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We consider the incompressible Navier-Stokes and Euler equations in a bounded domain with non-characteristic boundary condition, and study the energy dissipation near the outflow boundary in the zero-viscosity limit. We show that in a general setting, the energy dissipation rate is proportional to $\bar U \bar V ^2$, where $\bar U$ is the strength of the suction and $\bar V$ is the tangential component of the difference between Euler and Navier-Stokes on the outflow boundary. Moreover, we show that the enstrophy within a layer of order $\nu / \bar U$ is comparable with the total enstrophy. The rate of enstrophy production near the boundary is inversely proportional to $\nu$., Comment: 25 pages, 3 figures

In this paper, we show that a geometrical condition on $2\times2$ systems of conservation laws leads to non-uniqueness in the class of 1D continuous functions. This demonstrates that the Liu Entropy Condition alone is insufficient to guarantee uniqueness, even within the mono-dimensional setting. We provide examples of systems where this pathology holds, even if they verify stability and uniqueness for small BV solutions. Our proof is based on the convex integration process. Notably, this result represents the first application of convex integration to construct non-unique continuous solutions in one dimension.

The present paper is concerned with the well-posedness theory for non-homogeneous incompressible fluids exhibiting odd (non-dissipative) viscosity effects. Differently from previous works, we consider here the full odd viscosity tensor. Similarly to the work of Bresch and Desjardins in compressible fluid mechanics, we identify the presence of an effective velocity in the system, linking the velocity field of the fluid and the gradient of a suitable function of the density. By use of this effective velocity, we propose a new formulation of the original system of equations, thus highlighting a strong similarity with the equations of the ideal magnetohydrodynamics. By taking advantage of the new formulation of the equations, we establish a local in time well-posedness theory in Besov spaces based on $L^\infty$ and prove a lower bound for the lifespan of the solutions implying ``asymptotically global'' existence: in the regime of small initial density variations, $\rho_0-1= O(\varepsilon)$ for small $\varepsilon>0$, the corresponding solution is defined up to some time $T_\varepsilon>0$ satisfying the property $T_\varepsilon\,\longrightarrow\,+\infty$ when $\varepsilon\to0^+$., Comment: Submitted

In the realm of mathematical fluid dynamics, a formidable challenge lies in establishing inviscid limits from the Navier-Stokes equations to the Euler equations, wherein physically admissible solutions can be discerned. The pursuit of solving this intricate problem, particularly concerning singular solutions, persists in both compressible and incompressible scenarios. This article focuses on small $BV$ solutions to the barotropic Euler equation in one spatial dimension. Our investigation demonstrates that these solutions are inviscid limits for solutions to the associated compressible Navier-Stokes equation. Moreover, we extend our findings by establishing the well-posedness of such solutions within the broader class of inviscid limits of Navier-Stokes equations with locally bounded energy initial values., Comment: We fixed typos and retouched parts of the manuscript for better readability

We establish the time-asymptotic stability of solutions to the one-dimensional compressible Navier-Stokes-Fourier equations, with initial data perturbed from Riemann data that forms a generic Riemann solution. The Riemann solution under consideration is composed of a viscous shock, a viscous contact wave, and a rarefaction wave. We prove that the perturbed solution of Navier-Stokes-Fourier converges, uniformly in space as time goes to infinity, to a viscous ansatz composed of viscous shock with time-dependent shift, a viscous contact wave and an inviscid rarefaction wave. This is a first resolution of the challenging open problem associated with the generic Riemann solution. Our approach relies on the method of a-contraction with shifts, specifically applied to both the shock wave and the contact discontinuity wave. It enables the application of a global energy method for the generic combination of three waves., Comment: arXiv admin note: text overlap with arXiv:2104.06590

