arXiv subjects

Raphaël Danchin

Publications and source records attributed to Raphaël Danchin.

At least 37 records · Page 2Linked to original sources

arXiv2020

On the well-posedness of the Hall-magnetohydrodynamics system in critical spaces

We investigate the existence and uniqueness issues of the 3D incompressible Hall-magnetohydrodynamic system supplemented with initial velocity $u_0$ and magnetic field $B_0$ in critical regularity spaces.In the case where $u_0,$ $B_0$ and the current $J_0:=\nabla\times B_0$ belong to the homogeneous Besov space $\dot B^{\frac 3p-1}_{p,1},$ $\:1\leq p<\infty,$ and are small enough, we establish a global result and the conservation of higher regularity.If the viscosity is equal to the magnetic resistivity, then we obtain the global well-posedness provided $u_0,$ $B_0$ and $J_0$ are small enough in the \emph{larger} Besov space $\dot B^{\frac12}_{2,r},$ $r\geq1.$If $r=1,$ then we also establish the local existence for large data, and exhibit continuation criteria for solutions with critical regularity. Our results rely on an extended formulation of the Hall-MHD system, that has some similarities with the incompressibleNavier-Stokes equations.

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arXiv2020

Lorentz spaces in action on pressureless systems arising from models of collective behavior

We are concerned with global-in-time existence and uniqueness results for models of pressureless gases that come up in the description of phenomena in astrophysics or collective behavior. The initial data are rough: in particular, the density is only bounded. Our results are based on interpolation and parabolic maximal regularity, where Lorentz spaces play a key role. We establish a novel maximal regularity estimate for parabolic systems in $L_{q,r}(0,T;L_p(Ω))$ spaces.

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arXiv2019

The Global Solvability Of The Hall-magnetohydrodynamics System In Critical Sobolev Spaces

We are concerned with the 3D incompressible Hall-magnetohydro-dynamic system (Hall-MHD). Our first aim is to provide the reader with an elementary proof of a global well-posedness result for small data with critical Sobolev regularity, in the spirit of Fujita-Kato's theorem [10] for the Navier-Stokes equations. Next, we investigate the long-time asymptotics of global solutions of the Hall-MHD system that are in the Fujita-Kato regularity class. A weak-strong uniqueness statement is also proven. Finally, we consider the so-called 2 1/2 D flows for the Hall-MHD system, and prove the global existence of strong solutions, assuming only that the initial magnetic field is small.

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arXiv2019

The global existence issue for the compressible Euler system with Poisson or Helmholtz couplings

We consider the Cauchy problem for the barotropic Euler system coupled to Helmholtz or Poisson equations, in the whole space. We assume that the initial density is small enough, and that the initial velocity is close to some reference vector field u0 such that the spectrum of Du0 is bounded away from zero. We prove the existence of a global unique solution with (fractional) Sobolev regularity, and algebraic time decay estimates. Our work extends the papers by D. Serre and M. Grassin [20, 21, 43] dedicated to the compressible Euler system without coupling and integer regularity exponents.

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arXiv2018

Regular solutions to the fractional Euler alignment system in the Besov spaces framework

We here construct (large) local and small global-in-time regular unique solutions to the fractional Euler alignment system in the whole space ${\mathbb R}^d$, in the case where the deviation of the initial density from a constant is sufficiently small. Our analysis strongly relies on the use of Besov spaces of the type $L^1(0,T;\dot B^s_{p,1})$, which allow to get time independent estimates for the density even though it satisfies a transport equation with no damping. Our choice of a functional setting is not optimal but aims at providing a transparent and accessible argumentation.

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arXiv2018

Gevrey analyticity and decay for the compressible Navier-Stokes system with capillarity

We are concerned with an isothermal model of viscous and capillary compressible fluids derived by J. E. Dunn and J. Serrin (1985), which can be used as a phase transition model. Compared with the classical compressible Navier-Stokes equations, there is a smoothing effect on the density that comes from the capillary terms. First, we prove that the global solutions with critical regularity that have been constructed in [11] by the second author and B. Desjardins (2001), are Gevrey analytic. Second, we extend that result to a more general critical L p framework. As a consequence, we obtain algebraic time-decay estimates in critical Besov spaces (and even exponential decay for the high frequencies) for any derivatives of the solution. Our approach is partly inspired by the work of Bae, Biswas \& Tadmor [2] dedicated to the classical incompressible Navier-Stokes equations, and requires our establishing new bilinear estimates (of independent interest) involving the Gevrey regularity for the product or composition of functions. To the best of our knowledge, this is the first work pointing out Gevrey analyticity for a model of compressible fluids.

