arXiv subjects

Boris Haspot

Publications and source records attributed to Boris Haspot.

At least 37 records · Page 2Linked to original sources

arXiv2013

Porous media, Fast diffusion equations and the existence of global weak solution for the quasi-solution of compressible Navier-Stokes equations

In \cite{arxiv,arxiv1,Kor,cras1,cras2}, we have developed a new tool called \textit{quasi solutions} which approximate in some sense the compressible Navier-Stokes equation. In particular it allows us to obtain global strong solution for the compressible Navier-Stokes equations with \textit{large} initial data on the irrotational part of the velocity (\textit{large} in the sense that the smallness assumption is subcritical in terms of scaling, it turns out that in this framework we are able to obtain a family of large initial data in the energy space in dimension N=2). In this paper we are interested in proving the result anounced in \cite{cras3} concerning the existence of global weak solution for the quasi-solutions, we also observe that for some choice of initial data (irrotationnal) the quasi solutions verify the porous media, the heat equation or the fast diffusion equations in function of the structure of the viscosity coefficients. In particular it implies that it exists classical quasi-solutions in the sense that they are $C^{\infty}$ on $(0,T)\times\R^{N}$ for any $T>0$. Finally we show the convergence of the global weak solution of compressible Navier-Stokes equations to the quasi solutions in the case of a vanishing pressure limit process. In particular we show that for highly compressible equations the speed of propagation of the density is quasi finite when the viscosity corresponds to $μ(ρ)=ρ^α$ with $α>1$. Furthermore the density is not far from converging asymptotically in time to the Barrenblatt solution of mass the initial density $ρ_{0}$.

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arXiv2013

On a Lagrangian method for the convergence from a non-local to a local Korteweg capillary fluid model

In the present article we are interested in further investigations for the barotropic compressible Navier-Stokes system endowed with a non-local capillarity we studied in [7]. Thanks to an accurate study of the associated linear system using a Lagrangian change of coordinates, we provide more precise energy estimates in terms of hybrid Besov spaces naturally depending on a threshold frequency (which is determined in function of the physical parameter) distinguishing the low and the high regimes. It allows us in particular to prove the convergence of the solutions from the non-local to the local Korteweg system. Another mathematical interest of this article is the study of the effect of the Lagrangian change on the non-local capillary term.

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arXiv2012

Existence of global strong solution for the compressible Navier-Stokes system and the Korteweg system in two-dimension

This paper is dedicated to the study of viscous compressible barotropic fluids in dimension N=2. We address the question of the global existence of strong solutions with large initial data for compressible Navier-Stokes system and Korteweg system. In the first case we are interested by slightly extending a famous result due to V. A. Vaigant and A. V. Kazhikhov in \cite{VG} concerning the existence of global strong solution in dimension two for a suitable choice of viscosity coefficient ($μ(ρ)=μ>0$ and $λ(ρ)=λρ^β$ with $β>3$) in the torus. We are going to weaken the condition on $β$ by assuming only $β>2$ essentially by taking profit of commutator estimates introduced by Coifman et al in \cite{4M} and using a notion of \textit{effective velocity} as in \cite{VG}. In the second case we study the existence of global strong solution with large initial data in the sense of the scaling of the equations for Korteweg system with degenerate viscosity coefficient and with friction term.

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arXiv2012

Remarks on global controllability for the shallow-water system with two control forces

In this paper we deal with the compressible Navier-Stokes equations with a friction term in one dimension on an interval. We study the exact controllability properties of this equation with general initial condition when the boundary control is acting at both endpoints of the interval. Inspired by the work of Guerrero and Imanuvilov in \cite{GI} on the viscous Burger equation, we prove by choosing irrotational data and using the notion of effective velocity developed in \cite{cpde,cras} that the exact global controllability result does not hold for any time $T$.

