terms in (4.2) will require us to relate variations in
${N^{\downarrow}_{\pm}}$ and ${C^{\uparrow}}$ to the variation in
$u_{\ell}(t)$ and $u_{r}(t)$. The left and right hand states can be modified
by waves entering the region between the strong shocks, but also by waves
exiting it. On the other hand, we will show that the strength of the waves
exiting the domain is controlled by the strength of the waves entering the
domain. It is this last part that will make the analysis a little bit messier
than one would have hoped.

Figure 1: Illustration of the trajectories of the waves involved in a single
splitting-merging cycle. The $\alpha$ waves belong to the family $i$, the non-
convex family, and the $\beta$ waves are those belonging to families $j\neq
i$.

We now describe all the interactions with ${N^{\downarrow}_{\pm}}$ and
${C^{\uparrow}}$ that could modify the left and right-hand states. Let
$\alpha^{L}_{n}$, $n=1,\ldots,n^{L}$ be the $i$-waves that reach
${N^{\downarrow}_{\pm}}$ from the left and let $\alpha^{R}_{n}$,
$n=1,\ldots,n^{R}$ be the $i$-waves that interact with ${C^{\uparrow}}$ from
the right. The weak waves $\\{\alpha^{L}_{n}\\}_{n}$ have interactions of Case
4. with ${N^{\downarrow}_{\pm}}$ while the waves $\\{\alpha^{R}_{n}\\}_{n}$
have interactions with ${C^{\uparrow}}$ of Case 3. Similarly, let
$\widetilde{\alpha}^{R}_{n},n=1,\ldots,\widetilde{n}^{R}$ be the $i$-waves
crossing ${C^{\uparrow}}$ from the left. Since ${N^{\downarrow}_{\pm}}$ is an
undercompressive, no $i$-waves can interact with it from the right.

Let $\beta^{L}_{j,n}$, $n=1,\ldots,n^{L}_{j}$ be the weak $j$-waves with $j>i$
that interact with ${N^{\downarrow}_{\pm}}$ from the left and define
$\beta^{R}_{j,n}$, $n=1,\ldots,n^{R}_{j}$ be the $j$-waves with $j<i$
interacting with ${C^{\uparrow}}$ from the right. The waves $\beta^{L}_{j,n}$
have interactions described by Case 6. while the waves $\beta^{R}_{j,n}$ have
interactions studied in Case 7. Let
$\widetilde{\beta}^{L}_{j,n},n=1,\ldots,\widetilde{n}^{L}_{j}$ be the
$j$-waves with $j<i$ interacting with ${N^{\downarrow}_{\pm}}$ from the right
and let $\widetilde{\beta}^{R}_{j,n},n=1,\ldots,\widetilde{n}^{R}_{j}$ be the
$j$-waves with $j>i$ reaching ${C^{\uparrow}}$ from the left.

Observe that the interactions of ${N^{\downarrow}_{\pm}}$ and ${C^{\uparrow}}$
with the waves
$\alpha^{L},\alpha^{R},\widetilde{\alpha}^{R},\beta^{L},\widetilde{\beta}^{L},\beta^{R},\widetilde{\beta}^{R}$
generate new weak $i$-waves but all waves $\alpha^{*},\beta^{*}$ generate
$j$-waves between ${N^{\downarrow}_{\pm}}$ and ${C^{\uparrow}}$ These
secondary weak waves will need to be accounted for. Finally, recall our
earlier notation $y^{h}(t)$ for the position of ${N^{\downarrow}_{\pm}}$,
$z^{h}(t)$ for the position of ${C^{\uparrow}}$ and
$\Gamma\subset\mathbb{R}^{+}\times\mathbb{R}$ for the region between these
shocks.

The objective is to demonstrate that

$\big{[}W+KQ\big{]}_{t_{0}+}^{t_{f}-}\leq-\eta<0,$ (4.6)

since it will imply that if $W+KQ$ is a priori bounded in time, and therefore
that the splitting-merging cycles can only occur a finite number of times. As
Lemma 4.2 indicates, the quantity $-\eta$ can be bounded below by the total
change in $u_{\ell}$ and $u_{r}$, which is itself, up to linear terms, the sum
of the strengths of the waves crossing ${N^{\downarrow}_{\pm}}$ and
${C^{\uparrow}}$ during one splitting-merging cycle; see Lemma 4.3. The key
observation is that waves crossing ${N^{\downarrow}_{\pm}}$ from the left will
belong to families $j\geq i$ and waves crossing ${C^{\uparrow}}$ will belong
to families $j<i$, hence the signed variation of the $i$-waves crossing
${N^{\downarrow}_{\pm}}$ will be the same as the signed variation of the
$i$-waves interacting with ${C^{\uparrow}}$ from the left. The trick will be
to correctly account for changes in $Q$ that allow us to reduce the
interactions to linear superpositions.