We provide an unconditional $L^2$ upper bound for the boundary layer separation of Leray-Hopf solutions in a smooth bounded domain. By layer separation, we mean the discrepancy between a (turbulent) low-viscosity Leray-Hopf solution $u^\nu$ and a fixed (laminar) regular Euler solution $\bar u$ with similar initial conditions and body force. We show an asymptotic upper bound $C \|\bar u\|_{L^\infty}^3 T$ on the layer separation, anomalous dissipation, and the work done by friction. This extends the previous result when the Euler solution is a regular shear in a finite channel. The key estimate is to control the boundary vorticity in a way that does not degenerate in the vanishing viscosity limit., Comment: 33 pages, 2 figures

This paper deals with the space-homogenous Landau equation with very soft potentials, including the Coulomb case. This nonlinear equation is of parabolic type with diffusion matrix given by the convolution product of the solution with the matrix $a_{ij} (z)=|z|^\gamma (|z|^2 \delta_{ij} - z_iz_j)$ for $\gamma \in [-3,-2)$. We derive local truncated entropy estimates and use them to establish two facts. Firstly, we prove that the set of singular points (in time and velocity) for the weak solutions constructed as in [C. Villani, Arch. Rational Mech. Anal. 143 (1998), 273-307] has zero $\mathscr{P}^{m_\ast}$ parabolic Hausdorff measure with $m_\ast:= \frac72 |2+\gamma|$. Secondly, we prove that if such a weak solution is axisymmetric, then it is smooth away from the symmetry axis. In particular, radially symmetric weak solutions are smooth away from the origin., Comment: 58 pages. Hausdorff dimension improved

Consider the steady solution to the incompressible Euler equation $\bar u=Ae_1$ in the periodic tunnel $\Omega=\mathbb T^{d-1}\times(0,1)$ in dimension $d=2,3$. Consider now the family of solutions $u^\nu$ to the associated Navier-Stokes equation with the no-slip condition on the flat boundaries, for small viscosities $\nu=A/\mathsf{Re}$, and initial values in $L^2$. We are interested in the weak inviscid limits up to subsequences $u^\nu\rightharpoonup u^\infty$ when both the viscosity $\nu$ converges to 0, and the initial value $u^\nu_0$ converges to $Ae_1$ in $L^2$. Under a conditional assumption on the energy dissipation close to the boundary, Kato showed in 1984 that $u^\nu$ converges to $Ae_1$ strongly in $L^2$ uniformly in time under this double limit. It is still unknown whether this inviscid limit is unconditionally true. The convex integration method produces solutions $u _E$ to the Euler equation with the same initial values $Ae_1$ which verify at time $0

We prove the time-asymptotic stability of composite waves consisting of the superposition of a viscous shock and a rarefaction for the one-dimensional compressible barotropic Navier-Stokes equations. Our result solves a long-standing problem first mentioned in 1986 by Matsumura and Nishihara in [25]. The same authors introduced it officially as an open problem in 1992 in [26] and it was again described as very challenging open problem in 2018 in the survey paper [23]. The main difficulty is due to the incompatibility of the standard anti-derivative method, used to study the stability of viscous shocks, and the energy method used for the stability of rarefactions. Instead of the anti-derivative method, our proof uses the $a$-contraction with shifts theory recently developed by two of the authors. This method is energy based, and can seamlessly handle the superposition of waves of different kinds., Comment: 50 pages, some typos are corrected

In dimension $n=2$ and $3$, we show that for any initial datum belonging to a dense subset of the energy space, there exist infinitely many global-in-time admissible weak solutions to the isentropic Euler system whenever $1<\gamma\leq 1+\frac2n$. This result can be regarded as a compressible counterpart of the one obtained by Szekelyhidi--Wiedemann (ARMA, 2012) for incompressible flows. Similarly to the incompressible result, the admissibility condition is defined in its integral form. Our result is based on a generalization of a key step of the convex integration procedure. This generalization allows, even in the compressible case, to convex integrate any smooth positive Reynolds stress. A large family of subsolutions can then be considered. These subsolutions can be generated, for instance, via regularization of any weak inviscid limit of an associated compressible Navier--Stokes system with degenerate viscosities., Comment: 1 figure

