# Linear Degeneracy in a Class of
Nonlinear Second-Order Hyperbolic Systems

Heinrich Freistühler  Department of Mathematics, University of Konstanz, 78457
Konstanz, Germany; Supported by DFG Grant No. FR 822/11-1

(June 1, 2024)

##  1 Formation of singularities in first-order systems

A most prominent feature of quasilinear hyperbolic systems

$\displaystyle V_{t}+A(V)V_{x}=0,$ (1.1)

$A:\Omega\to\mathbb{R}^{n\times n},\Omega\subset\mathbb{R}^{n}$, is the
property that many smooth solutions to them develop singularities in finite
time. Such solutions exist whenever the system has a genuinely nonlinear mode,
i.e., there is a smooth (eigenvalue,eigenvector) pair
$(\lambda,R):\Omega^{*}\to\mathbb{R}\times\mathbb{R}^{n}\setminus\\{0\\}$,

$\displaystyle(-\lambda(V)I+A(V))R(V)=0,\quad V\in\Omega^{*},$ (1.2)

defined on an open neighborhood $\Omega^{*}\subset\Omega$ of some state
$V^{*}\in\Omega$ with

$\displaystyle(R(V)\cdot\nabla\lambda(V))|_{V=V^{*}}\neq 0.$ (1.3)

The condition (1.3) of _genuine nonlinearity_ goes back to Lax [10], and first
general results on singularity formation to him [11], John [8], and Liu [12].
For inhomogeneous systems

$\displaystyle V_{t}+A(V)V_{x}=G(V),$ (1.4)

with source $G:\Omega\to\mathbb{R}^{n}$, Bärlin has recently shown [1] that
under the sole assumptions that

$\displaystyle G(V^{*})=0$ (1.5)

and (1.3) hold, there exist, for any Euclidean neighborhood $\Omega^{*}$ of
$V^{*}$, smooth data $V_{0}:\mathbb{R}\to\Omega^{*}$ such that the associated
regular solution, while its values stay in $\Omega^{*}$, ceases to be
differentiable after finite time.

The present note has been prompted by the fact that this classical mechanism
of singularity formation due to genuine nonlinearity will generically transfer
to second-order hyperbolic systems. However, we show that this is not the case
for a class of systems of the special form

$B^{00}(U)U_{tt}+C^{j}(U)U_{tx^{j}}+B^{jk}(U)U_{x^{j}x^{k}}=H(U,U_{t},U_{x}).$

Systems of this form have recently been introduced as models for dissipative
relativistic fluid dynamics, and an application of the main result of this
paper – Proposition 1 below – is to provide an indication that solutions to
these models may avoid singularity formation, similarly to what is generically
expected for the hyperbolic-parabolic models of dissipative fluid dynamics
such as the Navier-Stokes-Fourier system for compressible fluids [9].

##  2 Dispersion relations and modes

We begin our study with a linear constant-coefficients system

$\displaystyle B^{00}U_{tt}+C^{j}U_{tx^{j}}+B^{jk}U_{x^{j}x^{k}}=0$ (2.1)

for $U:[0,\infty)\times\mathbb{R}^{d}\to\mathbb{R}^{n}$, where
$B^{00},C^{j},B^{jk},j,k=1,...,d,$ are real $n\times n$-matrices. Writing

$B(\xi_{0},{\bm{\xi}}):=\xi_{0}^{2}B^{00}+\xi_{0}C({\bm{\xi}})+B({\bm{\xi}}),\quad{\bm{\xi}}=(\xi_{1},...,\xi_{d}),$

with $C({\bm{\xi}})=C^{j}\xi_{j},B({\bm{\xi}})=B^{jk}\xi_{j}\xi_{k}$, we
assume that (2.1) is semi-strictly definitely hyperbolic in the sense that

  * •

$B^{00}<0$,

  * •

for any ${\bm{\xi}}\in S^{d-1}$, the polynomial

$\xi_{0}\mapsto p^{\bm{\xi}}(\xi_{0}):=\det(B(\xi_{0},{\bm{\xi}}))$

of degree $2n$ has exclusively real non-vanishing zeroes $\xi_{0}$, whose
multiplicities do not depend on ${\bm{\xi}}$, and

  * •

for any such zero of multiplicity $\nu$, the amplitude space

$X(\xi_{0},{\bm{\xi}})=\ker B(\xi_{0},{\bm{\xi}})$

has dimension $\nu$.

