arXiv subjects

Boris Haspot

Publications and source records attributed to Boris Haspot.

At least 19 recordsLinked to original sources

arXiv2025

Vanishing viscosity limit for $n\times n$ hyperbolic system of conservation laws in 1-d with nonlinear viscosity: Part-I Uniform BV estimates

We consider the following parabolic approximation for hyperbolic system of conservation laws in 1-D with non-singular viscosity matrix $B(u)$ and $A(u)$ strictly hyperbolic, \[u^\varepsilon_t+A(u^\varepsilon)u^\varepsilon_x=\varepsilon(B(u^\varepsilon)u^\varepsilon_x)_x.\] We prove global in time uniform $BV$ bound for solution to this parabolic system when $\varepsilon>0$ provided that the initial data is small in $BV$ and the matrix $A(u)$ and $B(u)$ commutate. Moreover, in the case where the system is conservative, we show that the sequence $(u^\varepsilon)_{\varepsilon>0}$ admits a limit $u$, which is the unique global weak solution to the limiting strictly hyperbolic system. We provide a concrete application of this result in the study of the visco-dispersive limit of the Navier-Stokes-Korteweg system.

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arXiv2025

Viscous approximation of triangular system in 1-d with nonlinear viscosity

We study the vanishing viscosity limit for $2\times2$ triangular system of hyperbolic conservation laws when the viscosity coefficients are non linear. In this article, we assume that the viscosity matrix $B(u)$ is commutating with the convective part $A(u)$. We show the existence of global smooth solution to the parabolic equation satisfying uniform total variation bound in $\varepsilon$ provided that the initial data is small in $BV$. This extends the previous result of Bianchini and Bressan [Commun. Pure Appl. Anal. (2002)] which was considering the case $B(u)=I$.

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arXiv2024

Existence of BV solutions for $2\times2$ hyperbolic balance laws for $L^\infty$ initial data

We prove the existence of BV solutions for $2\times 2$ system of hyperbolic balance laws in one space dimension. The flux is assumed to have two genuinely nonlinear characteristic fields. We consider a general force which may possibly depend on time and space variable as well. To prove the existence, we assume the initial data to be small in $L^\infty$. Furthermore, we also study qualitative behavior for entropy solutions to hyperbolic system of balance laws.

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arXiv2024

Vanishing viscosity limit for hyperbolic system of Temple class in 1-d with nonlinear viscosity

We consider hyperbolic system with nonlinear viscosity such that the viscosity matrix $B(u)$ is commutating with $A(u)$ the matrix associated to the convective term. The drift matrix is assumed to be Temple class. First, we prove the global existence of smooth solutions for initial data with small total variation. We show that the solution to the parabolic equation converges to a semi-group solution of the hyperbolic system as viscosity goes to zero. Furthermore, we prove that the zero diffusion limit coincides with the one obtained in [Bianchini and Bressan, Indiana Univ. Math. J. 2000].

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arXiv2023

Existence of BV solution for the Euler-Poisson system in one dimension with large initial data

This paper deals with the existence of BV solution for the Euler-Poisson system endowed with a $γ$ pressure law. More precisely, we prove the existence of weak solution in the BV framework with arbitrary large initial data when $γ=1+2ε$ satisfies a smallness condition. We use the Glimm scheme combined with a splitting method as introduced in [Poupaud, Rascle and Vila, J. Differential Equations, 1995]. Existence of BV solution of 1-D isentropic Euler equation for large data and $γ=1+2ε$ is proved in [Nishida and Smoller, Comm. Pure Appl. Math, 1973]. Due to the presence of electric field, the difficulty arises while controlling the Glimm functional for the Euler-Poisson system. It requires a subtle study of wave interaction. In the later part of this article, we discuss the initial-boundary value problem for the Euler-Poisson system. We prove the existence of $BV$ solution for the initial-boundary value problem with large initial and boundary data. By an explicit example, we also show ill-posedness of initial-boundary value problem for the isentropic Euler equation.

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arXiv2021

Vanishing capillarity limit of the Navier-Stokes-Korteweg system in one dimension with degenerate viscosity coefficient and discontinuous initial density

In the first main result of this paper we prove that one can approximate discontinious solutions of the 1d Navier Stokes system with solutions of the 1d Navier-Stokes-Korteweg system as the capilarity parameter tends to 0. Moreover, we allow the viscosity coefficients $μ$ = $μ$ ($ρ$) to degenerate near vaccum. In order to obtain this result, we propose two main technical novelties. First of all, we provide an upper bound for the density verifing NSK that does not degenerate when the capillarity coefficient tends to 0. Second of all, we are able to show that the positive part of the effective velocity is bounded uniformly w.r.t. the capillary coefficient. This turns out to be crucial in providing a lower bound for the density. The second main result states the existene of unique finite-energy global strong solutions for the 1d Navier-Stokes system assuming only that $ρ$0, 1/$ρ$0 $\in$ L $\infty$. This last result finds itself a natural application in the context of the mathematical modeling of multiphase flows.