The change in $W+KQ$ is the sum of the changes during all interactions
occuring between time $t_{0}+$ and $t_{f}-$. Our earlier work has already
shown that $[W+KQ]$ is always negative during these interactions but we will
examine only interactions involving at least one wave among
${N^{\downarrow}_{\pm}},{C^{\uparrow}}$ and the waves in $\Gamma$. The other
interactions will lead to arbitrary decreases in $W+KQ$ that cannot be
controlled.

#### Weak interactions with ${N^{\downarrow}_{\pm}}$

We begin by examining only interactions between $i$-waves $\alpha^{L}_{n}$ and
${N^{\downarrow}_{\pm}}$. These cross ${N^{\downarrow}_{\pm}}$ and a fraction
of the wave continues to travel in $\Gamma$. Secondary waves of families $j<i$
are created and propagate left of ${N^{\downarrow}_{\pm}}$ while secondary
waves of families $j>i$ propagate into $\Gamma$. Lemmas 3.11 and 3.18 quantify
these interactions, which are then summarized in (3.62),

$[W+KQ]\leq\big{(}-k_{L}+(1+C^{\flat\flat})k_{M}\big{)}\,|\alpha^{L}_{n}|.$
(4.7)

Similar calculations show that

$[\mu_{i}(u_{\ell})]=-\operatorname{sign}(\alpha^{L}_{n})|\alpha^{L}_{n}|+\mathcal{O}\big{(}|{N^{\downarrow}_{\pm}}|\cdot|\alpha^{L}_{n}|\big{)},\qquad[\mu_{j}(u_{\ell})]=\mathcal{O}\big{(}|{N^{\downarrow}_{\pm}}|\cdot|\alpha^{L}_{n}|\big{)},\quad
j\neq i,$ (4.8)

where $\operatorname{sign}(\alpha)=-1$ if $\alpha$ is a shock and $+1$ if
$\alpha$ is a rarefaction.

Treating now interactions between $j$-waves, $j>i$, and the strong wave
${N^{\downarrow}_{\pm}}$, as described in Lemmas 3.12 and 3.20, we obtain the
estimate (3.63)

$[W+KQ]\leq\big{(}-k_{L}^{>}+k_{M}^{>}\big{)}|\beta^{L}_{j,n}|.$ (4.9)

This weak wave also modifies the left-hand state by the amount

$[\mu_{j}(u_{\ell})]=-\operatorname{sign}(\beta^{L}_{j,n})|\beta^{L}_{j,n}|+\mathcal{O}\big{(}|{N^{\downarrow}_{\pm}}|\cdot|\beta^{L}_{j,n}|\big{)},\qquad[\mu_{k}(u_{\ell})]=\mathcal{O}\big{(}|{N^{\downarrow}_{\pm}}|\cdot|\beta^{L}_{j,n}|\big{)},\quad
k\neq j.$ (4.10)

Finally, there are $j$-waves $\widetilde{\beta}^{L}_{j,n}$ which cross
${N^{\downarrow}_{\pm}}$ from the right, after having entered $\Gamma$ and
these waves decrease the total variation by

$[W+KQ]\leq\big{(}-k_{M}^{<}+k_{L}^{<}\big{)}|\widetilde{\beta}^{L}_{j,n}|.$
(4.11)

and modify the left-hand state $\mu_{k}(u_{\ell})$ by the amount

$[\mu_{j}(u_{\ell})]=-\operatorname{sign}(\widetilde{\beta}^{L}_{j,n})|\widetilde{\beta}^{L}_{j,n}|+\mathcal{O}\big{(}|{N^{\downarrow}_{\pm}}|\cdot|\widetilde{\beta}^{L}_{j,n}|\big{)},\qquad[\mu_{k}(u_{\ell})]=\mathcal{O}\big{(}|{N^{\downarrow}_{\pm}}|\cdot|\widetilde{\beta}^{L}_{j,n}|\big{)},\quad
k\neq j.$ (4.12)

#### Weak interactions with ${C^{\uparrow}}$

The interactions between $j$-waves, and ${C^{\uparrow}}$ are simpler to treat
than those with ${N^{\downarrow}_{\pm}}$ since $i$-waves cannot cross
${C^{\uparrow}}$. On the other hand, it is important to understand that the
$i$-waves approaching ${C^{\uparrow}}$ from the left must be the remains of
the $i$-waves that previously crossed ${N^{\downarrow}_{\pm}}$.

