M. Milla Miranda

Weak solutions for a nonlinear dispersive equation

Abstract In this paper we study the existence, uniqueness, and regularity of the solutions for the Cauchy problem for the evolution equation u t + (f (u)) x − u xxt = g(x, t), ( ∗ ) where u = u(x, t), x is in (0, 1), 0 ⩽ t ⩽ T, T is an arbitrary positive real number,f(s)ϵC1 R , and g(x, t)ϵ L∞(0, T; L2(0, 1)). We prove the existence and uniqueness of the weak solutions for (∗) using the Galerkin method and a compactness argument such as that of J. L. Lions. We obtain regular solutions using eigenfunctions of the one-dimensional Laplace operator as a basis in the Galerkin method.

Weak solutions for a system of nonlinear Klein-Gordon equations

SummaryWe prove the existence and uniqueness of weak solutions of the mixed problem for a class of systems of nonlinear Klein-Gordon equations. Uniqueness is proved when the spatial dimension is either n=1, 2or 3.

Existence and boundary stabilization of solutions for the kirchhoff equation

This paper is concerned with the existence of local and global solutions of an initial-homogeneous boundary value problem for the Kirchhoff equation where is an open bounded set of Rn The boundary stability is also obtained. The fixed point method, Galerkinapproximations and energy functionals are used in the approach.

On Second-Order Differential Equations with Nonsmooth Second Member

In an abstract framework, we consider the following initial value problem: u′′

On the Navier–Stokes equations with variable viscosity in a noncylindrical domain

In this article, we study the existence of weak solutions when n≤ 4 of the mixed problem for the Navier–Stokes equations defined in a noncylindrical domain . We consider that the viscosity depends on the velocity of the fluid and is the image of a bounded cylinder Q of . The uniqueness of solutions for n≤ 3 is also analyzed.