Note that the equation

$\displaystyle p^{\bm{\xi}}(\xi_{0})=0$ (2.2)

is the dispersion relation of system (2.1).

With $U_{t}=:P,U_{x_{j}}=:Q_{j}$, system (2.1) implies

$\displaystyle\begin{aligned}
B^{00}P_{t}+C^{j}P_{x^{j}}+B^{jk}(Q_{j})_{x^{k}}&=0\\\
(Q_{j})_{t}-P_{x^{j}}&=0,\end{aligned}$ (2.3)

which we write as

$\displaystyle\begin{pmatrix}B^{00}&&&\\\ &I&&\\\ &&\ddots&\\\
&&&I\end{pmatrix}\begin{pmatrix}P\\\ Q_{1}\\\ \vdots\\\
Q_{d}\end{pmatrix}_{t}+\begin{pmatrix}C^{k}&B^{1k}&\cdots&B^{dk}\\\
-\delta_{1}^{k}I&0&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
-\delta_{d}^{k}I&0&\cdots&0\end{pmatrix}\begin{pmatrix}P\\\ Q_{1}\\\ \vdots\\\
Q_{d}\end{pmatrix}_{x^{k}}=0$ (2.4)

or briefly

$\displaystyle A^{0}V_{t}+A^{k}V_{x^{k}}=0.$ (2.5)

######  Lemma 2.1.

(i) The dispersion relation of (2.3) is

$\displaystyle 0=q^{\bm{\xi}}(\xi_{0}):=\xi_{0}^{(d-1)n}p^{\bm{\xi}}(\xi_{0})$
(2.6)

(ii) The non-zero modes $(\lambda({\bm{\xi}}),R({\bm{\xi}})),{\bm{\xi}}\in
S^{d-1},$ of the first-order formulation (2.3)/(2.4)/(2.5), i.e., the pairs
$(\lambda({\bm{\xi}}),R({\bm{\xi}}))$ satisfying $\lambda({\bm{\xi}})\neq 0$
and

$\\{0\\}\neq R({\bm{\xi}})=\ker(-\lambda({\bm{\xi}})A^{0}+A({\bm{\xi}})),\quad
A({\bm{\xi}})\equiv A^{k}\xi_{k},$

are given by

$(\lambda({\bm{\xi}}),R({\bm{\xi}}))=(-\xi_{0},\bar{X}(\xi_{0},{\bm{\xi}})),\quad{\bm{\xi}}\in
S^{d-1},$

with $\xi_{0}$ a zero of $p^{\bm{\xi}}$ and

$\bar{X}(\xi_{0},{\bm{\xi}})=\\{\begin{pmatrix}\xi_{0}x\\\ \xi_{1}x\\\
\vdots\\\ \xi_{d}x\end{pmatrix}:x\in X(\xi_{0},{\bm{\xi}})\\}.$

(iii) The zero mode is $(0,R_{0}({\bm{\xi}}))$ with some $R({\bm{\xi}})$ of
dimension $(d-1)n$.

###### Proof.

(i) With $d>1$ one has

$\displaystyle\begin{aligned}
q^{\bm{\xi}}(\xi_{0})=&\det(A^{\alpha}\xi_{\alpha})=\left|\begin{matrix}B^{00}\xi_{0}+C^{k}\xi_{k}&B^{1k}\xi_{k}&\cdots&B^{dk}\xi_{k}\\\
-\xi_{1}I&\xi_{0}I&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
-\xi_{d}I&0&\cdots&\xi_{0}I\\\ \end{matrix}\right|\\\
=&\det(\xi_{0}I)\left|\begin{matrix}B^{00}\xi_{0}+C^{k}\xi_{k}+\xi_{0}^{-1}\xi_{d}\xi_{k}B^{dk}&B^{1k}\xi_{k}&\cdots&B^{(d-1)k}\xi_{k}\\\
-\xi_{1}I&\xi_{0}I&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
-\xi_{d-1}I&0&\cdots&\xi_{0}I\end{matrix}\right|\\\
=&\ldots=(\det(\xi_{0}I))^{d}\left|\begin{matrix}B^{00}\xi_{0}+C^{k}\xi_{k}+\xi_{0}^{-1}\xi_{j}\xi_{k}B^{jk}\end{matrix}\right|\\\
=&\xi_{0}^{dn}\det(\xi_{0}^{-1}B(\xi_{0},{\bm{\xi}}))\\\
=&\xi_{0}^{(d-1)n}p^{\bm{\xi}}(\xi_{0})\end{aligned}$ (2.7)