In this paper, we study the $a$-contraction property of small extremal shocks for 1-d systems of hyperbolic conservation laws endowed with a single convex entropy, when subjected to large perturbations. We show that the weight coefficient $a$ can be chosen with amplitude proportional to the size of the shock. The main result of this paper is a key building block in the companion paper, [{arXiv:2010.04761}, 2020], in which uniqueness and BV-weak stability results for $2\times 2$ systems of hyperbolic conservation laws are proved., Comment: 41 pages, 2 figures, minor correction to lemma 4.2 and corresponding changes to section 4

We consider the Riemann problem composed of two shocks for the 1D Euler system. We show that the Riemann solution with two shocks is stable and unique in the class of weak inviscid limits of solutions to the Navier-Stokes equations with initial data with bounded energy. This work extends to the case of two shocks a previous result of the authors in the case of a single shock. It is based on the method of weighted relative entropy with shifts known as $a$-contraction theory. A major difficulty due to the method is that very little control is available on the shifts. A modification of the construction of the shifts is needed to ensure that the two shock waves are well separated, at the level of the Navier-Stokes system, even when subjected to large perturbations. This work put the foundations needed to consider a large family of interacting waves. It is a key result in the program to solve the Bianchini-Bressan conjecture, that is the inviscid limit of solutions to the Navier-Stokes equation to the unique BV solution of the Euler equation, in the case of small BV initial values.

Let a 1-d system of hyperbolic conservation laws, with two unknowns, be endowed with a convex entropy. We consider the family of small $BV$ functions which are global solutions of this equation. For any small $BV$ initial data, such global solutions are known to exist. Moreover, they are known to be unique among $BV$ solutions verifying either the so-called Tame Oscillation Condition, or the Bounded Variation Condition on space-like curves. In this paper, we show that these solutions are stable in a larger class of weak (and possibly not even $BV$) solutions of the system. This result extends the classical weak-strong uniqueness results which allow comparison to a smooth solution. Indeed our result extends these results to a weak-$BV$ uniqueness result, where only one of the solutions is supposed to be small $BV$, and the other solution can come from a large class. As a consequence of our result, the Tame Oscillation Condition, and the Bounded Variation Condition on space-like curves are not necessary for the uniqueness of solutions in the $BV$ theory, in the case of systems with 2 unknowns. The method is $L^2$ based. It builds up from the theory of a-contraction with shifts, where suitable weight functions $a$ are generated via the front tracking method., Comment: 25 pages, 1 figure

We study the second spatial derivatives of suitable weak solutions to the incompressible Navier-Stokes equations in dimension three. We show that it is locally $L ^{\frac43, q}$ for any $q > \frac43$, which improves from the current result $L ^{\frac43, \infty}$. Similar improvements in Lorentz space are also obtained for higher derivatives of the vorticity for smooth solutions. We use a blow-up technique to obtain nonlinear bounds compatible with the scaling. The local study works on the vorticity equation and uses De Giorgi iteration. In this local study, we can obtain any regularity of the vorticity without any a priori knowledge of the pressure. The local-to-global step uses a recently constructed maximal function for transport equations., Comment: 39 pages

It is still not known whether a solution to the incompressible Euler equation, endowed with a smooth initial value, can blow-up in finite time. In [{\em Comm. Math. Phys.}, 378:557--568, 2020] it has been shown that, if it exists, such a solution becomes linearly unstable close to the blow-up time. In this paper, we show that the same phenomenon holds even in the more rigid axisymmetric case. To obtain this result, we first prove a blow-up criterion involving only the toroidal component of the vorticity. The instability of blow-up profiles is also investigated., Comment: 14 pages

In this paper, we consider global weak solutions to com-pressible Navier-Stokes-Korteweg equations with density dependent viscosities , in a periodic domain $\Omega = \mathbb T^3$, with a linear drag term with respect to the velocity. The main result concerns the exponential decay to equilibrium of such solutions using log-sobolev type inequalities. In order to show such a result, the starting point is a global weak-entropy solutions definition introduced in D. Bresch, A. Vasseur and C. Yu [12]. Assuming extra assumptions on the shear viscosity when the density is close to vacuum and when the density tends to infinity, we conclude the exponential decay to equilibrium. Note that our result covers the quantum Navier-Stokes system with a drag term.