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arXiv2012

Global existence of strong solution for shallow water system with large initial data on the irrotational part

We show existence of global strong solutions with large initial data on the irrotational part for the shallow-water system in dimension $N\geq 2$. We introduce a new notion of \textit{quasi-solutions} when the initial velocity is assumed to be irrotational, these last one exhibit regularizing effects both on the velocity and in a very surprising way also on the density (indeed the density is a priori governed by an hyperbolic equation). We would like to point out that this smoothing effect is purely non linear and is absolutely crucial in order to deal with the pressure term as it provides new damping effects in high frequencies. In particular our result gives a first kind of answer to the problem of the existence of global weak solution for the shallow-water system. We conclude by giving new point wise decay estimates on the solution which improves the previous works \cite{HZ1,HZ2}.

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arXiv2011

Regularity of weak solutions of the compressible barotropic Navier-Stokes equations

Regularity and uniqueness of weak solutions of the compressible barotropic Navier-Stokes equations with constant viscosity coefficients is proven for small time in dimension $N=2,3$ under periodic boundary conditions. In this paper, the initial density is not required to have a positive lower bound and the pressure law is assumed to satisfy a condition that reduces to $P(ρ)=aρ^γ$ with $γ>1$ (in dimension three, additional conditions of size will be ask on $γ$). The second part of the paper is devoted to blow-up criteria for slightly subcritical initial data for the scaling of the equations when the viscosity coefficients $(μ,λ)$ are assumed constant provided that their ratio is large enough (in particular $0<λ<(5/4)μ$). More precisely we prove that under the condition $ρ$ belongs to $L^{\infty}((0,T)\times\T^{N})$ then we can extend the unique solution beyond $T>0$. Finally, we prove that weak solutions in the torus $\mathbb{T}^{N}$ turn out to be smooth as long as the density remains bounded in $L^{\infty}(0,T,L^{(N+1+\e)γ}(\mathbb{T}^{N}))$ with $\e>0$ arbitrary small. This result may be considered as a Prodi-Serrin theorem (see \cite{prodi} and \cite{serrin}) for compressible Navier-Stokes system.

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arXiv2011

Existence of global strong solutions for the shallow-water equations with large initial data

This work is devoted to the study of a viscous shallow-water system with friction and capillarity term. We prove in this paper the existence of global strong solutions for this system with some choice of large initial data when $N\geq 2$ in critical spaces for the scaling of the equations. More precisely, we introduce as in \cite{Hprepa} a new unknown,\textit{a effective velocity} $v=u+μ\n\ln h$ ($u$ is the classical velocity and $h$ the depth variation of the fluid) with $μ$ the viscosity coefficient which simplifies the system and allow us to cancel out the coupling between the velocity $u$ and the depth variation $h$. We obtain then the existence of global strong solution if $m_{0}=h_{0}v_{0}$ is small in $B^{\N-1}_{2,1}$ and $(h_{0}-1)$ large in $B^{\N}_{2,1}$. In particular it implies that the classical momentum $m_{0}^{'}=h_{0} u_{0}$ can be large in $B^{\N-1}_{2,1}$, but small when we project $m_{0}^{'}$ on the divergence field. These solutions are in some sense \textit{purely compressible}. We would like to point out that the friction term term has a fundamental role in our work inasmuch as coupling with the pressure term it creates a damping effect on the effective velocity.

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arXiv2011

Existence of global strong solution and vanishing capillarity-viscosity limit in one dimension for the Korteweg system

In the first part of this paper, we prove the existence of global strong solution for Korteweg system in one dimension. In the second part, motivated by the processes of vanishing capillarity-viscosity limit in order to select the physically relevant solutions for a hyperbolic system, we show that the global strong solution of the Korteweg system converges in the case of a $γ$ law for the pressure ($P(ρ)=aρ^γ$, $γ>1$) to entropic solution of the compressible Euler equations. In particular it justifies that the Korteweg system is suitable for selecting the physical solutions in the case where the Euler system is strictly hyperbolic. The problem remains open for a Van der Waals pressure because in this case the system is not strictly hyperbolic and in particular the classical theory of Lax and Glimm (see \cite{Lax,G}) can not be used.

