The Cauchy problem for first order system $L(t,x,{\partial }_t,{\partial }_x)$ is known to be well-posed in $L^2$ when it admits a microlocal symmetrizer $S(t,x,\xi )$ which is smooth in $\xi$ and Lipschitz continuous in $(t,x)$. This paper contains three main results. First we show that a Lipschitz smoothness globally in $(t,x,\xi )$ is sufficient. Second, we show that the existence of symmetrizers with a given smoothness is equivalent to the existence of full symmetrizers having the same smoothness. This notion was first introduced in [FL67]. This is the key point...

In this paper, we consider the global existence, uniqueness and $L^{\infty }$ estimates of weak solutions to quasilinear parabolic equation of $m$-Laplacian type $u_t-{\mathrm{div}\left(\right|\nabla u|}^{m-2}{\nabla u)=u|u|}^{\beta -1}{\int }_{\Omega }{\left|u\right|}^{\alpha }\mathrm{d}x$ in $\Omega \times (0,\infty )$ with zero Dirichlet boundary condition in $\partial \Omega$. Further, we obtain the $L^{\infty }$ estimate of the solution $u\left(t\right)$ and $\nabla u\left(t\right)$ for $t>0$ with the initial data $u_0\in L^q\left(\Omega \right)$$(q>...$

Sur ${ℂ}^n$ vu comme variété algébrique, soient $ℱ$ la transformation de Fourier pour les $𝒟$-modules, ${ℱ}^{+}$ la transformation de Fourier faisceautique de Brylinsky-Malgrange-Verdier, et $𝒮ol$ le foncteur “solutions”. On prouve alors que pour tout $𝒟$-module 1-spécialisable à l’infini $ℳ$, on a un isomorphisme $𝒮ol\left(ℱℳ\right)\equiv {ℱ}^{+}𝒮ol\left(ℳ\right)$. Le résultat a été conjecturé en 1988 par B. Malgrange, qui l’a prouvé pour $ℳ$ module de type fini sur l’algèbre de Weyl.

La variété caractéristique d’un système différentiel linéaire analytique possède les deux propriétés classiques suivantes :1. Indépendance de la filtration.2. Intégrabilité (i.e. stabilité par crochet de Poisson).On montre ici que la première propriété reste vraie hors de la section nulle pour les systèmes non linéaires. La seconde propriété reste vraie génériquement (ailleurs, la question reste ouverte).

The motion of a viscous incompressible fluid flow in bounded domains with a smooth boundary can be described by the nonlinear Navier-Stokes equations. This description corresponds to the so-called Eulerian approach. We develop a new approximation method for the Navier-Stokes equations in both the stationary and the non-stationary case by a suitable coupling of the Eulerian and the Lagrangian representation of the flow, where the latter is defined by the trajectories of the particles of the fluid....

The time-ordered exponential of a time-dependent matrix $𝖠\left(t\right)$ is defined as the function of $𝖠\left(t\right)$ that solves the first-order system of coupled linear differential equations with non-constant coefficients encoded in $𝖠\left(t\right)$. The authors have recently proposed the first Lanczos-like algorithm capable of evaluating this function. This algorithm relies on inverses of time-dependent functions with respect to a non-commutative convolution-like product, denoted by $*$. Yet, the existence of such inverses, crucial to...

Fundamental solutions to linear partial differential equations with constant coefficients are represented in the form of Laplace type integrals.

We prove the large time existence of solutions to the magnetohydrodynamics equations with slip boundary conditions in a cylindrical domain. Assuming smallness of the L₂-norms of the derivatives of the initial velocity and of the magnetic field with respect to the variable along the axis of the cylinder, we are able to obtain an estimate for the velocity and the magnetic field in $W{₂}^{2,1}$ without restriction on their magnitude. Then the existence follows from the Leray-Schauder fixed point theorem.