We prove the stability of contact discontinuities without shear, a family of special discontinuous solutions for the three-dimensional full Euler systems, in the class of vanishing dissipation limits of the corresponding Navier-Stokes-Fourier system. We also show that solutions of the Navier-Stokes-Fourier system converge to the contact discontinuity when the initial datum converges to the contact discontinuity itself. This implies the uniqueness of the contact discontinuity in the class that we are considering. Our results give an answer to the open question, whether the contact discontinuity is unique for the multi-D compressible Euler system. Our proof is based on the relative entropy method, together with the theory of $a$-contraction up to a shift.

We consider a one-dimensional system arising from a chemotaxis model in tumour angiogenesis, which is described by a Keller-Segel equation with singular sensitivity. This hyperbolic-parabolic system is known to allow viscous shocks (so-called traveling waves), and in literature, their nonlinear stabilities have been considered in the class of certain mean-zero small perturbations. We show the global existence of the solution without assuming the mean-zero condition for any initial data as arbitrarily large perturbations around traveling waves in the Sobolev space $H^1$ while the shock strength is assumed to be small enough. The main novelty of this paper is to develop the global well-posedness of any large $H^1$-perturbations of traveling wave connecting two different end states. The discrepancy of the end states is linked to the complexity of the corresponding flux, which requires a new type of an energy estimate. To overcome, we use the a priori contraction estimate of a weighted relative entropy functional up to a translation, which was proved by Choi-Kang-Kwon-Vasseur. The boundedness of the shift implies a priori bound of the relative entropy functional without a shift on any time interval of existence, which produces a $H^1$-estimate thanks to a De Giorgi type lemma. Moreover, to remove possibility of vacuum appearance, we use the lemma again.

We study the interaction between the stability, and the propagation of regularity, for solutions to the incompressible 3D Euler equation. It is still unknown whether a solution with smooth initial data can develop a singularity in finite time. This article explains why the prediction of such a blow-up, via direct numerical experiments, is so difficult. It is described how, in such a scenario, the solution becomes unstable as time approaches the blow-up time., Comment: 12 pages

We show the vanishing viscosity limit to entropy shocks for the fractal Burgers equation in one space dimension. More precisely, we quantify the rate of convergence of the inviscid limit in $L^2$ for large initial perturbations around the entropy shock on any bounded time interval. This is the first result on the inviscid limit to entropy shock for the fractal Burgers equation with the quantified convergence, for large initial perturbations.

We prove the global existence and uniqueness of smooth solutions to the one-dimensional barotropic Navier-Stokes system with degenerate viscosity $\mu(\rho)=\rho^\alpha$. We establish that the smooth solutions have possibly two different far-fields, and the initial density remains positive globally in time, for the initial data satisfying the same conditions. In addition, our result works for any $\alpha>0$, i.e., for a large class of degenerate viscosities. In particular, our models include the viscous shallow water equations. This extends the result of Constantin-Drivas-Nguyen-Pasqualotto \cite[Theorem 1.5]{CDNP} (on the case of periodic domain) to the case where smooth solutions connect possibly two different limits at the infinity on the whole space.

We prove that the set of singular times for weak solutions of the space homogeneous Landau equation with Coulomb potential constructed as in [C. Villani, Arch. Rational Mech. Anal. 143 (1998), 273-307] has Hausdorff dimension at most 1/2.

We consider the dissipative SQG equation in bounded domains, first introduced by Constantin and Ignatova in 2016. We show global Holder regularity up to the boundary of the solution, with a method based on the De Giorgi techniques. The boundary introduces several difficulties. In particular, the Dirichlet Laplacian is not translation invariant near the boundary, which leads to complications involving the Riesz transform., Comment: added proof of global existence, made several expositional changes based on referee report

