DPF2013-45 September 30, 2013

Measurements of $B\rightarrow DK$ decays to constrain the CKM Unitarity
Triangle angle $\gamma$ and related results at LHCb

Daniel Craik 111On behalf of the LHCb Collaboration.

Department of Physics
University of Warwick, Coventry, UK

> Constraints on the CKM angle $\gamma$ are presented from GLW, ADS, and GGSZ
> analyses of $B^{\pm}\rightarrow DK^{\pm}$ at the LHCb experiment. The
> branching fractions of $B^{0}\rightarrow\overline{D}^{0}K^{+}\pi^{-}$ and
> $B_{s}^{0}\rightarrow\overline{D}^{0}K^{-}\pi^{+}$ are also reported,
> measured relative to the related mode
> $B^{0}\rightarrow\overline{D}^{0}\pi^{+}\pi^{-}$.

> PRESENTED AT
>
>
>
>
> DPF 2013
> The Meeting of the American Physical Society
> Division of Particles and Fields
> Santa Cruz, California, August 13–17, 2013
>

##  1 Measurements of $\gamma$ from $B^{\pm}\rightarrow DK^{\pm}$

The CKM angle
$\gamma=\text{arg}(-V_{\text{ud}}V^{*}_{\text{ub}}/V_{\text{cd}}V^{*}_{\text{cb}})$
is currently the least well-constrained angle in the Unitarity Triangle. So
far, the most-sensitive measurements of $\gamma$ from a single experiment have
been performed by Belle [1] and BaBar [2]. These measurements yield values of
$\left(68^{+15}_{-14}\right)^{\circ}$ and
$\left(69^{+17}_{-16}\right)^{\circ}$, respectively.

Tree-level processes such as $B^{\pm}\rightarrow DK^{\pm}$ provide a
theoretically clean measurement of $\gamma$ with no contributions from new
physics processes. This measurement can be compared with measurements from
loop-mediated processes, which are sensitive to new physics, to provide a test
of the Standard Model. The current limits on the CKM Unitarity Triangle due to
tree-level and loop processes, as calculated by the CKMFitter group [3], are
shown in Fig. 1.

Figure 1: Constraints on the CKM Unitarity Triangle due to (left) tree-level
processes and (right) loop-mediated processes.

###  1.1 GLW/ADS analysis of $B^{\pm}\rightarrow DK^{\pm}$ and
$B^{\pm}\rightarrow D\pi^{\pm}$

The GLW method [4] uses $D$ decays to $C\\!P$ eigenstates such as $K^{+}K^{-}$
and $\pi^{+}\pi^{-}$. Decays can proceed either via a $D^{0}$ or a
$\overline{D}^{0}$ with a phase difference of $\delta_{B}+\gamma$. Suppression
in the decay via $D^{0}$ with respect to the $\overline{D}^{0}$ decay limits
interference to $\mathcal{O}(10\,\%)$ in $B^{\pm}\rightarrow DK^{\pm}$ and
$\mathcal{O}(1\,\%)$ in $B^{\pm}\rightarrow D\pi^{\pm}$.

The ADS method [5] uses $D$ decays to quasi-flavour-specific states such as
$\pi^{+}K^{-}$ and $\pi^{-}K^{+}\pi^{+}\pi^{-}$. Here the suppression of one
of the $B$ decays is partially balanced by the suppression of one of the $D$
decays, giving larger interference terms while also introducing an additional
phase shift of $\delta_{D}$.

Analyses have been performed on $B^{\pm}\rightarrow DK^{\pm}$ and
$B^{\pm}\rightarrow D\pi^{\pm}$ with the $D$ meson reconstructed from the
final states $K^{+}K^{-}$, $\pi^{+}\pi^{-}$, $K^{+}\pi^{-}$, $\pi^{+}K^{-}$,
$K^{-}\pi^{+}\pi^{+}\pi^{-}$ and $\pi^{-}K^{+}\pi^{+}\pi^{-}$ using LHCb data
corresponding to $1\,\text{fb}^{-1}$ of $pp$ collisions at a centre of mass
energy of 7 TeV [6, 7]. The invariant mass distributions of the two- and four-
body suppressed ADS modes are shown in Fig. 2 and Fig. 3, respectively.

Figure 2: Fits to the invariant mass distributions of the two-body suppressed
ADS mode $\pi^{\mp}K^{\pm}$ in (top) $B^{\mp}\rightarrow DK^{\mp}$ and
(bottom) $B^{\mp}\rightarrow D\pi^{\mp}$. The $B^{\mp}\rightarrow DK^{\mp}$
and $B^{\mp}\rightarrow D\pi^{\mp}$ components are shown in red and green,
respectively. The shaded component indicates partially reconstructed
backgrond, the dashed magenta line corresponds to partially reconstructed
$\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}h^{-}$ and the total shape also
includes a combinatoric background. Figure 3: Fits to the invariant mass
distributions of the four-body suppressed ADS mode
$\pi^{\mp}K^{\pm}\pi^{\pm}\pi^{\mp}$ in (top) $B^{\mp}\rightarrow DK^{\mp}$
and (bottom) $B^{\mp}\rightarrow D\pi^{\mp}$. The $B^{\mp}\rightarrow
DK^{\mp}$ and $B^{\mp}\rightarrow D\pi^{\mp}$ components are shown in red and
green, respectively. The shaded component indicates partially reconstructed
backgrond, the dashed magenta line corresponds to partially reconstructed
$B_{s}^{0}\rightarrow DK^{-}\pi^{+}$ and the total shape also includes a
combinatoric background.

The observables measured are the ratio of $DK$ to $D\pi$ for each $D$ final
state,

$R_{K/\pi}^{f}=\frac{\Gamma\left(B^{-}\rightarrow D\left[\rightarrow
f\right]K^{-}\right)+\Gamma\left(B^{+}\rightarrow
D\left[\rightarrow\overline{f}\right]K^{+}\right)}{\Gamma\left(B^{-}\rightarrow
D\left[\rightarrow f\right]\pi^{-}\right)+\Gamma\left(B^{+}\rightarrow
D\left[\rightarrow\overline{f}\right]\pi^{+}\right)}\,,$

the charge asymmetry for each final state,

$A_{h}^{f}=\frac{\Gamma\left(B^{-}\rightarrow D\left[\rightarrow
f\right]h^{-}\right)-\Gamma\left(B^{+}\rightarrow
D\left[\rightarrow\overline{f}\right]h^{+}\right)}{\Gamma\left(B^{-}\rightarrow
D\left[\rightarrow f\right]h^{-}\right)+\Gamma\left(B^{+}\rightarrow
D\left[\rightarrow\overline{f}\right]h^{+}\right)}\,,$

and the ratio of the suppressed to favoured modes for $D\rightarrow K\pi$ and
$D\rightarrow K\pi\pi\pi$,

$R_{h}^{\pm}=\frac{B^{\pm}\rightarrow
D\left[f_{\text{sup}}\right]h^{\pm}}{B^{\pm}\rightarrow
D\left[f\right]h^{\pm}}\,.$

The values obtained for each of these observables can be found in Refs. [6,
7]. These variables serve as inputs for the combined $\gamma$ measurements in
Section 1.3 and Section 1.4.

###  1.2 GGSZ analysis of $B^{\pm}\rightarrow DK^{\pm}$

The GGSZ method [8] exploits the variation of the strong phase $\delta_{D}$
across the Dalitz plot in $D$ decays to three-body self-conjugate states such
as $K_{S}^{0}\pi^{+}\pi^{-}$ and $K_{S}^{0}K^{+}K^{-}$. The Dalitz plot is
divided into bins, as shown in Fig. 4, chosen to maximise statistical
sensitivity. The populations of $B^{+}$ and $B^{-}$ decays in each bin are
given by

$N^{+}_{\pm i}=h_{B^{+}}\left[K_{\mp i}+(x^{2}_{+}+y^{2}_{+})K_{\pm
i}+2\sqrt{K_{i}K_{-i}}(x_{+}c_{\pm i}\mp y_{+}s_{\pm i})\right]\,,$
$N^{-}_{\pm i}=h_{B^{-}}\left[K_{\pm i}+(x^{2}_{-}+y^{2}_{-})K_{\mp
i}+2\sqrt{K_{i}K_{-i}}(x_{-}c_{\pm i}\pm y_{-}s_{\pm i})\right]\,,$

where $K_{\pm i}$ is the efficiency corrected yield in bin $\pm i$ due to
$D^{0}$ flavour tagged events from BaBar [9, 10] and $c_{\pm i}$ and $s_{\pm
i}$ are the cosine and sine of the strong phase $\delta_{D}$ in bin $\pm i$
from CLEO-c [11].