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arXiv2020

Fractional $BV$ solutions for $2\times 2$ systems of conservation laws with a linearly degenerate field

The class of $2\times 2$ nonlinear hyperbolic systems with one genuinely nonlinear field and one linearly degenerate field are considered. Existence of global weak solutions for small initial data in fractional BV spaces $BV^s$ is proved. The exponent $s$ is related to the usual fractional Sobolev derivative. Riemann invariants $w$ and $z$ corresponding respectively to the genuinely nonlinear component and to the linearly degenerate component play different key roles in this work. We obtain the existence of a global weak solution provided that the initial data written in Riemann coordinates $ (w_0,z_0)$ are small in $ BV^s \times L^\infty $, $1/3 \leq s<1$. The restriction on the exponent $s$ is due to a fundamental result of P.D. Lax, the variation of the Riemann invariant $z$ on the Lax shock curve depends in a cubic way of the variation of the other Riemann invariant $w$.

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arXiv2019

New effective pressure and existence of global strong solution for compressible Navier-Stokes equations with general viscosity coefficient in one dimension

In this paper we prove the existence of global strong solution for the Navier-Stokes equations with general degenerate viscosity coefficients. The cornerstone of the proof is the introduction of a new effective pressure which allows to obtain an Oleinik-type estimate for the so called effective velocity. In our proof we make use of additional regularizing effects on the velocity which requires to extend the technics developed by Hoff for the constant viscosity case.

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arXiv2019

Strong solution for Korteweg system in bmo$^{-1}(\mathbb{R}^N)$ with initial density in $L^\infty$

In this paper we investigate the question of the local existence of strong solution for the Korteweg system in critical spaces in dimension $N\geq 1$ provided that the initial data are small. More precisely the initial momentum $ρ_0 u_0$ belongs to $\mbox{bmo}_{T}^{-1}(\mathbb{R}^N)$ for $T>0$ and the initial density $ρ_0$ is in $L^\infty(\mathbb{R}^N)$ and far away from the vacuum. This result extends the so called Koch-Tataru Theorem for the incompressible Navier-Stokes equations to the case of the Korteweg system. It is also interesting to observe that any initial shock on the density is instantaneously regularized inasmuch as the density becomes Lipschitz for any $ρ(t,\cdot)$ with $t>0$. We also prove the existence of global strong solution for small initial data $(ρ_0-1,ρ_0u_0)$ in the homogeneous Besov spaces $(\dot{B}^{\frac{N}{2}-1}_{2,\infty} (\mathbb{R}^N) \cap \dot{B}^{\frac{N}{2}}_{2,\infty} (\mathbb{R}^N) \cap L^\infty(\mathbb{R}^N)) \times (\dot{B}^{\frac{N}{2}-1}_{2,\infty} (\mathbb{R}^N))^N$. This result allows in particular to extend in dimension $N=2$ the notion of Oseen solutions defined for incompressible Navier-Stokes equations to the case of the Korteweg system when the vorticity of the momentum $ρ_0 u_0$ is a Dirac mass $αδ_0$ with $α$ sufficiently small.

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arXiv2019

Global bmo$^{-1}(\mathbb{R}^N)$ radially symmetric solution for compressible Navier-Stokes equations with initial density in $L^\infty(\mathbb{R}^N)$