For $i$-waves reaching ${C^{\uparrow}}$ from the right, we find

$[W+KQ]\leq-k_{R}|\alpha^{R}_{n}|,$ (4.13)

and that the right-hand states are modified by

$[\mu_{i}(u_{r})]=-\operatorname{sign}(\alpha^{R}_{n})|\alpha^{R}_{n}|+\mathcal{O}\big{(}|{C^{\uparrow}}|\cdot|\alpha^{R}_{n}|\big{)},\qquad[\mu_{j}(u_{r})]=\mathcal{O}\big{(}|{C^{\uparrow}}|\cdot|\alpha^{R}_{n}|\big{)},\quad
j\neq i.$ (4.14)

When $i$-waves reach ${C^{\uparrow}}$ from the left, similar estimates hold,
namely

$[W+KQ]\leq-k_{M}|\widetilde{\alpha}^{R}_{n}|,$ (4.15)

and

$[\mu_{k}(u_{r})]=\mathcal{O}\big{(}|{C^{\uparrow}}|\cdot|\widetilde{\alpha}^{R}_{n}|\big{)},\quad\text{
for all }k.$ (4.16)

The interactions involving $j$-waves and ${C^{\uparrow}}$ are simple. If
$j<i$, the estimate (3.66) provides

$[W+KQ]\leq(-k_{R}^{<}+k_{M}^{<})\,|\beta^{R}_{j,n}|.$ (4.17)

and changes in the state on the right-hand side are easily seen to be

$[\mu_{j}(u_{r})]=-\operatorname{sign}(\beta^{R}_{j,n})|\beta^{R}_{j,n}|+\mathcal{O}\big{(}|{C^{\uparrow}}|\cdot|\beta^{R}_{j,n}|\big{)},\qquad[\mu_{k}(u_{r})]=\mathcal{O}\big{(}|{C^{\uparrow}}|\cdot|\beta^{R}_{j,n}|\big{)},\quad
k\neq j.$ (4.18)

On the other hand, for $j>i$, the estimate (3.65) is

$[W+KQ]\leq(-k_{M}^{>}+k_{R}^{>})\,|\widetilde{\beta}^{R}_{j,n}|,$ (4.19)

and the right-hand states change by

$[\mu_{j}(u_{r})]=-\operatorname{sign}(\widetilde{\beta}^{R}_{j,n})|\widetilde{\beta}^{R}_{j,n}|+\mathcal{O}\big{(}|{C^{\uparrow}}|\cdot|\widetilde{\beta}^{R}_{j,n}|\big{)},\qquad[\mu_{k}(u_{r})]=\mathcal{O}\big{(}|{C^{\uparrow}}|\cdot|\widetilde{\beta}^{R}_{j,n}|\big{)},\quad
k\neq j.$ (4.20)

#### Wave crossings between ${N^{\downarrow}_{\pm}}$ and ${C^{\uparrow}}$

The waves entering the region between the two strong shocks are either

  * •

$i$-waves $\alpha^{L}_{n}$ entering from the left;

  * •

$j$-waves $\beta^{L}_{j,n}$, $j>i$, entering from the left;

  * •

$j$-waves $\beta^{R}_{j,n}$, $j<i$, entering from the right.

Our goal is to relate the strength of these waves to the strength of the
outgoing waves $\widetilde{\alpha}^{R}_{n},\widetilde{\beta}^{R}_{j,n}$ and
$\widetilde{\beta}^{L}_{j,n}$.

If all interactions within $\Gamma$ were linear, then the $i$-waves exiting on
the right, namely $\widetilde{\alpha}^{R}_{n}$ would be the result of the
interactions among the fraction of the $i$-waves
$C^{\flat\flat}\alpha^{L}_{n}$ which had crossed ${N^{\downarrow}_{\pm}}$ from
the left. Similarly, since the only $j$-waves with $j>i$ ($j<i$) that entered
the domain were $\beta^{L}_{j,n}$ ($\beta^{R}_{j,n}$), then
$\widetilde{\beta}^{R}_{j,n}$ ($\widetilde{\beta}^{L}_{j,n}$) would be the
result of the interactions among $\beta^{L}_{j,n}$ ($\beta^{R}_{j,n}$). In the
linear case, this would justify the following three bounds