Figure 4: Binning schemes used for the Dalitz plots of (left) $D\rightarrow
K_{S}^{0}\pi^{+}\pi^{-}$ and (right) $D\rightarrow K_{S}^{0}K^{+}K^{-}$. Bins
in the top-left half of the plots
($m^{2}_{K_{S}^{0}h^{-}}>m^{2}_{K_{S}^{0}h^{+}}$) are identified as $+i$ and
bins in the bottom-right half are labeled $-i$.

The remaining parameters are left free in the fit to the data: $h_{B^{\pm}}$
are normalisation factors for $B^{\pm}$, and
$x_{\pm}=r_{B}\text{cos}(\delta_{B}\pm\gamma)$ and
$y_{\pm}=r_{B}\text{sin}(\delta_{B}\pm\gamma)$ are the Cartesian parameters,
which are sensitive to $\gamma$.

Analyses have been performed on $B^{\pm}\rightarrow DK^{\pm}$ with the $D$
meson reconstructed in the final states $K_{S}^{0}\pi^{+}\pi^{-}$ and
$K_{S}^{0}K^{+}K^{-}$ using LHCb data corresponding to $1\,\text{fb}^{-1}$ of
$pp$ collisions at a centre of mass energy of 7 TeV [12] and
$2\,\text{fb}^{-1}$ of $pp$ collisions at a centre of mass energy of 8 TeV
[13]. The values obtained for the Cartesian parameters in the 8 TeV analysis
are

$x_{+}=\left(-8.7\pm 3.1\text{(stat.)}\pm 1.6\text{(syst.)}\pm
0.6\text{(ext.)}\right)\times 10^{-2}\,,$ $x_{-}=\left(\phantom{-}5.3\pm
3.2\text{(stat.)}\pm 0.9\text{(syst.)}\pm 0.9\text{(ext.)}\right)\times
10^{-2}\,,$ $y_{+}=\left(\phantom{-}0.1\pm 3.6\text{(stat.)}\pm
1.4\text{(syst.)}\pm 1.9\text{(ext.)}\right)\times 10^{-2}\,,$
$y_{-}=\left(\phantom{-}9.9\pm 3.6\text{(stat.)}\pm 2.2\text{(syst.)}\pm
1.6\text{(ext.)}\right)\times 10^{-2}\,,$

where the third uncertainty is due to the CLEO-c strong phase measurements
used in the fit.

Combining these values with the results from the 7 TeV analysis and fitting
for $\gamma$, $r_{B}$ and $\delta_{B}$ yields values of $(57\pm 16)^{\circ}$,
$(8.8^{+2.3}_{-2.4})\times 10^{-2}$ and $(124^{+15}_{-17})^{\circ}$,
respectively, where the values for $\gamma$ and $\delta_{B}$ are modulo
$180^{\circ}$. Two-dimensional projections of the confidence regions for these
parameters are shown in Fig. 5.

Figure 5: Two-dimensional projections of the confidence regions onto the
(left) $(\gamma,r_{B})$ and (right) $(\gamma,\delta_{B})$ planes. Contours
indicate the $1$, $2$ and $3\sigma$ boundaries and diamonds mark the central
values.

###  1.3 Combination of results from $1\,\text{fb}^{-1}$ measurements

The results in Section 1.1 and Section 1.2 are combined using a frequentist
approach to obtain a more constraining measurement of $\gamma$ [14]. In
addition to these results further measurements are included to improve the
fit: measurements of the strong phases and coherence factors for $D\rightarrow
K\pi$ and $D\rightarrow K\pi\pi\pi$ decays from CLEO-c[15], $C\\!P$ asymmetry
measurements of the neutral $D$ mesons from the Heavy Flavour Averaging
Group[16] and charm mixing parameters from LHCb[17]. A likelihood is
constructed from the measured observables as

${\cal
L}\left(\vec{\alpha}\right)={\displaystyle\prod_{i}}\xi_{i}\left(\vec{A}_{i}^{\text{obs}}|\vec{\alpha}\right)\,,$

where the sum is over the different measurements, $\vec{\alpha}$ is the set of
parameters and $\xi_{i}$ denotes the likelihood probability density functions
(PDFs) of the observables $\vec{A}_{i}^{\text{obs}}$. For most observables a
Gaussian PDF is assumed, however, where highly non-Gaussian behaviour is
observed, the experimental likelihood is used.

A combined $\gamma$ measurement has been performed including the results from
Section 1.1 and a subset of the results from Section 1.2 corresponding to
$1\,\text{fb}^{-1}$ of $pp$ collisions at a centre of mass energy of 7 TeV
[12]. The best-fit values and confidence intervals (modulo $180^{\circ}$) of
$\gamma$ are given in Table 1 and the $1-\text{CL}$ curves for $\gamma$ are
shown in Fig. 6.

combination | $\gamma$ | 68 % CL | 95 % CL
---|---|---|---
$DK$ | $72.0^{\circ}$ | $[56.4,86.7]^{\circ}$ | $[42.6,99.6]^{\circ}$
$D\pi$ | $18.9^{\circ}$ | $[7.4,99.2]^{\circ}\cup[167.9,176.4]^{\circ}$ | -
$DK$ and $D\pi$ | $72.6^{\circ}$ | $[55.4,82.3]^{\circ}$ | $[40.2,92.7]^{\circ}$
Table 1: Best-fit values and confidence intervals for $\gamma$ from the
combination of $DK$ and $D\pi$ measurements.

Figure 6: $1-\text{CL}$ curves for $\gamma$ from the combined
$1\,\text{fb}^{-1}$ GLW/ADS and $1\,\text{fb}^{-1}$ GGSZ measurements using
(left) only $DK$, (centre) only $D\pi$ and (right) both decay modes.

###  1.4 Combination including $3\,\text{fb}^{-1}$ GGSZ measurement

Another combination [18] has been performed that incorporates all of the
results reported in Section 1.2 but only those observables from Section 1.1
corresponding to $B^{\pm}\rightarrow DK^{\pm}$ decays. Mixing in the neutral
$D$ mesons is also neglected in the equations used for the observables in this
combination.

The best-fit values and confidence intervals (all modulo $180^{\circ}$) for
$\gamma$, $r_{B}$ and $\delta_{B}$ are given in Table 2. Figure 7 and Figure 8
show the $1-\text{CL}$ curve for $\gamma$, and the 2D projection of the
likelihood in $\gamma$ and $r_{B}$, respectively.

quantity | value | 68 % CL | 95 % CL
---|---|---|---
$\gamma$ | $67.2^{\circ}$ | $[55.1,79.1]^{\circ}$ | $[43.9,89.5]^{\circ}$
$r_{B}$ | $0.0923$ | $[0.0843,0.1001]$ | $[0.0762,0.1075]$
$\delta_{B}$ | $114.3^{\circ}$ | $[101.3,126.3]^{\circ}$ | $[88.7,136.3]^{\circ}$
Table 2: Best-fit values and confidence intervals for $\gamma$, $r_{B}$ and
$\delta_{B}$ from the combination of $DK$ measurements including GGSZ
measurements from $3\,\text{fb}^{-1}$ of data. Figure 7: $1-\text{CL}$ curve
for $\gamma$ from the combined $1\,\text{fb}^{-1}$ GLW/ADS and
$3\,\text{fb}^{-1}$ GGSZ measurements. Figure 8: Two-dimensional projection of
the confidence regions onto the $(\gamma,r_{B})$ plane. Contours show the $1$
and $2\sigma$ boundaries and markers indicate the central values.

##  2 Measurement of $B_{(s)}^{0}\rightarrow DK\pi$ branching fractions

The decay mode $B^{0}\rightarrow DK^{+}\pi^{-}$ has potential for a
significant future measurement of $\gamma$ [19, 20, 21]. Sensitivity to
$\gamma$ comes from the interference of $b\rightarrow c$ and $b\rightarrow u$
amplitudes of a similar magnitude. $B_{s}^{0}\rightarrow DK^{-}\pi^{+}$ and
the related mode $B_{s}^{0}\rightarrow D^{*}K^{-}\pi^{+}$ form important
backgrounds to this mode, therefore, an understanding of these modes is
necessary.