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arXiv2011

Existence of strong solutions in a larger space for the shallow-water system

This paper is dedicated to the study of both viscous compressible barotropic fluids and Navier-Stokes equation with dependent density, when the viscosity coefficients are variable, in dimension $d\geq2$. We aim at proving the local and global well-posedness for respectively {\it large} and \textit{small} initial data having critical Besov regularity and more precisely we are interested in extending the class of initial data velocity when we consider the shallow water system, improving the results in \cite{CMZ1,H2} and \cite{arma}. Our result relies on the fact that the velocity $u$ can be written as the sum of the solution $u_{L}$ of the associated linear system and a remainder velocity term $\bar{u}$; then in the specific case of the shallow-water system the remainder term $\bar{u}$ is more regular than $u_{L}$ by taking into account the regularizing effects induced on the bilinear convection term. In particular we are able to deal with initial velocity in $\dot{H}^{\N-1}$ as Fujita and Kato for the incompressible Navier-Stokes equations (see \cite{FK}) with an additional condition of type $u_{0}\in B^{-1}_{\infty,1}$. We would like to point out that this type of result is of particular interest when we want to deal with the problem of the convergence of the solution of compressible system to the incompressible system when the Mach number goes to 0.

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arXiv2011

Blow-up criterion, ill-posedness and existence of strong solution for Korteweg system with infinite energy

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 strong solutions in finite time with discontinuous initial density, more precisely $\lnρ_{0}$ is in $B^{\N}_{2,\infty}(\R^{N})$. Our analysis improves the results of \cite{fDD} and \cite{fH1}, \cite{fH2} by working in space of infinite energy. In passing our result allow to consider initial data with discontinuous interfaces, whereas in all the literature the results of existence of strong solutions consider always initial density that are continuous. More precisely we investigate the existence of strong solution for Korteweg's system when we authorize jump in the pressure across some hypersurface. We obtain also a result of ill-posedness for Korteweg system and we derive a new blow-up criterion which is the main result of this paper. More precisely we show that if we control the vacuum (i.e $\frac{1}ρ\in L^{\infty}_{T}(\dot{B}^{0}_{N+\e,1}(\R^{N}))$ with $\e>0$) then we can extend the strong solutions in finite time. It extends substantially previous results obtained for compressible equations.

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arXiv2011

Convergence of capillary fluid models: from the non-local to the local Korteweg model

In this paper we are interested in the barotropic compressible Navier-Stokes system endowed with a non-local capillarity tensor depending on a small parameter $ε$ such that it heuristically tends to the local Korteweg system. After giving some physical motivations related to the theory of non-classical shocks (see [28]) we prove global well-posedness (in the whole space $R^d$ with $d\geq 2$) for the non-local model and we also prove the convergence, as $ε$ goes to zero, to the solution of the local Korteweg system.

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arXiv2010

Existence of global strong solutions in critical spaces for barotropic viscous fluids

This paper is dedicated to the study of viscous compressible barotropic fluids in dimension $N\geq2$. We address the question of the global existence of strong solutions for initial data close from a constant state having critical Besov regularity. In a first time, this article show the recent results of \cite{CD} and \cite{CMZ} with a new proof. Our result relies on a new a priori estimate for the velocity, where we introduce a new structure to \textit{kill} the coupling between the density and the velocity as in \cite{H2}. We study so a new variable that we call effective velocity. In a second time we improve the results of \cite{CD} and \cite{CMZ} by adding some regularity on the initial data in particular $ρ_{0}$ is in $H^{1}$. In this case we obtain global strong solutions for a class of large initial data on the density and the velocity which in particular improve the results of D. Hoff in \cite{5H4}. We conclude by generalizing these results for general viscosity coefficients.