$\displaystyle\sum_{n=1}^{\widetilde{n}^{R}}|\widetilde{\alpha}^{R}_{n}|\leq$
$\displaystyle\,C_{i}\sum_{n=1}^{n^{L}}|\alpha^{L}_{n}|,$ (4.21)
$\displaystyle\sum_{n=1}^{\widetilde{n}^{L}_{j}}|\widetilde{\beta}^{L}_{j,n}|\leq$
$\displaystyle\,\sum_{n=1}^{n^{R}_{j}}|\beta^{R}_{j,n}|,\quad\text{ for all
}j<i,$ (4.22)
$\displaystyle\sum_{n=1}^{\widetilde{n}^{R}_{j}}|\widetilde{\beta}^{R}_{j,n}|\leq$
$\displaystyle\,\sum_{n=1}^{n^{L}_{j}}|\beta^{L}_{j,n}|,\quad\text{ for all
}j>i.$ (4.23)

In the weakly nonlinear regime, these estimates are correct up to second-order
corrections weighted by the total strength of all of the secondary waves which
might be generated by interactions inside $\Gamma$. The correct upper bounds
would therefore be the same, except for the presence of weights of the form
$(1+\epsilon(t))$, which we can simply neglect for the purposes of this
section.

###  4.3 Total variation estimate for the splitting-merging cycle

We now combine all the bounds computed to produce a negative upper bound on
changes in $W+KQ$ during a single splitting-merging cycle. We begin by
returning to the quantity $\eta$ from Lemma 4.2 which was found to be strictly
negative during a cycle.

######  Lemma 4.3.

Suppose $C_{j}^{\sharp}$ is the Lipschitz constant for the map
$\mu_{j}\circ\Phi_{i}^{\sharp}$. Then the nucleation condition is bounded by
the signed variation of the incoming waves according to

$\displaystyle\eta\leq$
$\displaystyle\,\Big{(}C_{i}^{\sharp}+\mathcal{O}\big{(}|{N^{\downarrow}_{\pm}}|\big{)}\Big{)}\sum_{n=1}^{n^{L}}|\alpha^{L}_{n}|+\Big{(}1+\mathcal{O}\big{(}|{C^{\uparrow}}|\big{)}\Big{)}\sum_{n=1}^{n^{R}}|\alpha^{R}_{n}|$
(4.24)
$\displaystyle\,+\sum_{j>i}\Big{(}C_{j}^{\sharp}+\mathcal{O}\big{(}|{N^{\downarrow}_{\pm}}|\big{)}\Big{)}\sum_{n=1}^{n^{L}_{j}}|\beta^{L}_{j,n}|+\Big{(}1+\mathcal{O}\big{(}|{C^{\uparrow}}|\big{)}\Big{)}\sum_{j<i}\sum_{n=1}^{n^{R}_{j}}|\beta^{R}_{j,n}|.$

###### Proof.

We observe immediately that the Lipschitz continuity of the map
$\Phi^{\sharp}_{i}$ implies that there exists constants
$C^{\sharp}_{k},\,k=1,\ldots,n,$ such that

$\Big{|}\mu_{i}\big{(}\Phi^{\sharp}_{i}(u_{\ell}(t_{0}+)\big{)}-\mu_{i}\big{(}\Phi^{\sharp}_{i}(u_{\ell}(t_{f}-)\big{)}\Big{|}<\sum_{k=1}^{n}C^{\sharp}_{k}\Big{|}\mu_{k}\big{(}u_{\ell}(t_{0}+)\big{)}-\mu_{k}\big{(}u_{\ell}(t_{f}-)\big{)}\Big{|}\,.$

The first term of (4.2) is bounded by the sum (4.8), (4.10) and (4.12), while
the second term is bounded by the sum of (4.14), (4.16), (4.18) and (4.20).