Branching fraction measurements of $B^{0}\rightarrow DK^{+}\pi^{-}$ and
$B_{s}^{0}\rightarrow DK^{-}\pi^{+}$, relative to the normalisation mode
$B^{0}\rightarrow D\pi^{+}\pi^{-}$, have been made using LHCb data
corresponding to $1\,\text{fb}^{-1}$ of $pp$ collisions at a centre of mass
energy of 7 TeV [22]. The invariant mass distributions of $D\pi\pi$ and
$DK\pi$ candidates where the $D$ is reconstructed from
$\overline{D}^{0}\rightarrow K^{+}\pi^{-}$ are shown in Fig. 9. The measured
relative branching fractions are

$\frac{{\cal B}\left(B^{0}\to\overline{D}^{0}K^{+}\pi^{-}\right)}{{\cal
B}\left(B^{0}\to\overline{D}^{0}\pi^{+}\pi^{-}\right)}=0.106\pm
0.007\,\text{(stat.)}\pm 0.008\,\text{(syst.)}\,,$ $\frac{{\cal
B}\left(B_{s}^{0}\to\overline{D}^{0}K^{-}\pi^{+}\right)}{{\cal
B}\left(B^{0}\to\overline{D}^{0}\pi^{+}\pi^{-}\right)}=1.18\pm
0.05\,\text{(stat.)}\pm 0.12\,\text{(syst.)}\,.$

Figure 9: Fits to the $B^{0}_{(s)}$ candidate invariant mass distributions for
the (a) $D\pi\pi$ and (b) $DK\pi$ samples. Data points are shown in black, the
full fitted PDFs as solid blue lines and the components as detailed in the
legends.

These relative measurements yield absolute branching fractions of

${\cal B}\left(B^{0}\to\overline{D}^{0}K^{+}\pi^{-}\right)=(9.0\pm
0.6\,\text{(stat.)}\pm 0.7\,\text{(syst.)}\pm 0.9({\cal B}))\times 10^{-5}\,,$
${\cal B}\left(B_{s}^{0}\to\overline{D}^{0}K^{-}\pi^{+}\right)=(1.00\pm
0.04\,\text{(stat.)}\pm 0.10\,\text{(syst.)}\pm 0.10({\cal B}))\times
10^{-3}\,,$

where the third uncertainty arises from the uncertainties on ${\cal
B}(B^{0}\to\overline{D}^{0}\pi^{+}\pi^{-})$. This is the most precise
measurement of ${\cal B}(B^{0}\to\overline{D}^{0}K^{+}\pi^{-})$ to date and
the first measurement of ${\cal B}(B_{s}^{0}\to\overline{D}^{0}K^{-}\pi^{+})$.

Although no quantitative analysis of the Dalitz plots has yet been attempted,
the Dalitz plot distributions obtained (corrected for efficiency) are
presented in Fig. 10.

Figure 10: Efficiency corrected Dalitz plot distributions for (a)
$B^{0}\rightarrow\overline{D}^{0}\pi^{+}\pi^{-}$, (b)
$B^{0}\rightarrow\overline{D}^{0}K^{+}\pi^{-}$ and (c)
$B_{s}^{0}\rightarrow\overline{D}^{0}K^{-}\pi^{+}$ candidates obtained from
the signal weights.

##  3 Conclusions and prospects

The $B^{\pm}\rightarrow DK^{\pm}$ decay mode offers an excellent opportunity
to measure the CKM angle $\gamma$ from Standard Model processes. The
combination in Section 1.4 gives the most sensitive measurement of $\gamma$
from a single experiment so far, yielding a value of $(67\pm 12)^{\circ}$.
This measurement is expected to improve further with the completion of a
GLW/ADS analysis on the remaining $2\,\text{fb}^{-1}$ of LHCb data currently
available. In addition, other modes such as $B^{0}\rightarrow DK^{+}\pi^{-}$
offer great prospects for future $\gamma$ measurements.

ACKNOWLEDGMENTS

This work is funded in part by the European Research Council under FP7 and by
the United Kingdom’s Science and Technology Facilities Council.

## References

  * [1] K. Trabelsi [Belle Collaboration], arXiv:1301.2033 [hep-ex].
  * [2] J. P. Lees et al. [BaBar Collaboration], Phys. Rev. D 87 (2013) 052015 [arXiv:1301.1029 [hep-ex]].
  * [3] J. Charles et al. [CKMfitter Group Collaboration], Eur. Phys. J. C 41 (2005) 1 [hep-ph/0406184].
  * [4] M. Gronau and D. Wyler, Phys. Lett. B 265 (1991) 172.
  * [5] D. Atwood, I. Dunietz and A. Soni, Phys. Rev. Lett. 78 (1997) 3257 [hep-ph/9612433].
  * [6] R. Aaij et al. [LHCb Collaboration], Phys. Lett. B 712 (2012) 203 [Erratum-ibid. B 713 (2012) 351] [arXiv:1203.3662 [hep-ex]].
  * [7] R. Aaij et al. [LHCb Collaboration], Phys. Lett. B 723 (2013) 44 [arXiv:1303.4646 [hep-ex]].
  * [8] A. Giri, Y. Grossman, A. Soffer and J. Zupan, Phys. Rev. D 68 (2003) 054018 [hep-ph/0303187].
  * [9] B. Aubert et al. [BaBar Collaboration], Phys. Rev. D 78 (2008) 034023 [arXiv:0804.2089 [hep-ex]].
  * [10] P. del Amo Sanchez et al. [BaBar Collaboration], Phys. Rev. Lett. 105 (2010) 121801 [arXiv:1005.1096 [hep-ex]].
  * [11] J. Libby et al. [CLEO Collaboration], Phys. Rev. D 82 (2010) 112006 [arXiv:1010.2817 [hep-ex]].
  * [12] R. Aaij et al. [LHCb Collaboration], Phys. Lett. B 718 (2012) 43 [arXiv:1209.5869 [hep-ex]].
  * [13] R. Aaij et al. [LHCb Collaboration], LHCb-CONF-2013-004 (2013).
  * [14] R. Aaij et al. [LHCb Collaboration], arXiv:1305.2050 [hep-ex].
  * [15] N. Lowrey et al. [CLEO Collaboration], Phys. Rev. D 80 (2009) 031105 [arXiv:0903.4853 [hep-ex]].
  * [16] Y. Amhis et al. [Heavy Flavor Averaging Group Collaboration], arXiv:1207.1158 [hep-ex].
  * [17] R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett. 110 (2013) 101802 [arXiv:1211.1230 [hep-ex]].
  * [18] R. Aaij et al. [LHCb Collaboration], LHCb-CONF-2013-006 (2013).
  * [19] M. Gronau, Phys. Lett. B 557 (2003) 198 [hep-ph/0211282].
  * [20] T. Gershon, Phys. Rev. D 79 (2009) 051301 [arXiv:0810.2706 [hep-ph]].
  * [21] T. Gershon and M. Williams, Phys. Rev. D 80 (2009) 092002 [arXiv:0909.1495 [hep-ph]].
  * [22] R. Aaij et al. [LHCb Collaboration], Phys. Rev. D 87 (2013) 112009 [arXiv:1304.6317 [hep-ex]].


and establish the corresponding Hele-Shaw free boundary equation, we have
followed the same lines of proof as initiated in [54], with the gradient
estimate as in [24]. This has the advantage to also prove optimal second order
estimates. A fundamental new ingredient is a uniform $L^{1}$ estimate on the
time derivative of pressure. With this new estimate, another possible route to
establish the complementarity condition, the hard part of the problem, would
be to use the pure compactness method in [46, 55]. Still another possible
route is through the obstacle problem, see [33] and the references therein. We
have also established uniqueness, finite propagation speed, and limit energy
functional of solutions to this Hele-Shaw type free boundary problem. In
addition, we have studied the incompressible limit for the stationary state of
the PKS model, which is new for the diffusion-aggregation equations.