thus (2.6) for $\xi_{0}\neq 0$, and by continuity also for $\xi_{0}=0$. $d=1$
analogously.

(ii) Obviously,

$\displaystyle\begin{pmatrix}B^{00}\xi_{0}+C^{k}\xi_{k}&B^{1k}\xi_{k}&\cdots&B^{dk}\xi_{k}\\\
-\xi_{1}I&\xi_{0}I&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
-\xi_{d}I&0&\cdots&\xi_{0}I\\\ \end{pmatrix}\begin{pmatrix}\bar{x}_{0}\\\
\bar{x}_{1}\\\ \vdots\\\ \bar{x}_{d}\end{pmatrix}=\begin{pmatrix}0\\\ 0\\\
\vdots\\\ 0\end{pmatrix}$ (2.8)

is equivalent to $\bar{x}_{\alpha}=\xi_{\alpha}x$ with some $x\in
X(\xi_{0},{\bm{\xi}})$.

(iii) The left kernel of $A({\bm{\xi}})$, i.e.,
$\\{(0,l_{1},...,l_{d})\in\mathbb{R}^{(d+1)n}:\xi^{k}l_{k}=0\\}$ has dimension
$(d-1)n$, and thus also the right kernel $R({\bm{\xi}})$. ∎

##  3 Quasisemilinear second-order systems

From now on we consider nonlinear systems of the form

$\displaystyle
B^{00}(U)U_{tt}+C^{j}(U)U_{tx^{j}}+B^{jk}(U)U_{x^{j}x^{k}}=H(U,U_{t},U_{x}),$
(3.1)

i.e., the differential operator is linear in the highest, second-order,
derivatives, with coefficients depending only on the unknown functions itself,
not its first-order derivatives. The unknown/solution $U$ ranges in some open
subset $\mathcal{U}$ of $\mathbb{R}^{n}$.

We assume that the operator on the left-hand side of (3.1) is pointwise semi-
strictly definitely hyperbolic, in the sense that every linear operator

$U\mapsto
B^{00}(U^{*})U_{tt}+C^{j}(U^{*})U_{tx^{j}}+B^{jk}(U^{*})U_{x^{j}x^{k}},\quad
U^{*}\in\mathcal{U},$

is semi-strictly definitely hyperbolic

With again $Q=U_{x}$, system (3.1) has the first-order counterpart

$\displaystyle\begin{aligned}
B^{00}(U)P_{t}+C^{j}(U)P_{x^{j}}+B^{jk}(U)(Q_{j})_{x^{k}}&=H(U,P,Q)\\\
(Q_{j})_{t}-P_{x^{j}}&=0\\\ U_{t}&=P,\end{aligned}$ (3.2)

which we write as

$\displaystyle\mathcal{A}^{0}(\mathcal{V})\mathcal{V}_{t}+\mathcal{A}^{k}(\mathcal{V})\mathcal{V}_{x^{k}}=\mathcal{H}(\mathcal{V}).$
(3.3)

with

$\mathcal{V}=(P,Q_{1},...,Q_{d},U),$
$\mathcal{A}^{0}(\mathcal{V})=\begin{pmatrix}B^{00}(U)&&&&\\\ &I&&&\\\
&&\ddots&&\\\ &&&I&\\\
&&&&I\end{pmatrix},\mathcal{A}^{k}(\mathcal{V})=\begin{pmatrix}C^{k}(U)&B^{1k}(U)&\cdots&B^{dk}(U)&0\\\
-\delta_{1}^{k}I&0&\cdots&0&0\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\
-\delta_{d}^{k}I&0&\cdots&0&0\\\ 0&0&\cdots&0&0\end{pmatrix}$

and

$\mathcal{H}(\mathcal{V})=(H(U,P,Q),0,...,0,P).$

The modes of (3.2), i.e., pairs
$(\lambda(\mathcal{V},{\bm{\xi}}),\mathcal{R}(\mathcal{V},{\bm{\xi}}))$ with