$\displaystyle\eta<$
$\displaystyle\sum_{k=1}^{n}C^{\sharp}_{k}\Big{|}\mu_{k}\big{(}u_{\ell}(t_{0}+)\big{)}-\mu_{k}\big{(}u_{\ell}(t_{f}-)\big{)}\Big{|}+\Big{(}\mu_{i}\big{(}u_{r}(t_{f}-)\big{)}-\mu_{i}\big{(}u_{r}(t_{0}+)\big{)}\Big{)}$
$\displaystyle\leq$
$\displaystyle\,C_{i}^{\sharp}\Big{|}\sum_{n=1}^{n^{L}}-\operatorname{sign}(\alpha^{L}_{n})|\alpha^{L}_{n}|\Big{|}+\sum_{n=1}^{n^{L}}\mathcal{O}\big{(}|{N^{\downarrow}_{\pm}}(t_{n}^{L}-)|\cdot|\alpha^{L}_{n}|\big{)}$
$\displaystyle\,+\sum_{j>i}C_{j}^{\sharp}\Big{|}\sum_{n=1}^{n^{L}_{j}}-\operatorname{sign}(\beta^{L}_{j,n})|\beta^{L}_{j,n}|\Big{|}+\sum_{j>i}\sum_{n=1}^{n^{L}_{j}}\mathcal{O}\big{(}|{N^{\downarrow}_{\pm}}(t^{L}_{j,n}-)|\cdot|\beta^{L}_{j,n}|\big{)}$
$\displaystyle\,+\sum_{j<i}C_{j}^{\sharp}\Big{|}\sum_{n=1}^{\widetilde{n}^{L}_{j}}-\operatorname{sign}(\widetilde{\beta}^{L}_{j,n})|\widetilde{\beta}^{L}_{j,n}|\Big{|}+\sum_{j<i}\sum_{n=1}^{\widetilde{n}^{L}_{j}}\mathcal{O}\big{(}|{N^{\downarrow}_{\pm}}(\widetilde{t}^{L}_{j,n}-)|\cdot|\widetilde{\beta}^{L}_{j,n}|\big{)}$
$\displaystyle\,+\sum_{n=1}^{n^{R}}-\operatorname{sign}(\alpha^{R}_{n})|\alpha^{R}_{n}|+\sum_{n=1}^{n^{R}}\mathcal{O}\big{(}|{C^{\uparrow}}(t^{R}_{n}-)|\cdot|\alpha^{R}_{n}|\big{)}$
$\displaystyle\,+\sum_{n=1}^{\widetilde{n}^{R}}-\operatorname{sign}(\widetilde{\alpha}^{R}_{n})|\widetilde{\alpha}^{R}_{n}|+\sum_{n=1}^{\widetilde{n}^{R}}\mathcal{O}\big{(}|{C^{\uparrow}}(\widetilde{t}^{R}_{n}-)|\cdot|\widetilde{\alpha}^{R}_{n}|\big{)}$
$\displaystyle\,+\sum_{j<i}\sum_{n=1}^{n^{R}_{j}}-\operatorname{sign}(\beta^{R}_{j,n})|\beta^{R}_{j,n}|+\sum_{j<i}\sum_{n=1}^{n^{R}_{j}}\mathcal{O}\big{(}|{C^{\uparrow}}(t^{R}_{j,n}-)|\cdot|\beta^{R}_{j,n}|\big{)}$
$\displaystyle\,+\sum_{j>i}\sum_{n=1}^{\widetilde{n}^{R}_{j}}-\operatorname{sign}(\widetilde{\beta}^{R}_{j,n})|\widetilde{\beta}^{R}_{j,n}|+\sum_{j>i}\sum_{n=1}^{\widetilde{n}^{R}_{j}}\mathcal{O}\big{(}|{C^{\uparrow}}(\widetilde{t}^{R}_{j,n}-)|\cdot|\widetilde{\beta}^{R}_{j,n}|\big{)}$

Up to second-order terms, we may assume that the strength of the strong waves
is equal to the constant strength of the unperturbed waves in
$\overline{u}_{0}$. Our earlier bounds show that the strength of the waves
exiting $\Gamma$ are bounded by the strength of the waves entering it, and
therefore, after simplifying the expressions above, we can rewrite the upper
bound above as in (4.24). ∎

We complete this section with a proof of the main theorem. The proof resembles
the accounting that was necessary in the proof of the previous lemma, except
for the manipulation of the weights $k_{*}$ now appearing in $W+KQ$.

###### Proof of Theorem 4.1.

Variations in $W+KQ$ involving only waves in $V_{L}$ and $V_{R}$ always lead
to decreases in Glimm’s functional, and therefore it suffices to account for
changes involving interactions with at least one wave from among
${N^{\downarrow}_{\pm}}$ or ${C^{\uparrow}}$. We invoke the bounds (4.7),
(4.9), (4.11), (4.13), (4.15), (4.17) and (4.19).