We would like to point out that our analysis for the PKS model is compatible
with growth terms, as they appear naturally when dealing with mechanical
models of tumor growth, even though the technical details have to be checked.
Also we treated dimension $n\geq 3$ to avoid technical issues with the Sobolev
inequalities but we do not expect difficulties in two dimensions.

Several papers have treated of linear drift terms, see [38, 25]. But it is
difficult to extend these cases or Newtonian potential to more general
attractive potential, because our proof of the time derivative estimate of
pressure strongly depends on the structure of Newtonian potential. Among open
problems, let us also mention the convergence rate with $m\to\infty$, which
has been obtained in few papers, [1, 23]. Finally the case of systems is only
treated without drift, see [12, 46]. Large time asymptotic of solutions to the
Hele-Shaw system (2.6)–(2.7) is an interesting topic. [21] treated the
2-dimensional Hele-Shaw model when the initial density is a patch function.
But that for the n-dimensional ($n\geq 3$) Hele-Shaw case is still largely
open. The regularity of free boundary for a Hele-Shaw problem of tumor growth
was obtained in [52], also for the porous medium equation with an external
drift, cf. [38].

### Acknowledegements

The authors would like to thank Noemi David and Markus Schmidtchen for helpful
discussions and comments.
The research of Q. H. and H-L. L. was supported partially by National Natural
Science Foundation of China (No.11931010, 11671384 and 11871047), and by the
key research project of Academy for Multidisciplinary Studies, Capital Normal
University, and by the Capacity Building for Sci-Tech Innovation-Fundamental
Scientific Research Funds (No.007/20530290068).
B.P. has received funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (grant
agreement No.740623).

##  Appendix A Appendix: $L^{\infty}$ bound

Under the initial assumption (2.1), we show the solution to (1.1) is bounded
in any finite time using Moser’s iteration technique. In order to justify the
$L^{\infty}$ estimate of density, we introduce the following approximate
equations of the Cauchy problem (1.1):

$\begin{cases}\partial_{t}\rho_{\varepsilon}=\Delta\rho_{\varepsilon}^{m}+\varepsilon\Delta\rho_{\varepsilon}+\nabla\cdot\big{(}\rho_{\varepsilon}\nabla(J_{\varepsilon}\ast\mathcal{N}_{\varepsilon}\ast\rho_{\varepsilon})\big{)},&\\\
\rho_{\varepsilon}(x,0)=\rho_{0,\varepsilon}(x)=J_{\varepsilon}\ast\rho_{0},\\\
\|\rho_{\varepsilon}(t)\|_{L^{1}(\mathbb{R}^{n})}=\|\rho_{0,\varepsilon}\|_{L^{1}(\mathbb{R}^{n})}\leq\|\rho_{0}\|_{L^{1}(\mathbb{R}^{n})},\end{cases}$
(A.1)

where $J_{1}\in C^{\infty}(\mathbb{R}^{n})$ with both
$\|J_{1}\|_{L^{1}(\mathbb{R}^{n})}=1$ and ${\rm supp}(J_{1})\subset B_{1}$ is
the standard mollifier and
$J_{\varepsilon}:=\frac{1}{\varepsilon^{d}}J_{1}(\frac{x}{\varepsilon})$. We
have

######  Lemma A.1.

Let the initial assumption (2.1) hold. Then, the solution $\rho_{\varepsilon}$
to the Cauchy problem Eq. (A.1) satisfies for any $T>0$, $r\in[1,\infty)$,

$\sup\limits_{0\leq t\leq
T}\|\rho_{\varepsilon}(t)\|_{L^{r}(\mathbb{R}^{n})}\leq
C(M,T,m,n,r,\|\rho_{0}\|_{L^{\infty}(\mathbb{R}^{n})}).$

###### Proof.

Multiplying Eq. (A.1) by $\rho_{\varepsilon}^{r-1}$ and integrating on
$\mathbb{R}^{n}$, we find

$\displaystyle\frac{d}{dt}\|\rho_{\varepsilon}\|_{L^{r}}^{r}+\frac{4mr(r-1)}{(r+m-1)^{2}}\int_{\mathbb{R}^{n}}|\nabla\rho_{\varepsilon}^{\frac{r+m-1}{2}}|^{2}dx\leq$
$\displaystyle-r\int_{\mathbb{R}^{n}}\rho_{\varepsilon}\nabla\mathcal{N}\ast(J_{\varepsilon}\ast\rho_{\varepsilon})\cdot\nabla\rho_{\varepsilon}^{r-1}dx$
(A.2) $\displaystyle\leq$
$\displaystyle(r-1)\int_{\mathbb{R}^{n}}\rho_{\varepsilon}^{r}J_{\varepsilon}\ast\rho_{\varepsilon}dx$
$\displaystyle\leq$
$\displaystyle(r-1)\int_{\mathbb{R}^{n}}\rho_{\varepsilon}^{r+1}dx,$

because it follows from Holder’s inequality and convolution-type Young’s
inequality that

$\displaystyle\int_{\mathbb{R}^{n}}\rho_{\varepsilon}^{r}J_{\varepsilon}\ast\rho_{\varepsilon}dx$
$\displaystyle\leq(\int_{\mathbb{R}^{n}}\rho_{\varepsilon}^{r+1}dx)^{r/(r+1)}(\int_{\mathbb{R}^{n}}(J_{\varepsilon}\ast\rho_{\varepsilon})^{r+1}dx)^{1/(r+1)}$
$\displaystyle\leq\|\rho_{\varepsilon}\|_{L^{r+1}(\mathbb{R}^{n})}^{r}\|J_{\varepsilon}\|_{L^{1}(\mathbb{R}^{n})}\|\rho_{\varepsilon}\|_{L^{r+1}(\mathbb{R}^{n})}$
$\displaystyle=\|\rho_{\varepsilon}\|_{L^{r+1}(\mathbb{R}^{n})}^{r+1}.$

Thanks to the interpolation inequality (Theorem C.2) and Sobolev’s inequality
for gradient (Theorem C.3), we have

$\displaystyle\|\rho_{\varepsilon}\|_{L^{r+1}(\mathbb{R}^{n})}^{r+1}$
$\displaystyle\leq(\int_{\mathbb{R}^{n}}\rho_{\varepsilon}dx)^{\frac{r+mn-2n+1}{rn+mn-2n+1}}(\int_{\mathbb{R}^{n}}\rho_{\varepsilon}^{\frac{(r+m-1)n}{n-1}}dx)^{\frac{r(n-1)}{rn+mn-2n+1}}$
$\displaystyle\leq
C(M,m,n,r)(\frac{4mr}{(r+m-1)^{2}}\int_{\mathbb{R}^{n}}|\nabla\rho_{\varepsilon}^{\frac{m+r-1}{2}}|^{2}dx)^{\frac{rn}{rn+mn-2n+1}}$
$\displaystyle\leq
C(M,m,n,r)+\frac{4mr}{(r+m-1)^{2}}\int_{\mathbb{R}^{n}}|\nabla\rho_{\varepsilon}^{\frac{m+r-1}{2}}|^{2}dx,$

and it follows from (A.2) that

$\frac{d}{dt}\|\rho_{\varepsilon}\|_{L^{r}}^{r}\leq C(M,m,n,r).$ (A.3)

But the initial data is independent of $\varepsilon$ since for $r\geq 1$, we
have

$\|J_{\varepsilon}\ast\rho_{0}\|_{L^{r}(\mathbb{R}^{n})}\leq\|J_{\varepsilon}\|_{L^{1}(\mathbb{R}^{n})}\|\rho_{0}\|_{L^{r}(\mathbb{R}^{n})}\leq\|\rho_{0}\|_{L^{1}(\mathbb{R}^{n})}^{\frac{1}{r}}\|\rho_{0}\|_{L^{\infty}(\mathbb{R}^{n})}^{\frac{r-1}{r}}\leq
C.$

Therefore, integrating (A.3) on $[0,t]$ the proof of the lemma is completed. ∎

The $L^{\infty}$ estimate of the density $\rho_{\varepsilon}$ is obtained by
adapting the proof in [59].