In this paper, we extend considerably the global existence results of entropy-weak solutions related to compressible Navier-Stokes system with density dependent viscosities obtained, independently (using different strategies), by Vasseur-Yu [Inventiones mathematicae (2016) and arXiv:1501.06803 (2015)] and by Li-Xin [arXiv:1504.06826 (2015)].More precisely we are able to consider a physical symmetric viscous stress tensor $\sigma=2\mu(\rho)\,{\mathbb{D}}(u)+\bigl(\lambda(\rho){\rm div}u -P(\rho)\bigr)\, {\rm Id}$ where ${\mathbb D}(u) = [\nabla u + \nabla^T u]/2$ with a shear and bulk viscosities (respectively $\mu(\rho)$ and $\lambda(\rho)$) satisfying the BD relation $\lambda(\rho)=2(\mu'(\rho)\rho - \mu(\rho))$ and a pressure law $P(\rho)=a\rho^\gamma$ (with $a>0$ a given constant) for any adiabatic constant $\gamma>1$. The nonlinear shear viscosity $\mu(\rho)$ satisfies some lower and upper bounds for low and high densities (our mathematical result includes the case $\mu(\rho)= \mu\rho^\alpha$ with $2/3 < \alpha < 4$ and $\mu>0$ constant). This provides an answer to a longstanding mathematical question on compressible Navier-Stokes equations with density dependent viscosities as mentioned for instance by F. Rousset in the Bourbaki 69\`eme ann\'ee, 2016--2017, no 1135.

We revisit a model for three-dimensional, inviscid quasi-geostrophic flow on bounded, cylindrical domains introduced by the authors in \cite{nv18}. We prove the local-in-time existence of classical solutions.

We consider a hyperbolic-parabolic system arising from a chemotaxis model in angiogenesis, which is described by a Keller-Segel equation with singular sensitivity. It is known to allow viscous shocks (so-called traveling waves). We introduce a relative entropy of the system, which can capture how close a solution at a given time is to a given shock wave in almost $L^2$-sense. When the shock strength is small enough, we show the functional is non-increasing in time for any large initial perturbation. The contraction property holds independently of the strength of the diffusion.

In this paper, we show uniqueness and stability for the piecewise-smooth solutions to the Burgers--Hilbert equation constructed in Bressan and Zhang [Commun. Math. Sci., 15(1):165--184, 2017]. The Burgers--Hilbert equation is $u_t+(\frac{u^2}{2})_x=\mathbf{H}[u]$ where $\mathbf{H}$ is the Hilbert transform, a nonlocal operator. We show stability and uniqueness for solutions amongst a larger class than the uniqueness result in Bressan and Zhang. The solutions we consider are measurable and bounded, satisfy at least one entropy condition, and verify a strong trace condition. We do not have smallness assumptions. We use the relative entropy method and theory of shifts (see Vasseur [Handbook of Differential Equations: Evolutionary Equations, 4:323 -- 376, 2008])., Comment: 46 pages

We prove the uniqueness and stability of entropy shocks to the isentropic Euler systems among all vanishing viscosity limits of solutions to associated Navier-Stokes systems. To take into account the vanishing viscosity limit, we show a contraction property for any large perturbations of viscous shocks to the Navier-Stokes system. The contraction estimate does not depend on the strength of the viscosity. This provides a good control on the inviscid limit process. We prove that, for any initial value, there exists a vanishing viscosity limit to solutions of the Navier-Stokes system. The convergence holds in a weak topology. However, this limit satisfies some stability estimates measured by the relative entropy with respect to an entropy shock. In particular, our result provides the uniqueness of entropy shocks to the shallow water equation in a class of inviscid limits of solutions to the viscous shallow water equations., Comment: arXiv admin note: text overlap with arXiv:1712.07348

We present a formal derivation of the inviscid 3D quasi-geostrophic system (QG) from primitive equations on a bounded, cylindrical domain. A key point in the derivation is the treatment of the lateral boundary and the resulting boundary conditions it imposes on solutions. To our knowledge, these boundary conditions are new and differentiate our model from closely related models which have been the object of recent study. These boundary conditions are natural for a variational problem in a particular Hilbert space. We construct solutions and prove an elliptic regularity theorem corresponding to the variational problem, allowing us to show the existence of global weak solutions to (QG)., Comment: 29 pages

