The purpose of this article is to introduce for dispersive partial differential equations with random initial data, the notion of well-posedness (in the Hadamard-probabilistic sense). We restrict the study to one of the simplest examples of such equations: the periodic cubic semi-linear wave equation. Our contributions in this work are twofold: first we break the algebraic rigidity involved in our previous works and allow much more general randomizations (general infinite product measures v.s. Gibbs...

We prove an existence result for equations of the form where the coefficients $a_{ij}(x,s)$ satisfy the usual ellipticity conditions and hypotheses weaker than the continuity with respect to the variable $s$. Moreover, we give a counterexample which shows that the problem above may have no solution if the coefficients $a_{ij}(x,s)$ are supposed only Borel functions

We report on new results concerning the global well-posedness, dissipativity and attractors for the quintic wave equations in bounded domains of ${ℝ}^3$ with damping terms of the form ${(-{\Delta }_x)}^{\theta }{\partial }_{t}u$, where $\theta =0$ or $\theta =1/2$. The main ingredient of the work is the hidden extra regularity of solutions that does not follow from energy estimates. Due to the extra regularity of solutions existence of a smooth attractor then follows from the smoothing property when $\theta =1/2$. For $\theta =0$ existence of smooth attractors is more complicated and follows...

In this paper, we consider a 2nd order semilinear parabolic initial boundary value problem (IBVP) on a bounded domain $\Omega \subset {ℝ}^N$, with nonstandard boundary conditions (BCs). More precisely, at some part of the boundary we impose a Neumann BC containing an unknown additive space-constant $\alpha \left(t\right)$, accompanied with a nonlocal (integral) Dirichlet side condition. We design a numerical scheme for the approximation of a weak solution to the IBVP and derive error estimates for the approximation of the solution $u$ and...

We present a new class of averaging lemmas directly motivated by the question of regularity for different nonlinear equations or variational problems which admit a kinetic formulation. In particular they improve the known regularity for systems like $\gamma =3$ in isentropic gas dynamics or in some variational problems arising in thin micromagnetic films. They also allow to obtain directly the best known regularizing effect in multidimensional scalar conservation laws. The new ingredient here is to use velocity...

We present a new class of averaging lemmas directly motivated by the question of regularity for different nonlinear equations or variational problems which admit a kinetic formulation. In particular they improve the known regularity for systems like γ = 3 in isentropic gas dynamics or in some variational problems arising in thin micromagnetic films. They also allow to obtain directly the best known regularizing effect in multidimensional scalar conservation laws. The new ingredient here is to...

The main object of this paper is to study the regularity with respect to the parameter h of solutions of the problem $du/dt+A_h\left(t\right)u\left(t\right)=f_h\left(t\right)$, $u\left(0\right)=u_h^0$. The continuity of u with respect to both h and t has been considered in [6].

A class of parabolic initial-boundary value problems is considered, where admissible coefficients are given in certain intervals. We are looking for maximal values of the solution with respect to the set of admissible coefficients. We give the abstract general scheme, proposing how to solve such problems with uncertain data. We formulate a general maximization problem and prove its solvability, provided all fundamental assumptions are fulfilled. We apply the theory to certain Fourier obstacle type...