######  Lemma A.2.

Under the initial assumptions (2.1), the solution $\rho_{\varepsilon}$ to the
Cauchy problem Eq. (A.1) satisfies for all $T>0$

$\sup\limits_{0\leq t\leq
T}\|\rho_{\varepsilon}(t)\|_{L^{\infty}(\mathbb{R}^{n})}\leq
C(M,T,m,n,\|\rho_{0}\|_{L^{\infty}(\mathbb{R}^{n})}).$

###### Proof.

Similar to (3.8), by Young’s inequality we have, for $t\in[0,T]$,

$\displaystyle|\nabla(J_{\varepsilon}\ast\mathcal{N}\ast\rho_{\varepsilon}(t))|=$
$\displaystyle|\nabla\mathcal{N}\ast(J_{\varepsilon}\ast\rho_{\varepsilon}(t))|$
$\displaystyle\leq$ $\displaystyle
C(n)\Big{(}\int_{\mathbb{R}^{n}}(J_{\varepsilon}\ast\rho_{\varepsilon})^{2n-1}(y,t)dy\Big{)}^{\frac{1}{2n-1}}+C(n)\|J_{\varepsilon}\ast\rho_{\varepsilon}(t)\|_{L^{1}(\mathbb{R}^{n})}$
$\displaystyle\leq$ $\displaystyle
C(n)(\|\rho_{\varepsilon}(t)\|_{L^{2n-1}(\mathbb{R}^{n})}+\|\rho_{\varepsilon}(t)\|_{L^{1}(\mathbb{R}^{n})})$
$\displaystyle\leq$ $\displaystyle
C(M,T,m,n,\|\rho_{0}\|_{L^{\infty}(\mathbb{R}^{n})}).$

Multiplying Eq. (A.1) by $\rho_{\varepsilon}^{r-1}$ and integrating on
$\mathbb{R}^{n}$, it follows that, for $r\geq m\geq 1$,

$\displaystyle\frac{1}{r}\frac{d}{dt}$
$\displaystyle\|\rho_{\varepsilon}\|_{L^{r}}^{r}+\frac{4m(r-1)}{(r+m-1)^{2}}\int_{\mathbb{R}^{n}}|\nabla\rho_{\varepsilon}^{\frac{r+m-1}{2}}|^{2}dx\leq(r-1)\int_{\mathbb{R}^{n}}\rho_{\varepsilon}^{r-1}\nabla\rho_{\varepsilon}\cdot\nabla(J_{\varepsilon}\ast\mathcal{N}\ast\rho_{\varepsilon})dx$
$\displaystyle\leq$
$\displaystyle\frac{2m(r-1)}{(r+m-1)^{2}}\int_{\mathbb{R}^{n}}|\nabla\rho_{\varepsilon}^{\frac{r+m-1}{2}}|^{2}dx+\frac{(r-1)}{2m}\|\nabla(J_{\varepsilon}\ast\mathcal{N}\ast\rho_{\varepsilon})(t)\|_{L^{\infty}(\mathbb{R}^{n})}^{2}\int_{\mathbb{R}^{n}}\rho_{\varepsilon}^{r+1-m}dx.$

Furthermore, it holds by Hölder’s and Young’s inequalities that

$\displaystyle\frac{1}{r}\frac{d}{dt}\|\rho_{\varepsilon}\|_{L^{r}}^{r}+$
$\displaystyle\frac{2m(r-1)}{(r+m-1)^{2}}\int_{\mathbb{R}^{n}}|\nabla\rho_{\varepsilon}^{\frac{r+m-1}{2}}|^{2}dx$
(A.4) $\displaystyle\leq$
$\displaystyle(r-1)C(M,T,m,n,\|\rho_{0}\|_{L^{\infty}(\mathbb{R}^{n})})(\|\rho_{\varepsilon}\|_{L^{1}(\mathbb{R}^{n})})^{\frac{m-1}{r-1}}(\|\rho_{\varepsilon}\|_{L^{r}(\mathbb{R}^{n})}^{r})^{\frac{r-m}{r-1}}$
$\displaystyle\leq$ $\displaystyle
C^{r}(M,T,m,n,\|\rho_{0}\|_{L^{\infty}(\mathbb{R}^{n})})+r^{2}\|\rho_{\varepsilon}\|_{L^{r}(\mathbb{R}^{n})}^{r},\text{
for }r+1\geq 2m.$

By means of the interpolation inequality (Theorem C.2), Sobolev’s inequality
for the gradient (Theorem C.3) and Young’s inequality, we also have

$\displaystyle r^{2}\|\rho_{\varepsilon}\|_{L^{r}(\mathbb{R}^{n})}^{r}$
$\displaystyle\leq
r^{2}\|\rho_{\varepsilon}\|_{L^{\frac{r}{4}}(\mathbb{R}^{n})}^{\frac{[r+(m-1)n]r}{(3n+1)r+4(m-1)n}}\|\rho_{\varepsilon}\|_{L^{\frac{(m+r-1)n}{n-1}}(\mathbb{R}^{n})}^{\frac{[3nr+3(m-1)n]r}{(3n+1)r+4(m-1)n}}$
$\displaystyle=r^{2}\|\rho_{\varepsilon}\|_{L^{\frac{r}{4}}(\mathbb{R}^{n})}^{\frac{[r+(m-1)n]r}{(3n+1)r+4(m-1)n}}(S_{n}^{-2}\int_{\mathbb{R}^{n}}|\nabla\rho_{\varepsilon}^{\frac{m+r-1}{2}}|^{2}dx)^{\frac{3nr}{(3n+1)r+4(m-1)n}}$
$\displaystyle\leq\frac{m(r-1)}{(m+r-1)^{2}}\|\nabla\rho_{\varepsilon}^{\frac{r+m-1}{2}}\|_{L^{2}(\mathbb{R}^{n})}^{2}+\frac{r+4(m-1)n}{(3n+1)r+4(m-1)n}r^{\frac{2(3n+1)r+8(m-1)n}{r+4(m-1)n}}$
$\displaystyle\qquad\qquad(S_{n}^{-2}\frac{(r+m-1)^{2}}{m(r-1)}\frac{3nr}{(3n+1)r+4(m-1)n})^{\frac{3nr}{r+4(m-1)n}}\|\rho_{\varepsilon}\|_{L^{\frac{r}{4}}(\mathbb{R}^{n})}^{\frac{[r+(m-1)n]r}{r+4(m-1)n}}$
$\displaystyle\leq\frac{m(r-1)}{(m+r-1)^{2}}\|\nabla\rho_{\varepsilon}^{\frac{r+m-1}{2}}\|_{L^{2}(\mathbb{R}^{n})}^{2}+r^{N}\|\rho_{\varepsilon}\|_{L^{\frac{r}{4}}(\mathbb{R}^{n})}^{r}+C^{r}(m,n),$
(A.5)

where $N:=9n+2$, $r\geq\max\\{4,2m\\}$, and in the last inequality we use,

$\big{(}S_{n}^{-2}\frac{(r+m-1)^{2}}{mr(r-1)}\frac{3nr}{(3n+1)r+4(m-1)n}\big{)}^{\frac{3n}{m-1}}\to(\frac{S_{n}^{-2}3n}{m(3n+1)})^{\frac{3n}{m-1}},\quad\text{as
}r\to\infty.$

Inserting (A.5) into (A.4), it holds

$\frac{d}{dt}\|\rho_{\varepsilon}\|_{L^{r}(\mathbb{R}^{n})}^{r}\leq
rC^{r}(m,n)+r^{N+1}\|\rho_{\varepsilon}\|_{L^{\frac{r}{4}}(\mathbb{R}^{n})}.$
(A.6)

Integrating (A.6) on $[0,t]$ for any $t\in(0,T]$, we have

$\displaystyle\sup\limits_{0\leq t\leq
T}\|\rho_{\varepsilon}(t)\|_{L^{r}(\mathbb{R}^{n})}\leq$ $\displaystyle
r^{\frac{N+1}{r}}\sup\limits_{0\leq t\leq
T}\|\rho_{\varepsilon}(t)\|_{L^{\frac{r}{4}}(\mathbb{R}^{n})}+\|\rho_{\varepsilon,0}\|_{L^{r}(\mathbb{R}^{n})}+r^{\frac{1}{r}}C(m,n)$
$\displaystyle\leq$ $\displaystyle r^{\frac{N+1}{r}}\sup\limits_{0\leq t\leq
T}\|\rho_{\varepsilon}(t)\|_{L^{\frac{r}{4}}(\mathbb{R}^{n})}+\|\rho_{0}\|_{L^{r}(\mathbb{R}^{n})}+r^{\frac{1}{r}}C(m,n)$
$\displaystyle\leq$ $\displaystyle
r^{\frac{N+1}{r}}\max\\{M,\|\rho_{0}\|_{L^{\infty}(\mathbb{R}^{n})},C(T,m,n),\sup\limits_{0\leq
t\leq T}\|\rho_{\varepsilon}(t)\|_{L^{\frac{r}{4}}(\mathbb{R}^{n})}\\}.$

