arXiv subjects

Raphaël Danchin

Publications and source records attributed to Raphaël Danchin.

47 records · Page 3Linked to original sources

arXiv2014

On the well-posedness of the full compressible Navier-Stokes system in critical Besov spaces

We are concerned with the Cauchy problem of the full compressible Navier-Stokes equations satisfied by viscous and heat conducting fluids in $\mathbb{R}^n.$ We focus on the so-called critical Besov regularity framework. In this setting, it is natural to consider initial densities $ρ_0,$ velocity fields $u_0$ and temperatures $θ_0$ with $a_0:=ρ_0-1\in\dot B^{\frac np}_{p,1},$ $u_0\in\dot B^{\frac np-1}_{p,1}$ and $θ_0\in\dot B^{\frac np-2}_{p,1}.$ After recasting the whole system in Lagrangian coordinates, and working with the \emph{total energy along the flow} rather than with the temperature, we discover that the system may be solved by means of Banach fixed point theorem in a critical functional framework whenever the space dimension is $n\geq2,$ and $1<p<2n.$ Back to Eulerian coordinates, this allows to improve the range of $p$'s for which the system is locally well-posed, compared to Danchin, Comm. Partial Differential Equations 26 (2001).

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arXiv2013

The well-posedness issue for the density-dependent Euler equations in endpoint Besov spaces

This work is the continuation of the recent paper \cite{D2} devoted to the density-dependent incompressible Euler equations. Here we concentrate on the well-posedness issue in Besov spaces of type $B^s_{\infty,r}$ embedded in the set of Lipschitz continuous functions, a functional framework which contains the particular case of Hölder spaces and of the endpoint Besov space $B^1_{\infty,1}.$ For such data and under the nonvacuum assumption, we establish the local well-posedness and a continuation criterion in the spirit of that of Beale, Kato and Majda in \cite{BKM}. In the last part of the paper, we give lower bounds for the lifespan of a solution. In dimension two, we point out that the lifespan tends to infinity when the initial density tends to be a constant. This is, to our knowledge, the first result of this kind for the density-dependent incompressible Euler equations.

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arXiv2013

The Oberbeck-Boussinesq approximation in critical spaces

In this paper we study the validity of the so-called Oberbeck-Boussinesq approximation for compressible viscous perfect gases in the whole three-dimensional space. Both the cases of uids with positive heat conductivity and zero conductivity are considered. For small perturbations of a constant equilibrium, we establish the global existence of unique strong solutions in a critical regularity functional framework. Next, taking advantage of Strichartz estimates for the associated system of acoustic waves, and of uniform estimates with respect to the Mach number, we obtain all-time convergence to the Boussinesq system with a explicit decay rate.

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arXiv2012

New maximal regularity results for the heat equation in exterior domains, and applications

This paper is dedicated to the proof of new maximal regularity results involving Besov spaces for the heat equation in the half-space or in bounded or exterior domains of R^n. We strive for time independent a priori estimates in regularity spaces of type L^1(0,T;X) where X stands for some homogeneous Besov space. In the case of bounded domains, the results that we get are similar to those of the whole space or of the half-space. For exterior domains, we need to use mixed Besov norms in order to get a control on the low frequencies. Those estimates are crucial for proving global-in-time results for nonlinear heat equations in a critical functional framework.

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arXiv2012

Incompressible flows with piecewise constant density

We investigate the incompressible Navier-Stokes equations with variable density. The aim is to prove existence and uniqueness results in the case of discontinuous ini- tial density. In dimension n = 2, 3, assuming only that the initial density is bounded and bounded away from zero, and that the initial velocity is smooth enough, we get the local-in-time existence of unique solutions. Uniqueness holds in any dimension and for a wider class of velocity fields. Let us emphasize that all those results are true for piecewise constant densities with arbitrarily large jumps. Global results are established in dimension two if the density is close enough to a positive constant, and in n-dimension if, in addition, the initial velocity is small. The Lagrangian formula- tion for describing the flow plays a key role in the analysis that is proposed in the present paper.

