Almost global well-posedness of Kirchhoff equation with Gevrey data

The aim of this note is to present the almost global well-posedness result for the Cauchy problem for the Kirchhoff equation with large data in Gevrey spaces. We also briefly discuss the corresponding results in bounded and in exterior domains.     

Global well-posedness for the 2-D nonhomogeneous incompressible MHD equations with large initial data

In this paper, we consider the 2-D nonhomogeneous incompressible magnetohydrodynamic equations with variable viscosity and variable conductivity. We obtain the global existence of solutions for this system with initial data in the scaling invariant Besov spaces and without size restriction for the initial velocity and magnetic field.     

Monte-Carlo approximations for 2d Navier-Stokes equations with measure initial data

We are interested in proving Monte-Carlo approximations for 2d Navier-Stokes equations with initial data u ₀ belonging to the Lorentz space L ^2,∞ and such that curl u ₀ is a finite measure. Giga, Miyakawaand Osada [7] proved that a solution u exists and that u=K* curl u, where K is the Biot-Savartkernel and v = curl u is solution of a nonlinear equation in dimension one, called the vortex equation. In this paper, we approximate a solution v of this vortex equationby a stochastic interacting particlesystem and deduce a Monte-Carlo approximation for a solution of the Navier-Stokesequation. That gives in this case a pathwise proofof the vortex algorithm introducedby Chorin and consequently generalizes the works ofMarchioro-Pulvirenti [12] and Méléardv [15] obtained in the case of a vortex equation with bounded density initial data. Received: 6 October 1999 / Revised version: 15 September 2000 / Published online: 9 October 2001     

Nonlinear elliptic equations with measure data

In this paper we prove the existence of solutions of nonlinear equations of the type-div(a(x, u, Du)+H(x, u, Du)=f, wherea andH are Caratheodory functions andf is a bounded Radon measure. We remark that the operator can be not coercive. We give also some regularity results.     

Aggregation equations with gradient potential as Radon measure and initial data in Besov-Morrey spaces

In this work, we present conditions to obtain a global-in-time existence of solutions to a class of nonlinear viscous transport equations describing aggregation phenomena in biology with sufficiently small initial data in Besov-Morrey spaces and gradient potential as a Radon measure. We also study the self-similarity and asymptotic stability of solutions at large times.     

Almost optimal local well-posedness for the (3+1)-dimensional Maxwell-Klein-Gordon equations

We prove that the evolution problem for the Maxwell-Klein-Gordon system is locally well posed when the initial data belong to the Sobolev space for any 0$">. This is in spite of a complete failure of the standard model equations in the range . The device that enables us to obtain inductive estimates is a new null structure which involves cancellations between the elliptic and hyperbolic terms in the full equations.

     

Almost everywhere summability of multiple Walsh-Fourier series

We prove that for the N-dimensional Walsh-Fourier series the maximal operator of the Marcinkiewicz means is of weak type (1,1). Moreover, the Marcinkiewicz means σ_nf of the function f∈L¹ converge a.e. to f as n→∞.     

Nonlinear degenerate parabolic equations with measure data

In this paper, we study the existence of solution to nonlinear degenerate parabolic equations with measure data.     

Doubly nonlinear parabolic equations with measure data

The existence of solutions to the initial boundary value problem for the equation $$u_{t}-{\rm div}(u^m|Du|^{p-2}Du)=\lambda|Du^q|^{l}+u^{\alpha},$$ with zero-Dirichlit boundary condition and Radon measure as initial condition is studied, where m > 0, p > 1, λ, q, l, and α in various situations.     

Almost everywhere convergence of Fourier integrals revisited

A condition of proved worth guarantees almost everywhere convergence of Fourier integrals of functions from an essentially wider class than known earlier.     

Almost everywhere summability of multiple Fourier integrals

We obtain sufficient conditions for the Riesz means of spectral expansions converge to the function to be expanded.     

Degenerate parabolic equations with initial data measures

We address the problem of existence of solutions to degenerate (and nondegenerate) parabolic equations under optimal assumptions on the initial data, which are assumed to be measures. The requirements imposed on the initial data are connected both with the degeneracy of the principal part of the equation, and with the form of the nonlinear forcing term. The latter depends on the space gradient of a power of the solution. Applications to related problems are also outlined.

     

Almost everywhere convergence of inverse Fourier transforms

We show that if , then the inverse Fourier transform of converges almost everywhere. Here the partial integrals in the Fourier inversion formula come from dilates of a closed bounded neighbourhood of the origin which is star shaped with respect to 0. Our proof is based on a simple application of the Rademacher-Menshov Theorem. In the special case of spherical partial integrals, the theorem was proved by Carbery and Soria. We obtain some partial results when and . We also consider sequential convergence for general elements of .