Let $r=4^{p}$ for $p\in\mathbb{N}^{+}$, by iteration, we obtain

$\displaystyle\sup\limits_{0\leq t\leq T}$
$\displaystyle\|\rho_{\varepsilon}(t)\|_{L^{4^{p}}(\mathbb{R}^{n})}\leq
4^{[\frac{p}{4^{p}}+\frac{p-1}{4^{(p-1)}}]}\max\\{M,\|\rho_{0}\|_{L^{\infty}(\mathbb{R}^{n})},C(T,m,n),\sup\limits_{0\leq
t\leq T}\|\rho_{\varepsilon}\|_{L^{4^{(p-2)}}(\mathbb{R}^{n})}\\}$
$\displaystyle\leq$ $\displaystyle
4^{(N+1)[\frac{p}{4^{p}}+\frac{p-1}{4^{(p-1)}}+\cdots+\frac{p^{*}}{4^{p^{*}}}]}\max\\{M,\|\rho_{0}\|_{L^{\infty}(\mathbb{R}^{n})},C(T,m,n),\sup\limits_{0\leq
t\leq T}\|\rho_{\varepsilon}(t)\|_{L^{4}(\mathbb{R}^{n})}\\}$
$\displaystyle\leq$ $\displaystyle
C(M,T,m,n,\|\rho_{0}\|_{L^{\infty}(\mathbb{R}^{n})}),$

where $\sup\limits_{0\leq t\leq
T}\|\rho_{\varepsilon}(t)\|_{L^{4^{p^{*}}}(\mathbb{R}^{n})}\leq
C(M,T,m,n,\|\rho_{0}\|_{L^{\infty}(\mathbb{R}^{n})})$ for
$p^{*}:=\inf\\{p\in\mathbb{N}^{+}:4^{p}\geq\max\\{4,2m\\}\\}$ holds from Lemma
A.1. Taking $p\to\infty$, we find the announced bound. ∎

##  Appendix B Compact support property

We prove the $m$ dependent compact support property

######  Lemma B.1.

Assume (2.1), then, the global weak solution to the Cauchy problem Eq. (1.1)
in the sense of Def. 1.1 is compactly supported

$\begin{cases}{\rm supp}(\rho_{m}(t))\subset
B_{\mathcal{R}_{m}(T,t)},\qquad\quad\forall\ t\in[0,T],\,\forall T>0,\\\
\mathcal{R}_{m}(T,t):=(\mathcal{R}_{m,0}+\frac{n\|\nabla\mathcal{N}\ast\rho_{m}\|_{L^{\infty}(Q_{T})}}{A_{m,T}})e^{\frac{A_{m,T}t}{n}}-\frac{n\|\nabla\mathcal{N}\ast\rho_{m}\|_{L^{\infty}(Q_{T})}}{A_{m,T}},\end{cases}$

with $\mathcal{R}_{m,0}=2\max(R_{m},\sqrt{nP_{m,0}})$ and
$A_{m,T}:=\max\big{(}\|\rho_{m}\|_{L^{\infty}(Q_{T})},1\big{)}$.

Set $V_{m}(x,t):=\nabla\mathcal{N}\ast\rho_{m}$, it follows from the
$L^{\infty}$ bound in Lemma A.2 that

$\displaystyle\sup\limits_{0\leq t\leq
T}\|V_{m}(t)\|_{L^{\infty}(\mathbb{R}^{n})}\leq
C(M,T,m,n,\|\rho_{m,0}\|_{L^{\infty}(\mathbb{R}^{n})}),$
$\displaystyle\sup\limits_{0\leq t\leq T}\|\nabla\cdot
V_{m}(t)\|_{L^{\infty}(\mathbb{R}^{n})}\leq
C(M,T,m,n,\|\rho_{m,0}\|_{L^{\infty}(\mathbb{R}^{n})}),$

where the first inequality is obtained as (3.8). We rewrite Eq. (1.1) as

$\partial_{t}\rho_{m}=\Delta\rho_{m}^{m}+\nabla\cdot(\rho_{m}V_{m}),$

which is now considered as a porous medium equation with a given drift
$V_{m}$. Then, similar to [21, Lemma 3.8], we can construct a viscosity sup-
solution with a uniform compact support in any finite time for the equation of
pressure $P_{m}=\frac{m}{m-1}\rho_{m}^{m-1}$,

$\partial_{t}P_{m}=(m-1)P_{m}(\Delta P_{m}+\nabla\cdot V_{m})+|\nabla
P_{m}|^{2}+\nabla P_{m}\cdot V_{m}.$

Hence, the solution $\rho_{m}$ to the Cauchy problem (1.1) under the initial
assumption (2.1) is compactly supported on $[0,T]$ for any $T>0$.

More precisely, let
$\mathcal{R}_{m}(T,t):=(\mathcal{R}_{m,0}+\frac{n\|\nabla\mathcal{N}\ast\rho_{m}\|_{L^{\infty}(Q_{T})}}{A_{m,T}})e^{\frac{A_{m,T}t}{n}}-\frac{n\|\nabla\mathcal{N}\ast\rho_{m}\|_{L^{\infty}(Q_{T})}}{A_{m,T}}$
for any $T>0$ with $\mathcal{R}_{m,0}\geq 1$ satisfying
$2R_{m}\leq\mathcal{R}_{m.0}$ and
$P_{m,0}\leq\frac{\mathcal{R}_{m,0}^{2}}{4n}$, and
$A_{m,T}:=\max\\{\|\rho_{m}\|_{L^{\infty}(Q_{T})},1\\}$. Set
$\phi_{m,T}(x,t):=\frac{A_{m,T}\mathcal{R}_{m,T}^{2}\big{(}1-\frac{|x|^{2}}{\mathcal{R}_{m,T}^{2}}\big{)}_{+}}{2n}$,
it is easy to verify on the support of $\phi_{m,T}$ that

$\partial_{t}\phi_{m,T}\geq(m-1)\phi_{m,T}(\Delta\phi_{m,T}+\nabla\cdot
V_{m})+|\nabla\phi_{m,T}|^{2}+\nabla\phi_{m,T}\cdot V_{m}\quad\text{for
}t\in[0,T]$

with $P_{m,0}\leq\phi_{m,T}(x,0)$ in $\mathbb{R}^{n}$. Therefore, it holds by
the comparison principle that

$\displaystyle P_{m}(x,t)\leq\phi_{m,T}(x,t),\quad$ $\displaystyle\forall\
(x,t)\in\mathbb{R}^{n}\times[0,T],$ $\displaystyle{\rm
supp}(\rho_{m}(t))\subset B_{\mathcal{R}_{m}(T,t)},\quad$
$\displaystyle\forall\ t\in[0,T].$

The proof of the compact support property is completed.

##  Appendix C Appendix: various functional inequalities

######  Lemma C.1 (Singular integral for Newtonian potential).

[57, 3] Let $\mathcal{N}$ be the Newtonian potential. For $f(x)\in
L^{p}(\mathbb{R}^{n}),1<p<\infty$, we have

$\|\nabla^{2}\mathcal{N}\ast f(x)\|_{L^{p}(\mathbb{R}^{n})}\leq
C(p,n)\|f(x)\|_{L^{p}(\mathbb{R}^{n})},$

where $C(p,n)\sim\frac{1}{p-1}$ for $0<p-1\ll 1$ and $C(p,n)\sim p$ for $p\gg
1$.