In this paper we investigate the question of the existence of global weak solution for the compressible Navier Stokes equations provided that the initial momentum $ρ_0 u_0$ belongs to $\mbox{bmo}^{-1}(\mathbb{R}^N)$ with $N= 2,3$ and is radially symmetric. More precisely we deal with the so called viscous shallow water system when the viscosity coefficients verify $μ(ρ)=μρ$, $λ(ρ)=0$ with $μ>0$. We prove then a equivalent of the so called Koch-Tataru theorem for the compressible Navier-Stokes equations. In addition we assume that the initial density $ρ_0$ is only bounded in $L^\infty(\mathbb{R}^N)$, it allows us in particular to consider initial density admitting shocks. Furthermore we show that if the coupling between the density and the velocity is sufficiently strong, then the initial density which admits initially shocks is instantaneously regularizing inasmuch as the density becomes Lipschitz. This coupling is expressed via the regularity of the so called effective velocity $v=u+2μ\nabla\lnρ$. In our case $v_0$ belongs to $L^2(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$, it is important to point out that this choice on the initial data implies that we work in a setting of infinite energy on the initial data $(ρ_0,u_0)$, it extends in particular the results of \cite{V}. In a similar way, we consider also the case of the dimension $N=1$ where the momentum $ρ_0 u_0$ belongs to $bmo^{-1}(\mathbb{R})$ without any geometric restriction.\\ To finish we prove the global existence of strong solution for large initial data provided that the initial data are radially symmetric and sufficiently regular in dimension $N=2,3$ for $γ$ law pressure.

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arXiv2019

Vortex solutions for the compressible Navier-Stokes equations with general viscosity coefficients in 1D: regularizing effects or not on the density

We consider Navier-Stokes equations for compressible viscous fluids in the one-dimensional case with general viscosity coefficients. We prove the existence of global weak solution when the initial momentum $ρ_0 u_0$ belongs to the set of the finite measure ${\cal M}(\mathbb{R})$ and when the initial density $ρ_0$ is in the set of bounded variation functions $BV(\mathbb{R})$. In particular it allows to deal with initial momentum which are Dirac masses and initial density which admit shocks. We can observe in particular that this type of initial data have infinite energy. Furthermore we show that if the coupling between the density and the velocity is sufficiently strong then the initial density which admits initially shocks is instantaneously regularized and becomes continuous. This coupling is expressed via the regularity of the so called effective velocity $v=u+\frac{μ(ρ)}{ρ^2}\partial_x ρ$ with $μ(ρ)$ the viscosity coefficient. Inversely if the coupling between the initial density and the initial velocity is too weak (typically $ρ_0 v_0\in{\cal M}(\mathbb{R})$) then we prove the existence of weak energy solution in finite time but the density remains a priori discontinuous on the time interval of existence.

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arXiv2018

Existence of global strong solution for the compressible Navier-Stokes equations with degenerate viscosity coefficients in 1D

We consider Navier-Stokes equations for compressible viscous fluids in one dimension. We prove the existence of global strong solution with large initial data for the shallow water system. The key ingredient of the proof relies to a new formulation of the compressible equations involving a new effective velocity $v$ (see \cite{cras,para,CPAM,CPAM1}) such that the density verifies a parabolic equation. We estimate $v$ in $L^\infty$ norm which enables us to control the vacuum on the density.

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arXiv2016

Global strong solution for the Korteweg system with quantum pressure in dimension $N\geq 2$

This work is devoted to prove the existence of global strong solution in dimension $N\geq 2$ for a isothermal model of capillary fluids derived by J.E Dunn and J.Serrin (1985) (see \cite{fDS}), which can be used as a phase transition model. We will restrict us to the case of the so called compressible Navier-Stokes system with quantum pressure which corresponds to consider the capillary coefficient $κ(ρ)=\frac{κ_1}ρ$ with $κ_1>0$. In a first part we prove the existence of strong solution in finite time for large initial data with a precise bound by below on the life span $T^*$. This one depends on the norm of the initial data $(ρ_0,v_0)$. The second part consists in proving the existence of global strong solution with particular choice on the capillary coefficient ( where $κ_1=μ^2$) and on the viscosity tensor which corresponds to the viscous shallow water case $-2μ{\rm div}(ρDu)$. To do this we derivate different energy estimate on the density and the effective velocity $v$ which ensures that the strong solution can be extended beyond $T^*$. The main difficulty consists in controlling the vacuum or in other words to estimate the $L^\infty$ norm of $\frac{1}ρ$. The proof relies mostly on a method introduced by De Giorgi \cite{DG} (see also Ladyzhenskaya et al in \cite{La} for the parabolic case) to obtain regularity results for elliptic equations with discontinuous diffusion coefficients and a suitable bootstrap argument.

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arXiv2016

Global well-posedness of the Euler-Korteweg system for small irrotational data

The Euler-Korteweg equations are a modification of the Euler equations that takes into account capillary effects. In the general case they form a quasi-linear system that can be recast as a degenerate Schrödinger type equation. Local well-posedness (in subcritical Sobolev spaces) was obtained by Benzoni-Danchin-Descombes in any space dimension, however, except in some special case (semi-linear with particular pressure) no global well-posedness is known. We prove here that under a natural stability condition on the pressure, global well-posedness holds in dimension $d\geq 3$ for small irrotational initial data. The proof is based on a modified energy estimate, standard dispersive properties if $d\geq 5$, and a careful study of the nonlinear structure of the quadratic terms in dimension $3$ and $4$ involving the theory of space time resonance.