$\\{0\\}\neq\mathcal{R}(\mathcal{V},{\bm{\xi}})=\ker(-\lambda(\mathcal{V},{\bm{\xi}})\mathcal{A}^{0}(\mathcal{V})+\mathcal{A}(\mathcal{V},{\bm{\xi}})),\quad\mathcal{A}(\mathcal{V},{\bm{\xi}})\equiv\mathcal{A}^{k}(\mathcal{V})\xi_{k},$

now depend on the state $\mathcal{V}$, with
$\mathcal{R}(\mathcal{V},{\bm{\xi}})$ consisting of all solutions
$(\bar{x}_{0},\bar{x}_{1},...,\bar{x}_{d},y)\in\mathbb{R}^{(d+1)n}$ of

$\displaystyle\begin{pmatrix}B^{00}(U)\xi_{0}+C^{k}(U)\xi_{k}&B^{1k}(U)\xi_{k}&\cdots&B^{dk}(U)\xi_{k}&0\\\
-\xi_{1}I&\xi_{0}I&\cdots&0&0\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\
-\xi_{d}I&0&\cdots&\xi_{0}I&0\\\
0&0&\cdots&0&\xi_{0}I\end{pmatrix}\begin{pmatrix}\bar{x}_{0}\\\ \bar{x}_{1}\\\
\vdots\\\ \bar{x}_{d}\\\ y\end{pmatrix}=\begin{pmatrix}0\\\ 0\\\ \vdots\\\
0\\\ 0\end{pmatrix}$ (3.4)

with $-\lambda=\xi_{0}$ equal to $0$ or a zero of $p^{\bm{\xi}}_{U}=\ker
B(U,.,{\bm{\xi}})$.

In view of the close proximity of (3.4) and (2.8), the non-zero modes are of
the form

$\displaystyle\mathcal{R}(\mathcal{V},{\bm{\xi}})=R(U,{\bm{\xi}})\times\\{0\\},$
(3.5)

while

$\mathcal{R}_{0}(\mathcal{V},{\bm{\xi}})=(R_{0}(U,{\bm{\xi}})\times\\{0\\})+(\\{0\\}\times\mathbb{R}^{n}).$

Following Lax [10], a smooth mode
$(\lambda(\mathcal{V},{\bm{\xi}}),\mathcal{R}(\mathcal{V},{\bm{\xi}}))$ of a
hyperbolic system (3.3) is called linearly degenerate if the directional
derivative, in state space, of the characteristic speed $\lambda$ with respect
to the state $\mathcal{V}$ vanishes along the associated amplitude space
$\mathcal{R}(\mathcal{V},{\bm{\xi}})$.

$\mathcal{R}(\mathcal{V},{\bm{\xi}})\cdot\frac{\partial}{\partial\mathcal{V}}(\lambda(\mathcal{V},{\bm{\xi}}))=0.$

The point of this note is the following.

######  Proposition 1.

All modes of the first order formulation (3.2) of the quasisemilinear system
(3.1) are linearly degenerate.

###### Proof.

For the zero mode, the linear degeneracy trivial. For the non-zero modes it
follows from (3.5) and the fact that for every ${\bm{\xi}}\in S^{d-1}$,
$\lambda(.,{\bm{\xi}})$ is a function of $U$ alone. ∎

######  Corollary 1.

The first-order versions of all models of dissipative relativistic fluid
dynamics considered in [5, 6, 7, 2, 3, 4] have exclusively linearly degenerate
modes.

Could it be that solutions to any of these models for smooth data generally
avoid singularity formation?

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