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arXiv2010

Regularity of weak solutions of the compressible isentropic Navier-Stokes equation

Regularity and uniqueness of weak solution of the compressible isentropic Navier-Stokes equations is proven for small time in dimension $N=2,3$ under periodic boundary conditions. In this paper, the initial density is not required to have a positive lower bound and the pressure law is assumed to satisfy a condition that reduces to $γ>1$ when $N=2,3$ and $P(ρ)=aρ^γ$. In a second part we prove a condition of blow-up in slightly subcritical initial data when $ρ\in L^{\infty}$. We finish by proving that weak solutions in $\T^{N}$ turn out to be smooth as long as the density remains bounded in $L^{\infty}(L^{(N+1+\e)γ})$ with $\e>0$ arbitrary small.

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arXiv2009

Well-posedness in critical spaces for the system of Navier-Stokes compressible

This paper is dedicated to the study of viscous compressible barotropic fluids in dimension $N\geq2$. We address the question of well-posedness for {\it large} data having critical Besov regularity. Our result improve the analysis of R. Danchin and of B. Haspot, by the fact that we choose initial density more general in $B^{\NN}_{p,1}$ with $1\leq p<+\infty$. Our result relies on a new a priori estimate for the velocity, where we introduce a new structure to \textit{kill} the coupling between the density and the velocity. In particular our result is the first where we obtain uniqueness without imposing hypothesis on the gradient of the density.

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arXiv2009

Well-posedness in critical spaces for barotropic viscous fluids

This paper is dedicated to the study of viscous compressible barotropic fluids in dimension $N\geq2$. We address the question of well-posedness for {\it large} data having critical Besov regularity. %Our sole additional assumption is that %the initial density be bounded away from zero. This improves the analysis of \cite{DL} where the smallness of $ρ_{0} %-\barρ$ for some positive constant $\barρ$ was needed. Our result improve the analysis of R. Danchin by the fact that we choose initial density more general in $B^{\NN}_{p,1}$ with $1\leq p<+\infty$. Our result relies on a new a priori estimate for the velocity, where we introduce a new structure to kill the coupling between the density and the velocity. In particular our result is the first where we obtain uniqueness without imposing hypothesis on the gradient of the density.

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arXiv2009

Local well-posedness results for density-dependent incompressible fluids

This paper is dedicated to the study of the initial value problem for density dependent incompressible viscous fluids in $\R^{N}$ with $N\geq2$. We address the question of well-posedness for {\it large} data having critical Besov regularity and we aim at stating well-posedness in functional spaces as close as possible to the ones imposed in the incompressible Navier Stokes system by Cannone, Meyer and Planchon where $u_{0}\in B^{\NN-1}_{p,\infty}$ with $1\leq p<+\infty$. This improves the analysis of H. Abidi, R. Danchin and M. Paicu where $u_{0}$ is considered belonging to $B^{\NN-1}_{p,1}$ with $1\leq p<2N$. Our result relies on a new a priori estimate for transport equation introduce by Bahouri, Chemin and Danchin when the velocity $u$ is not considered Lipschitz.

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arXiv2009

Existence of weak solution for compressible fluid models of Korteweg type

This work is devoted to prove existence of global weak solutions for a general isothermal model of capillary fluids derived by J.- E Dunn and J. Serrin (1985) [6], which can be used as a phase transition model. We improve the results of [5] by showing the existence of global weak solution in dimension two for initial data in the energy space, close to a stable equilibrium and with specific choices on the capillary coefficients. In particular we are interested in capillary coefficients approximating a constant capillarity coefficient. To finish we show the existence of global weak solution in dimension one for a specific type of capillary coefficients with large initial data in the energy space.

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arXiv2008

Weak solution for compressible fluid models of Korteweg type

This work is devoted to proving existence of global weak solutions for a general isothermal model of capillary fluids derived by J.E Dunn and J.Serrin (1985) \cite{3DS}, which can be used as a phase transition model. We distinguish two cases when the dimension N=2 and N=1, in the first case we need that $\frac{1}ρ\in L^{\infty}$, when N=1 we get a weak solution in finite time in the energy space.

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