arXiv subjects

Alexis F. Vasseur

Publications and source records attributed to Alexis F. Vasseur.

At least 19 recordsLinked to original sources

arXiv2025

From Navier-Stokes to BV solutions of the barotropic Euler equations

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.

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arXiv2025

$L^2$-contraction and asymptotic stability of large shock for scalar viscous conservation laws

We investigate $L^2$-contraction and time-asymptotic stability of large shock for scalar viscous conservation laws with polynomial flux. For the strictly convex flux $f(u)=u^p $ with $2\leq p \leq 4$, we can prove $L^2$-contraction and time-asymptotic stability of arbitrarily large viscous shock profile in $H^1$-framework by using $a$-contraction method with time-dependent shift and suitable weight function. Additionally, if the initial perturbation belongs to $L^1$, then $L^2$ time-asymptotic decay rate $t^{-\frac{1}{4}}$ can be obtained.

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arXiv2025

Energy dissipation near the outflow boundary in the vanishing viscosity limit

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 the Euler and the Navier-Stokes solutions on the outflow boundary. Moreover, we show that the enstrophy within a layer of order $ν/ \bar U$ is comparable with the total enstrophy. The rate of enstrophy production near the boundary is inversely proportional to $ν$.

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arXiv2024

Non-uniqueness for continuous solutions to 1D hyperbolic systems

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.

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arXiv2024

Layer separation of the 3D incompressible Navier-Stokes equation in a bounded domain

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 $u^ν$ 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.

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arXiv2024

Effective velocity and $L^\infty$-based well-posedness for incompressible fluids with odd viscosity

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, $ρ_0-1= O(\varepsilon)$ for small $\varepsilon>0$, the corresponding solution is defined up to some time $T_\varepsilon>0$ satisfying the property $T_\varepsilon\,\longrightarrow\,+\infty$ when $\varepsilon\to0^+$.

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arXiv2024

Local regularity for the space-homogeneous Landau equation with very soft potentials

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 $a_{ij} (z)=|z|^γ(|z|^2 δ_{ij} - z_iz_j)$ for $γ\in [-3,-2)$. 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 $m_\ast:= \frac72 |2+γ|$. 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.

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arXiv2021

Boundary Vorticity Estimates for Navier-Stokes and Application to the Inviscid Limit

Consider the steady solution to the incompressible Euler equation $\bar u=Ae_1$ in the periodic tunnel $Ω=\mathbb T^{d-1}\times(0,1)$ in dimension $d=2,3$. Consider now the family of solutions $u^ν$ to the associated Navier-Stokes equation with the no-slip condition on the flat boundaries, for small viscosities $ν=A/\mathsf{Re}$, and initial values in $L^2$. We are interested in the weak inviscid limits up to subsequences $u^ν\rightharpoonup u^\infty$ when both the viscosity $ν$ converges to 0, and the initial value $u^ν_0$ converges to $Ae_1$ in $L^2$. Under a conditional assumption on the energy dissipation close to the boundary, Kato showed in 1984 that $u^ν$ 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 $0<T<T_0$: $\|u_E(T)-Ae_1\|_{L^2(Ω)}^2\approx A^3T$. This predicts the possibility of a layer separation with an energy of order $A^3 T$. We show in this paper that the energy of layer separation associated with any asymptotic $u^\infty$ obtained via double limits cannot be more than $\|u^\infty(T)-Ae_1\|_{L^2 (Ω)}^2\lesssim A^3T$. This result holds unconditionally for any weak limit of Leray-Hopf solutions of the Navier-Stokes equation. Especially, it shows that, even if the limit is not unique, the shear flow pattern is observable up to time $1/A$. This provides a notion of stability despite the possible non-uniqueness of the limit predicted by the convex integration theory. The result relies on a new boundary vorticity estimate for the Navier-Stokes equation. This new estimate, inspired by previous work on higher regularity estimates for Navier-Stokes, provides a nonlinear control scalable through the inviscid limit.

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arXiv2021

Time-asymptotic stability of composite waves of viscous shock and rarefaction for barotropic Navier-Stokes equations

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.

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arXiv2021

Global ill-posedness for a dense set of initial data to the Isentropic system of gas dynamics

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 $1<γ\leq 1+\frac2n$. 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.

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arXiv2020

Uniqueness and weak-BV stability for $2\times 2$ conservation laws

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.

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arXiv2020

Instability for Axisymmetric Blow-up Solutions to Incompressible Euler Equations

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.

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arXiv2019

Stability and uniqueness for piecewise smooth solutions to Burgers-Hilbert among a large class of solutions

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]).

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arXiv2018

On Uniqueness of Solutions to Conservation Laws Verifying a Single Entropy Condition

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.

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arXiv2017

Global in Time Classical Solutions to the 3D Quasi-geostrophic System for Large Initial Data

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 $\mathring{B}_{\infty,\infty}^1$.

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arXiv2017

De Giorgi Techniques Applied to Hamilton-Jacobi Equations with Unbounded Right-Hand Side

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].

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arXiv2015

$L^2$-contraction for shock waves of scalar viscous conservation laws

We consider the $L^2$-contraction up to a shift for viscous shocks of scalar viscous conservation laws with strictly convex fluxes in one space dimension. In the case of a flux which is a small perturbation of the quadratic burgers flux, we show that any viscous shock induces a contraction in $L^2$, up to a shift. That is, the $L^2$ norm of the difference of any solution of the viscous conservation law, with an appropriate shift of the shock wave, does not increase in time. If, in addition, the difference between the initial value of the solution and the shock wave is also bounded in $L^1$, the $L^2$ norm of the difference converges at the optimal rate $t^{-1/4}$. Both results do not involve any smallness condition on the initial value, nor on the size of the shock. In this context of small perturbations of the quadratic Burgers flux, the result improves the Choi and Vasseur's result in [7]. However, we show that the $L^2$-contraction up to a shift does not hold for every convex flux. We construct a smooth strictly convex flux, for which the $L^2$-contraction does not hold any more even along any Lipschitz shift.

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arXiv2015

Global weak solutions to compressible quantum Navier-Stokes equations with damping

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.

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