######  Theorem C.1 (Hardy-Littlewood-Sobolev inequality).

[43] Let $p,r>1$ and $0<\lambda<n$ with
$\frac{1}{p}+\frac{\lambda}{n}+\frac{1}{r}=2$. Let $f\in
L^{p}(\mathbb{R}^{n})$ and $h\in L^{r}(\mathbb{R}^{n})$. Then, we have

$\Big{|}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}f(x)|x-y|^{-\lambda}h(y)dxdy\Big{|}\leq
C(n,\lambda,p)||f||_{L^{p}(\mathbb{R}^{n})}||h||_{L^{r}(\mathbb{R}^{n})},$
(C.1)

where the constant satisfies
$C(n,\lambda,p)=\frac{n}{n-\lambda}(\frac{|\mathbf{S}^{n-1}|}{n})^{\frac{\lambda}{n}}\frac{1}{pr}\big{(}(\frac{p\lambda}{np-n})^{\frac{\lambda}{n}}+(\frac{r\lambda}{nr-n})^{\frac{\lambda}{n}}\big{)}$.

For $\lambda=n-2$ and $p=r=\frac{2n}{n+2}$, we have
$C(n,p,\lambda)=C(n)=\pi^{\frac{n-2}{2}}\frac{\Gamma(1)}{\Gamma(\frac{n+2}{2})}\\{\frac{\Gamma(\frac{n}{2})}{\Gamma(n)}\\}^{-\frac{2}{n}}.$

######  Theorem C.2 (Interpolation inequality).

Let $\Omega\subset\mathbb{R}^{n}$ be a measurable domain, let $f(x)\in
L^{p}(\Omega)\cap L^{q}(\Omega)$, then, with $0\leq\beta=\frac{q-r}{q-p}\leq
1$, we have

$\int_{\Omega}|f|^{r}dx\leq(\int_{\Omega}|f|^{p}dx)^{\beta}(\int_{\Omega}|f|^{q}dx)^{1-\beta}.$

######  Theorem C.3 (Sobolev’s inequality for gradient).

[43] For $n\geq 3$, let $f\in L^{1}_{loc}(\mathbb{R}^{n})$ with $\|\nabla
f\|_{L^{2}}<\infty$. Then $f\in L^{q}(\mathbb{R}^{n})$ and the following
inequality holds:

$S_{n}\|f\|_{L^{q}(\mathbb{R}^{n})}^{2}\leq\|\nabla
f\|_{L^{2}(\mathbb{R}^{n})}^{2},\qquad q=\frac{2n}{n-2},$ (C.2)

where
$S_{n}:=\frac{n(n-1)}{4}2^{\frac{2}{n}}\pi^{1+\frac{2}{n}}\Gamma(\frac{n+1}{2})^{-\frac{1}{n}}$
is the optimal constant for (C.2).

######  Theorem C.4 (Gagliardo-Nirenberg-Sobolev inequality).

Assume $q,r$ satisfy $1\leq q,r\leq\infty$ and $j,m\in\mathbb{Z}^{+}$ satisfy
$0\leq j<m$. For any $f(x)\in C_{0}^{\infty}(\mathbb{R}^{n})$, then we have

$\|D^{j}f(x)\|_{L^{p}(\mathbb{R}^{n})}\leq
C\|D^{m}f(x)\|_{L^{r}(\mathbb{R}^{n})}^{\alpha}\|u\|_{L^{q}(\mathbb{R}^{n})}^{1-\alpha},$
(C.3)

where
$\frac{1}{p}-\frac{j}{n}=\alpha(\frac{1}{r}-\frac{m}{n})+(1-\alpha)\frac{1}{q}$,$\frac{j}{m}\leq\alpha\leq
1$ and C depends on $m,n,j,q,r,\alpha$. If $m-j-\frac{n}{r}$ is a nonnegative
integer, then (C.3)is established for $\frac{j}{m}\leq\alpha<1$.