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arXiv2015

From the highly compressible Navier-Stokes equations to the Porous Medium equation - rate of convergence

We consider the one-dimensional Cauchy problem for the Navier-Stokes equations with degenerate viscosity coefficient in highly compressible regime. It corresponds to the compressible Navier-Stokes system with large Mach number equal to $\frac{1}{\sqrt{\varepsilon}}$ for $\varepsilon$ going to $0$. When the initial velocity is related to the gradient of the initial density, a solution to the continuity equation-$ρ_\varepsilon$ converges to the unique solution to the porous medium equation [13,14]. For viscosity coefficient $μ(ρ_\varepsilon)=ρ_\varepsilon^α$ with $α>1$, we obtain a rate of convergence of $ρ_\varepsilon$ in $L^\infty(0,T; H^{-1}(\mathbb{R}))$; for $1<α\leq\frac{3}{2}$ the solution $ρ_\varepsilon$ converges in $L^\infty(0,T;L^2(\mathbb{R}))$. For compactly supported initial data, we prove that most of the mass corresponding to solution $ρ_\varepsilon$ is located in the support of the solution to the porous medium equation. The mass outside this support is small in terms of $\varepsilon$.

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arXiv2014

Weak-Strong uniqueness for compressible Navier-Stokes system with degenerate viscosity coefficient and vacuum in one dimension

We prove weak-strong uniqueness results for the compressible Navier-Stokes system with degenerate viscosity coefficient and with vacuum in one dimension. In other words, we give conditions on the weak solution constructed in \cite{Jiu} so that it is unique. The novelty consists in dealing with initial density $ρ_0$ which contains vacuum. To do this we use the notion of relative entropy developed recently by Germain, Feireisl et al and Mellet and Vasseur (see \cite{PG,Fei,15}) combined with a new formulation of the compressible system (\cite{cras,CPAM,CPAM1,para}) (more precisely we introduce a new effective velocity which makes the system parabolic on the density and hyperbolic on this velocity).

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arXiv2014

New formulation of the compressible Navier-Stokes equations and parabolicity of the density

In this paper we give a new formulation of the compressible Navier-Stokes by introducing an suitable effective velocity $v=u+\n\va(ρ)$ provided that the viscosity coefficients verify the algebraic relation of \cite{BD}. We give in particular a very simple proof of the entropy discovered in \cite{BD}, in addition our argument show why the algebraic relation of \cite{BD} appears naturally. More precisely the system reads in a very surprising way as two parabolic equation on the density $ρ$ and the vorticity ${\rm curl}v$, and as a transport equation on the divergence ${\rm div}v$. We show the existence of strong solution with large initial data in finite time when $(ρ_0-1)\in B^{\NN}_{p,1}$. A remarkable feature of this solution is the regularizing effects on the density. We extend this result to the case of global strong solution with small initial data.

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arXiv2013

Existence of global strong solution for Korteweg system with large infinite energy initial data

This work is devoted to the study of the initial boundary value problem for a general isothermal model of capillary fluids derived by J.E Dunn and J.Serrin (1985), which can be used as a phase transition model. We will prove the existence of local and global (under a condition of smallness on the initial data) strong solutions with discontinuous initial density when $\lnρ_{0}$ belongs in the Besov space $B^{\N}_{2,\infty}(\R^{N})$.The main difficulty concerns the proof of new estimate of maximum principle type for the linear system associated to the Korteweg system, the proof is based on a characterization of the Besov space in terms of the semi group associated to this linear system. Let also point out that we prove the existence of global strong solution with a smallness hypothesis which is subcritical in terms of the scaling of the equations, it allows us to exhibit a family of large energy initial data for the scaling of the equations providing global strong solution. In particular for the first time up our knowledge we show the existence of global strong solution for some large energy initial data when N=2. We finish this paper by introducing the notion of quasi-solutions for the Korteweg's system (a tool which has been developed in the framework of the compressible Navier-Stokes equations \cite{arxiv,arxiv1,hal,cras1,cras2}) which enables us to improve the previous result and to obtain the existence of global strong solution with large initial velocity in $B^{\N-1}_{2,\infty}$. As a corollary, we get global existence (and uniqueness) for highly compressible Korteweg system when $N\geq2$.

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