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

Almost classical solutions to the total variation flow

The paper examines one-dimensional total variation flow equation with Dirichlet boundary conditions. Thanks to a new concept of "almost classical" solutions we are able to determine evolution of facets -- flat regions of solutions. A key element of our approach is the natural regularity determined by nonlinear elliptic operator, for which $x^2$ is an irregular function. Such a point of view allows us to construct solutions. We apply this idea to implement our approach to numerical simulations for typical initial data. Due to the nature of Dirichlet data any monotone function is an equilibrium. We prove that each solution reaches such steady state in a finite time.Comment: 3 figure

Stability of 2D incompressible flows in R 3

Abstract. We investigate the global in time stability of regular solutions with large velocity vectors to the evolutionary Navier-Stokes equation in R 3. The class of stable flows contains all two dimensional weak solutions. The only assumption which is required is smallness of the L2norm of initial perturbation or its derivative with respect to the ‘z’-coordinate in the same norm. The magnitude of the rest of the norm of initial datum is not restricted. MSC: 35Q30, 76D05. Key words: global in time solutions, large data, stability, the Navier-Stokes equations

On lifespan of solutions to the Einstein equations

We investigate the issue of existence of maximal solutions to the vacuum Einstein solutions for asymptotically flat spacetime. Solutions are established globally in time outside a domain of influence of a suitable large compact set, where singularities can appear. Our approach shows existence of metric coefficients which obey the following behavior: $g_{\alpha\beta}=\eta_{\alpha\beta}+O(r^{-\delta})$ for a small fixed $\delta > 0$ at infinity (where $\eta_{\alpha\beta}$ is the Minkowski metric). The system is studied in the harmonic (wavelike) gauge

Problème de Stokes et système de Navier-Stokes incompressible à densité variable dans le demi-espace

International audienceWe study the incompressible Navier-Stokes system with variable density in the half-space. We consider initial data with critical regularity. We establish that if the initial density is close to a strictly positive constant in ${L^{\infty}\cap\dot W^{1,n}}$, and if the initial velocity is small with respect to the viscosity in the homogeneous Besov space $\dot B^0_{n,1}$, then the Navier-Stokes system has a unique global solution. The proof is based on some new maximal regularity estimates for solutions of the Stokes system in the half-spac

Divergence

International audienceThis note is dedicated to a few questions related to the divergence equation which have been motivated by recent studies concerning the Neumann problem for the Laplace equation or the (evolutionary) Stokes system in domains. For simplicity, we focus on the classical Sobolev spaces framework in bounded domains, but our results have natural and simple extensions to the Besov spaces framework in more general domains

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