## References

  * [1] Damon Alexander, Inwon Kim, and Yao Yao.  Quasi-static evolution and congested crowd transport.  Nonlinearity, 27(4):823–858, 2014.
  * [2] Donald G. Aronson and Philippe Bénilan.  Régularité des solutions de l’équation des milieux poreux dans ${\bf R}^{N}$.  C. R. Acad. Sci. Paris Sér. A-B, 288(2):A103–A105, 1979.
  * [3] Jacob Bedrossian, Nancy Rodríguez, and Andrea L. Bertozzi.  Local and global well-posedness for aggregation equations and Patlak-Keller-Segel models with degenerate diffusion.  Nonlinearity, 24(6):1683–1714, 2011.
  * [4] Philippe Bénilan and Michael G. Crandall.  The continuous dependence on $\varphi$ of solutions of $u_{t}-\Delta\varphi(u)=0$.  Indiana Univ. Math. J., 30(2):161–177, 1981.
  * [5] Philippe Bénilan and Noureddine Igbida.  La limite de la solution de $u_{t}=\Delta_{p}u^{m}$ lorsque $m\to\infty$.  C. R. Acad. Sci. Paris Sér. I Math., 321(10):1323–1328, 1995\.
  * [6] Andrea L. Bertozzi and Dejan Slepčev.  Existence and uniqueness of solutions to an aggregation equation with degenerate diffusion.  Commun. Pure Appl. Anal., 9(6):1617–1637, 2010.
  * [7] Shen Bian and Jian-Guo Liu.  Dynamic and steady states for multi-dimensional Keller-Segel model with diffusion exponent $m>0$.  Comm. Math. Phys., 323(3):1017–1070, 2013.
  * [8] Shen Bian, Jian-Guo Liu, and Chen Zou.  Ultra-contractivity for Keller-Segel model with diffusion exponent $m>1-2/d$.  Kinet. Relat. Models, 7(1):9–28, 2014.
  * [9] Adrien Blanchet, José A. Carrillo, and Philippe Laurençot.  Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions.  Calc. Var. Partial Differential Equations, 35(2):133–168, 2009\.
  * [10] Lucio Boccardo and Thierry Gallouët.  Nonlinear elliptic and parabolic equations involving measure data.  J. Funct. Anal., 87(1):149–169, 1989.
  * [11] Didier Bresch, Thierry Colin, Emmanuel Grenier, Benjamin Ribba, and Olivier Saut.  Computational modeling of solid tumor growth: the avascular stage.  SIAM J. Sci. Comput., 32(4):2321–2344, 2010.
  * [12] Federica Bubba, Benoît Perthame, Camille Pouchol, and Markus Schmidtchen.  Hele-Shaw limit for a system of two reaction-(cross-)diffusion equations for living tissues.  Arch. Ration. Mech. Anal., 236(2):735–766, 2020.
  * [13] Luis Caffarelli and Sandro Salsa.  A geometric approach to free boundary problems, volume 68 of Graduate Studies in Mathematics.  American Mathematical Society, Providence, RI, 2005.
  * [14] Vincent Calvez and José A. Carrillo.  Volume effects in the Keller-Segel model: energy estimates preventing blow-up.  J. Math. Pures Appl. (9), 86(2):155–175, 2006.
  * [15] Vincent Calvez, José Antonio Carrillo, and Franca Hoffmann.  The geometry of diffusing and self-attracting particles in a one-dimensional fair-competition regime.  In Nonlocal and nonlinear diffusions and interactions: new methods and directions, volume 2186 of Lecture Notes in Math., pages 1–71. Springer, Cham, 2017.
  * [16] Vincent Calvez and Lucilla Corrias.  Blow-up dynamics of self-attracting diffusive particles driven by competing convexities.  Discrete Contin. Dyn. Syst. Ser. B, 18(8):2029–2050, 2013.
  * [17] J. A. Carrillo, S. Hittmeir, B. Volzone, and Y. Yao.  Nonlinear aggregation-diffusion equations: radial symmetry and long time asymptotics.  Invent. Math., 218(3):889–977, 2019.
  * [18] José A. Carrillo and Yoshie Sugiyama.  Compactly supported stationary states of the degenerate Keller-Segel system in the diffusion-dominated regime.  Indiana Univ. Math. J., 67(6):2279–2312, 2018.
  * [19] José Antonio Carrillo, Daniele Castorina, and Bruno Volzone.  Ground states for diffusion dominated free energies with logarithmic interaction.  SIAM J. Math. Anal., 47(1):1–25, 2015.
  * [20] Li Chen and Jinhuan Wang.  Exact criterion for global existence and blow up to a degenerate Keller-Segel system.  Doc. Math., 19:103–120, 2014.
  * [21] Katy Craig, Inwon Kim, and Yao Yao.  Congested aggregation via Newtonian interaction.  Arch. Ration. Mech. Anal., 227(1):1–67, 2018.
  * [22] Noemi David.  Phenotypic heterogeneity in a model of tumor growth: existence of solutions and incompressible limit.  preprint, April 2022.
  * [23] Noemi David, Tomasz Dębiec, and Benoît Perthame.  Convergence rate for the incompressible limit of nonlinear diffusion–advection equations.  Annales de l’Institut Henri Poincaré C, 2022.
  * [24] Noemi David and Benoît Perthame.  Free boundary limit of a tumor growth model with nutrient.  J. Math. Pures Appl. (9), 155:62–82, 2021.
  * [25] Noemi David and Markus Schmidtchen.  On the incompressible limit for a tumour growth model incorporating convective effects.  arXiv preprint arXiv:2103.02564, 2021.
  * [26] Tomasz Dębiec, Benoît Perthame, Markus Schmidtchen, and Nicolas Vauchelet.  Incompressible limit for a two-species model with coupling through Brinkman’s law in any dimension.  J. Math. Pures Appl. (9), 145:204–239, 2021.
  * [27] Tomasz Dębiec and Markus Schmidtchen.  Incompressible limit for a two-species tumour model with coupling through Brinkman’s law in one dimension.  Acta Appl. Math., 169:593–611, 2020.
  * [28] Matias G. Delgadino, Xukai Yan, and Yao Yao.  Uniqueness and nonuniqueness of steady states of aggregation-diffusion equations.  Comm. Pure Appl. Math., 75(1):3–59, 2022.
  * [29] Emmanuele DiBenedetto and Avner Friedman.  The ill-posed Hele-Shaw model and the Stefan problem for supercooled water.  Trans. Amer. Math. Soc., 282(1):183–204, 1984.
  * [30] C. M. Elliott, M. A. Herrero, J. R. King, and J. R. Ockendon.  The mesa problem: diffusion patterns for $u_{t}=\nabla\cdot(u^{m}\nabla u)$ as $m\to+\infty$.  IMA J. Appl. Math., 37(2):147–154, 1986.
  * [31] C. M. Elliott and V. Janovský.  A variational inequality approach to Hele-Shaw flow with a moving boundary.  Proc. Roy. Soc. Edinburgh Sect. A, 88(1-2):93–107, 1981.
  * [32] O. Gil and F. Quirós.  Convergence of the porous media equation to Hele-Shaw.  Nonlinear Anal., 44(8, Ser. A: Theory Methods):1111–1131, 2001\.
  * [33] Nestor Guillen, Inwon Kim, and Antoine Mellet.  A Hele-Shaw limit without monotonicity.  Arch. Ration. Mech. Anal., 243(2):829–868, 2022.
  * [34] Piotr Gwiazda, Benoît Perthame, and Agnieszka Świerczewska Gwiazda.  A two-species hyperbolic-parabolic model of tissue growth.  Comm. Partial Differential Equations, 44(12):1605–1618, 2019.
  * [35] N. Igbida.  ${L}^{1}$-Theory for reaction-diffusion Hele-Shaw flow with linear drift.  arXiv preprint arXiv:2105.00182, 2021.
  * [36] Noureddine Igbida.  The mesa-limit of the porous-medium equation and the Hele-Shaw problem.  Differential Integral Equations, 15(2):129–146, 2002.
  * [37] Inwon Kim and Yao Yao.  The Patlak-Keller-Segel model and its variations: properties of solutions via maximum principle.  SIAM J. Math. Anal., 44(2):568–602, 2012.
  * [38] Inwon Kim and Yuming Paul Zhang.  Porous medium equation with a drift: free boundary regularity.  Arch. Ration. Mech. Anal., 242(2):1177–1228, 2021.
  * [39] Inwon C. Kim and Helen K. Lei.  Degenerate diffusion with a drift potential: a viscosity solutions approach.  Discrete Contin. Dyn. Syst., 27(2):767–786, 2010.
  * [40] Inwon C. Kim, Benoît Perthame, and Panagiotis E. Souganidis.  Free boundary problems for tumor growth: a viscosity solutions approach.  Nonlinear Anal., 138:207–228, 2016.
  * [41] Maxime Laborde.  On cross-diffusion systems for two populations subject to a common congestion effect.  Appl. Math. Optim., 81(3):989–1020, 2020.
  * [42] Hai-Liang Li, Benoít Perthame, and Xinmei Wen.  Sharp long-time asymptotics for chemotaxis with free boundary.  SIAM J. Math. Anal., 53(2):2027–2083, 2021.
  * [43] Elliott H. Lieb and Michael Loss.  Analysis, volume 14 of Graduate Studies in Mathematics.  American Mathematical Society, Providence, RI, second edition, 2001.
  * [44] Elliott H. Lieb and Horng-Tzer Yau.  The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics.  Comm. Math. Phys., 112(1):147–174, 1987.
  * [45] P.-L. Lions.  The concentration-compactness principle in the calculus of variations. The locally compact case. I.  Ann. Inst. H. Poincaré Anal. Non Linéaire, 1(2):109–145, 1984.
  * [46] Jian-Guo Liu and Xiangsheng Xu.  Existence and incompressible limit of a tissue growth model with autophagy.  SIAM J. Math. Anal., 53(5):5215–5242, 2021.
  * [47] J. S. Lowengrub, H. B. Frieboes, F. Jin, Y.-L. Chuang, X Li, P. Macklin, S. M. Wise, and V. Cristini.  Nonlinear modelling of cancer: bridging the gap between cells and tumours.  Nonlinearity, 23(1):R1–R91, 2010.
  * [48] Stephan Luckhaus and Yoshie Sugiyama.  Asymptotic profile with the optimal convergence rate for a parabolic equation of chemotaxis in super-critical cases.  Indiana Univ. Math. J., 56(3):1279–1297, 2007.
  * [49] Bertrand Maury, Aude Roudneff-Chupin, and Filippo Santambrogio.  A macroscopic crowd motion model of gradient flow type.  Math. Models Methods Appl. Sci., 20(10):1787–1821, 2010.
  * [50] Bertrand Maury, Aude Roudneff-Chupin, and Filippo Santambrogio.  Congestion-driven dendritic growth.  Discrete Contin. Dyn. Syst., 34(4):1575–1604, 2014.
  * [51] Bertrand Maury, Aude Roudneff-Chupin, Filippo Santambrogio, and Juliette Venel.  Handling congestion in crowd motion modeling.  Netw. Heterog. Media, 6(3):485–519, 2011.
  * [52] Antoine Mellet, Benoît Perthame, and Fernando Quirós.  A Hele-Shaw problem for tumor growth.  J. Funct. Anal., 273(10):3061–3093, 2017.
  * [53] Frank Morgan.  A round ball uniquely minimizes gravitational potential energy.  Proc. Amer. Math. Soc., 133(9):2733–2735, 2005.
  * [54] Benoît Perthame, Fernando Quirós, and Juan Luis Vázquez.  The Hele-Shaw asymptotics for mechanical models of tumor growth.  Arch. Ration. Mech. Anal., 212(1):93–127, 2014.
  * [55] Brock C. Price and Xiangsheng Xu.  Global existence theorem for a model governing the motion of two cell populations.  Kinet. Relat. Models, 13(6):1175–1191, 2020.
  * [56] Clément Sire and Pierre-Henri Chavanis.  Critical dynamics of self-gravitating Langevin particles and bacterial populations.  Phys. Rev. E (3), 78(6):061111, 22, 2008.
  * [57] Elias M. Stein.  Singular integrals and differentiability properties of functions.  Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970.
  * [58] Gerhard Ströhmer.  Stationary states and moving planes.  81:501–513, 2008.
  * [59] Yoshie Sugiyama.  Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems.  Differential Integral Equations, 19(8):841–876, 2006.
  * [60] Yoshie Sugiyama.  Time global existence and asymptotic behavior of solutions to degenerate quasi-linear parabolic systems of chemotaxis.  Differential Integral Equations, 20(2):133–180, 2007.
  * [61] Yao Yao.  Asymptotic behavior for critical Patlak-Keller-Segel model and a repulsive-attractive aggregation equation.  Ann. Inst. H. Poincaré C Anal. Non Linéaire, 31(1):81–101, 2014.