This paper is dedicated to the construction of a pseudo-norm, for which small shock profiles of the barotropic Navier-Stokes equation have a contraction property. This contraction property holds in the class of any large 1D weak solutions to the barotropic Navier-Stokes equation. It implies a stability condition which is independent of the strength of the viscosity. The proof is based on the relative entropy method, and is reminiscent to the notion of a-contraction first introduced by the authors in the hyperbolic case., Comment: Accepted in J. Eur. Math. Soc. (JEMS)

We establish the boundedness of solutions of reaction-diffusion systems with quadratic (in fact slightly super-quadratic) reaction terms that satisfy a natural entropy dissipation property, in any space dimension N>2. This bound imply the smoothness of the solutions. This result extends the theory which was restricted to the two-dimensional case. The proof heavily uses De Giorgi's iteration scheme, which allows us to obtain local estimates. The arguments rely on duality reasonings in order to obtain new estimates on the total mass of the system, both in $L^{(N+1)/N}$ norm and in a suitable weak norm. The latter uses $C^\alpha$ regularization properties for parabolic equations.

For hyperbolic systems of conservation laws, uniqueness of solutions is still largely open. We aim to expand the theory of uniqueness for systems of conservation laws. One difficulty is that many systems have only one entropy. This contrasts with scalar conservation laws, where many entropies exist. It took until 1994 to show that one entropy is enough to ensure uniqueness of solutions for the scalar conservation laws (see Panov [Mat. Zametki, 55(5):116--129, 159, 1994]). This single entropy result was proven again by De Lellis, Otto and Westdickenberg about 10 years later [Quart. Appl. Math., 62(4):687--700, 2004]. These two proofs both rely on the special connection between Hamilton--Jacobi equations and scalar conservation laws in one space dimension. However, this special connection does not extend to systems. In this paper, we prove the single entropy result for scalar conservation laws without using Hamilton--Jacobi. Our proof lays out new techniques that are promising for showing uniqueness of solutions in the systems case., Comment: 34 pages

In this paper, we prove the existence of global weak solutions to the compressible two-fluid Navier-Stokes equations in three dimensional space. The pressure depends on two different variables from the continuity equations. We develop an argument of variable reduction for the pressure law. This yields to the strong convergence of the densities, and provides the existence of global solutions in time, for the compressible two-fluid Navier-Stokes equations, with large data in three dimensional space.

In this article we obtain Holder estimates for solutions to second-order Hamilton-Jacobi equations with super-quadratic growth in the gradient and unbounded source term. The estimates are uniform with respect to the smallness of the diffusion and the smoothness of the Hamiltonian. Our work is in the spirit of a result by P. Cardaliaguet and L. Silvestre [5]. We utilize De Giorgi's method, which was introduced to this class of equations in [6].

In this paper, the authors show the existence of global in time classical solutions to the 3D quasi-geostrophic system with Ekman pumping for any smooth initial value (possibly large). This system couples an inviscid transport equation in $\mathbb{R}^3_+$ with an equation on the boundary satisfied by the trace. The proof combines the De Giorgi regularization effect on the boundary $z=0$ -similar to the so called surface quasi-geostrophic equation- with Beale-Kato-Majda techniques to propagate regularity for $z>0$. A potential theory argument is used to strengthen the regularization effect on the trace up to the Besov space $\mathring{B}_{\infty,\infty}^1$.

We consider a $L^2$-contraction of large viscous shock waves for the multi-dimensional scalar viscous conservation laws, up to a suitable shift. The shift function depends on the time and space variables. It solves a parabolic equation with inhomogeneous coefficients reflecting the perturbation. We consider a suitably small $L^2$-perturbation around a viscous planar shock wave of arbitrarily large strength. However, we do not impose any condition on the anti-derivative variables of the perturbation around shock profile. More precisely, it is proved that if the initial perturbation around the viscous shock wave is suitably small in the $L^2$ norm, then the $L^2$-contraction holds true for the viscous shock wave up to a shift function which may depend on the temporal and spatial variables. Moreover, as the time $t$ tends to infinity, the $L^2$-contraction holds true up to a time-dependent shift function. In particular, if we choose some special initial perturbation, then we can prove a $L^2$ convergence of the solutions towards the associated shock profile up to a time-dependent shift.