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arXiv2012

On the well-posedness of the full low-Mach number limit system in general critical Besov spaces

This work is devoted to the well-posedness issue for the low-Mach number limit system obtained from the full compressible Navier-Stokes system, in the whole space. In the case where the initial temperature (or density) is close to a positive constant, we establish the local existence and uniqueness of a solution in critical homogeneous Besov spaces. If, in addition, the initial velocity is small then we show that the solution exists for all positive time. In the fully nonhomogeneous case, we establish the local well-posedness in nonhomogeneous Besov spaces (still with critical regularity) for arbitrarily large data with positive initial temperature. Our analysis strongly relies on the use of a modified divergence-free velocity which allows to reduce the system to a nonlinear coupling between a parabolic equation and some evolutionary Stokes system. The Lebesgue exponents of the Besov spaces for the temperature and the (modified) velocity, need not be the same. This enables us to consider initial data in Besov spaces with a negative index of regularity.

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arXiv2012

Remarks on the lifespan of the solutions to some models of incompressible fluid mechanics

We give lower bounds for the lifespan of a solution to the inviscid Boussinesq system. In dimension two, we point out that it tends to infinity when the initial (relative) temperature tends to zero. This is, to the best of our knowledge, the first result of this kind for the inviscid Boussinesq system. In passing, we provide continuation criteria (of independent interest) in the $N$-dimensional case. In the second part of the paper, our method is adapted to handle the axisymmetric incompressible Euler equations with swirl.

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arXiv2011

Madelung, Gross-Pitaevskii and Korteweg

This paper surveys various aspects of the hydrodynamic formulation of the nonlinear Schrodinger equation obtained via the Madelung transform in connexion to models of quantum hydrodynamics and to compressible fluids of the Korteweg type.

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arXiv2011

A Lagrangian approach for the incompressible Navier-Stokes equations with variable density

Here we investigate the Cauchy problem for the inhomogeneous Navier-Stokes equations in the whole $n$-dimensional space. Under some smallness assumption on the data, we show the existence of global-in-time unique solutions in a critical functional framework. The initial density is required to belong to the multiplier space of $\dot B^{n/p-1}_{p,1}(\R^n)$. In particular, piecewise constant initial densities are admissible data \emph{provided the jump at the interface is small enough}, and generate global unique solutions with piecewise constant densities. Using Lagrangian coordinates is the key to our results as it enables us to solve the system by means of the basic contraction mapping theorem. As a consequence, conditions for uniqueness are the same as for existence.

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arXiv2009

On the well-posedness of the incompressible density-dependent Euler equations in the $L^p$ framework

The present paper is devoted to the study of the well-posedness issue for the density-dependent Euler equations in the whole space. We establish local-in-time results for the Cauchy problem pertaining to data in the Besov spaces embedded in the set of Lipschitz functions, including the borderline case $B^{\frac Np+1}_{p,1}(\R^N).$ A continuation criterion in the spirit of the celebrated one by Beale-Kato-Majda for the classical Euler equations, is also proved. In contrast with the previous work dedicated to this system in the whole space, our approach is not restricted to the $L^2$ framework or to small perturbations of a constant density state: we just need the density to be bounded away from zero. The key to that improvement is a new a priori estimate in Besov spaces for an elliptic equation with nonconstant coefficients.

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arXiv2008

Global Existence Results for the Anisotropic Boussinesq System in Dimension Two

We study the global existence issue for the two-dimensional Boussinesq system with horizontal viscosity in only one equation. We first examine the case where the Navier-Stokes equation with no vertical viscosity is coupled with a transport equation. Second, we consider a coupling between the classical two-dimensional incompressible Euler equation and a transportdiffusion equation with diffusion in the horizontal direction only. For the both systems and for arbitrarily large data, we construct global weak solutions `a la Leray. Next, we state global wellposedness results for more regular data. Our results strongly rely on the fact that the diffusion occurs in a direction perpendicular to the buoyancy force.

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