$\displaystyle[W+KQ]\leq$
$\displaystyle\,\sum_{n=1}^{n^{L}}\big{(}-k_{L}+(1+C^{\flat\flat})k_{M}\big{)}\,|\alpha^{L}_{n}|$
$\displaystyle\,+\sum_{j>i}\sum_{n=1}^{n^{L}_{j}}\big{(}-k_{L}^{>}+k_{M}^{>}\big{)}\,|\beta^{L}_{j,n}|+\sum_{j<i}\sum_{n=1}^{\widetilde{n}^{L}_{j}}\big{(}-k_{M}^{<}+k_{L}^{<}\big{)}\,|\widetilde{\beta}^{L}_{j,n}|$
$\displaystyle\,+\sum_{n=1}^{n^{R}}-k_{R}|\alpha^{R}_{n}|+\sum_{n=1}^{\widetilde{n}^{R}}-k_{M}|\widetilde{\alpha}^{R}_{n}|$
$\displaystyle\,+\sum_{j<i}\sum_{n=1}^{n^{R}_{j}}\big{(}-k_{R}^{<}+k_{M}^{<}\big{)}\,|\beta^{R}_{j,n}|+\sum_{j>i}\sum_{n=1}^{\widetilde{n}^{R}_{j}}\big{(}-k_{M}^{>}+k_{R}^{>}\big{)}\,|\widetilde{\beta}^{R}_{j,n}|$

We observe that up to second-order terms

$(-k_{L}+(1+C^{\flat\flat})k_{M})\sum_{n=1}^{n^{L}}|\alpha^{L}_{n}|-k_{M}\sum_{n=1}^{\widetilde{n}^{R}}|\widetilde{\alpha}^{R}_{n}|\leq(-k_{L}+k_{M})\sum_{n=1}^{n^{L}}|\alpha^{L}_{n}|.$

Similarly, since the weights are negative, the bounds (4.21)-(4.23) imply that

$\displaystyle\sum_{j<i}\sum_{n=1}^{\widetilde{n}^{L}_{j}}(-k_{M}^{<}+k_{L}^{<})|\widetilde{\beta}^{L}_{j,n}|\leq$
$\displaystyle\,\sum_{j<i}\sum_{n=1}^{n^{L}_{j}}(-k_{M}^{<}+k_{L}^{<})|\beta^{L}_{j,n}|,$
$\displaystyle\sum_{j>i}\sum_{n=1}^{\widetilde{n}^{R}_{j}}(-k_{M}^{>}+k_{R}^{>})|\widetilde{\beta}^{R}_{j,n}|\leq$
$\displaystyle\,\sum_{j>i}\sum_{n=1}^{n^{R}_{j}}(-k_{M}^{>}+k_{R}^{>})|\beta^{R}_{j,n}|.$

These bounds allow us to further simplify the estimates to

$\displaystyle[W+KQ]\leq$
$\displaystyle\,\big{(}-k_{L}+k_{M}\big{)}\sum_{n=1}^{n^{L}}|\alpha^{L}_{n}|-k_{R}\sum_{n=1}^{n^{R}}|\alpha^{R}_{n}|$
$\displaystyle\,+\big{(}-k_{L}^{>}+k_{R}^{>}\big{)}\,\sum_{j>i}\sum_{n=1}^{n^{L}_{j}}|\beta^{L}_{j,n}|+\big{(}-k_{R}^{<}+k_{L}^{<}\big{)}\,\sum_{j<i}\sum_{n=1}^{n^{R}_{j}}|\beta^{R}_{j,n}|.$
(4.25)

For the choices of weights $k_{*}$ proposed in Lemma 3.22, we find that
$k_{R}=1$ and

$\displaystyle-k_{L}+k_{M}=$
$\displaystyle\,-(1+C^{\flat\flat}+\zeta)+1=-C^{\flat\flat}-\zeta,$
$\displaystyle-k_{L}^{>}+k_{R}^{>}=$
$\displaystyle\,-(1+\zeta)+(1-\zeta)=-2\zeta,$ $\displaystyle-
k_{R}^{<}+k_{L}^{<}=$ $\displaystyle\,-(1+\zeta)+(1-\zeta)=-2\zeta.$

In order to bound above the negative quantity (4.25) by a multiple of $-\eta$,
it suffices to observe that the weights on (4.24) are all bounded uniformly,
and hence for every fixed positive value of $\zeta$, there exists a positive
constant $c$ such that

$[W+KQ]\leq-c\eta.$

Splitting and mergings cannot proceed indefinitely since each cycle leads
$W+KQ$ to decrease by a constant amount. The constant $c\eta$ being uniform as
the front-tracking scheme converges to a weak solution, the result also holds
for that weak solution. ∎

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