This paper is dedicated to the construction of global weak solutions to the quantum Navier-Stokes equation, for any initial value with bounded energy and entropy. The construction is uniform with respect to the Planck constant. This allows to perform the semi-classical limit to the associated compressible Navier-Stokes equation. One of the difficulty of the problem is to deal with the degenerate viscosity, together with the lack of integrability on the velocity. Our method is based on the construction of weak solutions that are renormalized in the velocity variable. The existence, and stability of these solutions do not need the Mellet-Vasseur inequality.

We study a porous medium equation with right hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator. The derivative in time is also fractional of Caputo-type and which takes into account "memory''. The precise model is \[ D_t^{\alpha} u - \text{div}(u(-\Delta)^{-\sigma} u) = f, \quad 0<\sigma <1/2. \] We pose the problem over $\{t\in {\mathbb R}^+, x\in {\mathbb R}^n\}$ with nonnegative initial data $u(0,x)\geq 0 $ as well as right hand side $f\geq 0$. We first prove existence for weak solutions when $f,u(0,x)$ have exponential decay at infinity. Our main result is H\"older continuity for such weak solutions.

This paper is dedicated to the application of the DeGiorgi-Nash-Moser regularity theory to the kinetic Fokker-Planck equation. This equation is hypoelliptic. It is parabolic only in the velocity variable, while the Liouville transport operator has a mixing effect in the position/velocity phase space. The mixing effect is incorporated in the classical DeGiorgi method via the averaging lemmas. The result can be seen as a H\"{o}lder regularity version of the classical averaging lemmas., Comment: 21 pages. Statement of Lemma 3.2 corrected. Proof of Lemma 3.2 modified accordingly

We consider the zero heat conductivity limit to a contact discontinuity for the mono-dimensional full compressible Navier-Stokes-Fourier system. The method is based on the relative entropy method, and do not assume any smallness conditions on the discontinuity, nor on the $BV$ norm of the initial data. It is proved that for any viscosity $\nu\geq0$, the solution of the compressible Navier-Stokes-Fourier system (with well prepared initial value) converges, when the heat conductivity $\kappa$ tends to zero, to the contact discontinuity solution to the corresponding Euler system. We obtain the decay rate $\kappa^{\frac12}$. It implies that the heat conductivity dominates the dissipation in the regime of the limit to a contact discontinuity. This is the first result, based on the relative entropy, of an asymptotic limit to a discontinuous solutions for a system.

We study the contraction properties (up to shift) for admissible Rankine-Hugoniot discontinuities of $n\times n$ systems of conservation laws endowed with a convex entropy. We first generalize the criterion developed in [47], using the spatially inhomogeneous pseudo-distance introduced in [50]. Our generalized criterion guarantees the contraction property for extremal shocks of a large class of systems, including the Euler system. Moreover, we introduce necessary conditions for contraction, specifically targeted for intermediate shocks. As an application, we show that intermediate shocks of the two-dimensional isentropic magnetohydrodynamics do not verify any of our contraction properties. We also investigate the contraction properties, for contact discontinuities of the Euler system, for a certain range of contraction weights. All results do not involve any smallness condition on the initial perturbation, nor on the size of the shock.

The global-in-time existence of weak solutions to the barotropic compressible quantum Navier-Stokes equations with damping is proved for large data in three dimensional space. The model consists of the compressible Navier-Stokes equations with degenerate viscosity, and a nonlinear third-order differential operator, with the quantum Bohm potential, and the damping terms. The global weak solutions to such system is shown by using the Faedo-Galerkin method and the compactness argument. This system is also a very important approximated system to the compressible Navier-Stokes equations. It will help us to prove the existence of global weak solutions to the compressible Navier-Stokes equations with degenerate viscosity in three dimensional space., Comment: This paper provides the existence of the approximation in arXiv:1501.06803