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arXiv2018

A well-posedness result for viscous compressible fluids with only bounded density

We are concerned with the existence and uniqueness of solutions with only bounded density for the barotropic compressible Navier-Stokes equations. Assuming that the initial velocity has slightly sub-critical regularity and that the initial density is a small perturbation (in the $L^\infty$ norm) of a positive constant, we prove the existence of local-in-time solutions. In the case where the density takes two constant values across a smooth interface (or, more generally, has striated regularity with respect to some nondegenerate family of vector-fields), we get uniqueness. This latter result supplements the work by D. Hoff in [26] with a uniqueness statement, and is valid in any dimension $d\geq2$ and for general pressure laws.

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arXiv2017

From compressible to incompressible inhomogeneous flows in the case of large data

This paper is concerned with the mathematical derivation of the inhomoge-neous incompressible Navier-Stokes equations (INS) from the compressible Navier-Stokes equations (CNS) in the large volume viscosity limit. We first prove a result of large time existence of regular solutions for (CNS). Next, as a consequence, we establish that the solutions of (CNS) converge to those of (INS) when the volume viscosity tends to infinity. Analysis is performed in the two dimensional torus, for general initial data. In particular, we are able to handle large variations of density.

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arXiv2017

The incompressible navier-stokes equations in vacuum

We are concerned with the existence and uniqueness issue for the inhomogeneous incompressible Navier-Stokes equations supplemented with H^1 initial velocity and only bounded nonnegative density. In contrast with all the previous works on that topics, we do not require regularity or positive lower bound for the initial density, or compatibility conditions for the initial velocity, and still obtain unique solutions. Those solutions are global in the two-dimensional case for general data, and in the three-dimensional case if the velocity satisfies a suitable scaling invariant smallness condition. As a straightforward application, we provide a complete answer to Lions' question in [25], page 34, concerning the evolution of a drop of incompressible viscous fluid in the vacuum.

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arXiv2017

Optimal decay estimates in the critical $L^p$ framework for flows of compressible viscous and heat-conductive gases

The global existence issue in critical regularity spaces for the full Navier-Stokes equationssatisfied by compressible viscous and heat-conductive gases has been first addressed in \cite{D2}, then recently extended to the general $L^p$ framework in \cite{DH}.In the present work, we establish decay estimates for the global solutions constructed in \cite{DH}, under anadditional mild integrability assumption that is satisfied if the low frequencies of the initial data are in $L^{p/2}({\mathbb{R}}^d).$As a by-product we recover in dimension three the classical decay rate $t^{-\frac34}$ for $t\to+\infty$ that has been observed by A. Matsumura and T. Nishida in \cite{MN2} for solutions with high Sobolev regularity. Compared to a recent paper of us \cite{DX} dedicated to the barotropic case, not only we are able to treat the full system, but we also improve the decay rates for the high frequencies of the solution. We believe the correspondant decay exponents to be optimal.

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arXiv2016

On the persistence of Hölder regular patches of density for the inhomogeneous Navier-Stokes equations

In our recent work dedicated to the Boussinesq equations [Danchin and Zhang 2016], we established the persistence of solutions with piecewise constant temperature along interfaces with Hölder regularity. We here address the same problem for the inhomogeneous Navier-Stokes equations satisfied by a viscous incompressibleand inhomogeneous fluid. We establish that, indeed, in the slightly inhomogeneous case, patches of densities with $\mathcal{C}^{1, \varepsilon}$ regularity propagate for all time. As in [Danchin and Zhang 2016], our result follows from the conservation of Hölder regularityalong vector fields moving with the flow. The proof of that latter result is based on commutator estimates involving para-vector fields, and multiplier spaces. The overall analysis is more complicated than in [Danchin and Zhang 2016] however, since the coupling between the mass and velocity equations in the inhomogeneous Navier-Stokes equations is \emph{quasilinear} while it is linear for the Boussinesq equations.

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arXiv2016

Optimal time-decay estimates for the compressible navier-stokes equations in the critical l p framework

The global existence issue for the isentropic compressible Navier-Stokes equations in the critical regularity framework has been addressed in [7] more than fifteen years ago. However, whether (optimal) time-decay rates could be shown in general critical spaces and any dimension d $\ge$ 2 has remained an open question. Here we give a positive answer to that issue not only in the L 2 critical framework of [7] but also in the more general L p critical framework of [3, 6, 14]. More precisely, we show that under a mild additional decay assumption that is satisfied if the low frequencies of the initial data are in e.g. L p/2 (R d), the L p norm (the slightly stronger $\dot B^0_{p,1}$ norm in fact) of the critical global solutions decays like t --d(1 p -- 1 4) for t $\rightarrow$ +$\infty$, exactly as firstly observed by A. Matsumura and T. Nishida in [23] in the case p = 2 and d = 3, for solutions with high Sobolev regularity. Our method relies on refined time weighted inequalities in the Fourier space, and is likely to be effective for other hyperbolic/parabolic systems that are encountered in fluid mechanics or mathematical physics.

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arXiv2016

Global persistence of geometrical structures for the boussinesq equation with no diffusion

Here we investigate the so-called temperature patch problem for the incompressible Boussinesq system with partial viscosity, in the whole space $\mathbb{R}^N$ $(N \geq 2)$, where the initial temperature is the characteristic function of some simply connected domain with $C^{1, \varepsilon}$ H{ö}lder regularity. Although recent results in [1, 15] ensure that an initially $C^1$ patch persists through the evolution, whether higher regularity is preserved has remained an open question. In the present paper, we give a positive answer to that issue globally in time, in the 2-D case for large initial data and in the higher dimension case for small initial data.

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arXiv2016

Compressible Navier-Stokes system : large solutions and incompressible limit

Here we prove the existence of global in time regular solutions to the two-dimensional compressible Navier-Stokes equations supplemented with arbitrary large initial velocity $v\_0$ and almost constantdensity $\varrho\_0$, for large volume (bulk) viscosity. The result is generalized to the higher dimensional case under the additional assumption that the strong solution of the classical incompressible Navier-Stokes equations supplemented with the divergence-freeprojection of $v\_0,$ is global. The systems are examined in $R^d$ with $d \geq 2$, in the critical $\dot B^s\_{2,1}$ Besov spaces framework.

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arXiv2015

Diffusive limits for a barotropic model of radiative flow

Here we aim at justifying rigorously different types of physically relevant diffusive limits for radiative flows. For simplicity, we consider the barotropic situation, and adopt the so-called P1-approximation of the radiative transfer equation. In the critical functional framework, we establish the existence of global-in-time strong solutions corresponding to small enough data, and exhibit uniform estimates with respect to the coefficients of the system. Combining with standard compactness arguments, this enables us to justify rigorously the convergence of the solutions to the expected limit systems. Our results hold true in the whole space as well as in a periodic box in dimension n $\ge$ 2.

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arXiv2015

Fourier analysis methods for the compressible Navier-Stokes equations

In the last three decades, Fourier analysis methods have known a growing importance in the study of linear and nonlinear PDE's. In particular, techniques based on Littlewood-Paley decomposition and paradifferential calculus have proved to be very efficient for investigating evolutionary fluid mechanics equations in the whole space or in the torus. We here give an overview of results that we can get by Fourier analysis and paradifferential calculus, for the compressible Navier-Stokes equations. We focus on the Initial Value Problem in the case where the fluid domain is the whole space or the torus in dimension at least two, and also establish some asymptotic properties of global small solutions. The time decay estimates in the critical regularity framework that are stated at the end of the survey are new, to the best of our knowledge.

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arXiv2014

A Lagrangian approach for the compressible Navier-Stokes equations

Here we investigate the Cauchy problem for the barotropic Navier-Stokes equations in R^n, in the critical Besov spaces setting. We improve recent results as regards the uniqueness condition: initial velocities in critical Besov spaces with (not too) \emph{negative} indices generate a unique local solution. Apart from (critical) regularity, the initial density just has to be bounded away from 0 and to tend to some positive constant at infinity. Density-dependent viscosity coefficients may be considered. Using Lagrangian coordinates is the key to our statements 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, and Lipschitz continuity of the flow map (in Lagrangian coordinates) is established.

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arXiv2014

The incompressible limit in $L^p$ type critical spaces

This paper aims at justifying the low Mach number convergence to the incompressible Navier-Stokes equations for viscous compressible flows in the ill-prepared data case. The fluid domain is either the whole space, or the torus. A number of works have been dedicated to this classical issue, all of them being, to our knowledge, related to $L^2$ spaces and to energy type arguments. In the present paper, we investigate the low Mach number convergence in the $L^p$ type critical regularity framework. More precisely, in the barotropic case, the divergence-free part of the initial velocity field just has to be bounded in the critical Besov space $\dot B^{d/p-1}_{p,r}\cap\dot B^{-1}_{\infty,1}$ for some suitable $(p,r)\in[2,4]\times[1,+\infty].$ We still require $L^2$ type bounds on the low frequencies of the potential part of the velocity and on the density, though, an assumption which seems to be unavoidable in the ill-prepared data framework, because of acoustic waves. In the last part of the paper, our results are extended to the full Navier-Stokes system for heat conducting fluids.

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