Using as a main tool the time-regularizing convolution operator introduced by R. Landes, we obtain regularity results for entropy solutions of a class of parabolic equations with irregular data. The results are obtained in a very general setting and include known previous results.

In this note we give some summability results for entropy solutions of the nonlinear parabolic equation ut - div ap (x, ∇u) = f in ] 0,T [xΩ with initial datum in L1(Ω) and assuming Dirichlet's boundary condition, where ap(.,.) is a Carathéodory function satisfying the classical Leray-Lions hypotheses, f ∈ L1 (]0,T[xΩ) and Ω is a domain in RN. We find spaces of type Lr(0,T;Mq(Ω)) containing the entropy solution and its gradient. We also include some summability results when f = 0 and the p-Laplacian...

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...

We study regularity properties of the free boundary for the thin one-phase problem which consists of minimizing the energy functional $E(u,\Omega )={\int }_{\Omega }{\left|\nabla u\right|}^{2}dX+{\mathscr{H}}^n(\{u>0\}\cap \{x_{n+1}=0\}),\Omega \subset {ℝ}^{n+1},$ among all functions $u\ge 0$ which are fixed on $\partial \Omega$.

We prove boundedness and continuity for solutions to the Dirichlet problem for the equation $$-\mathrm{div}\left(a(x,\nabla u)\right)=h(x,u)+\mu ,\text{in}\Omega \subset {ℝ}^N,$$ where the left-hand side is a Leray-Lions operator from $W_0^{1,p}\left(\Omega \right)$ into $W^{-1,p^{\text{'}}}\left(\Omega \right)$ with $1<p<N$, $h(x,s)$ is a Carathéodory function which grows like ${\left|s\right|}^{p-1}$ and $\mu$ is a finite Radon measure. We prove that renormalized solutions, though not globally bounded, are Hölder-continuous far from the support of $\mu$.

We consider an initial-boundary problem for a sixth order nonlinear parabolic equation, which arises in oil-water-surfactant mixtures. Using Schauder type estimates and Campanato spaces, we prove the global existence of classical solutions for the problem in two space dimensions.

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].

We study a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. The precise model is $u_t=\nabla ·\left(u\nabla {(-\Delta )}^{-s}u\right),0<s<1$. The problem is posed in $\{x\in {ℝ}^n,t\in ℝ\}$ with nonnegative initial data $u(x,0)$ that are integrable and decay at infinity. A previous paper has established the existence of mass-preserving, nonnegative weak solutions satisfying energy estimates and finite propagation. As main results we establish the boundedness and $C^{\alpha }$ regularity of such weak solutions. Finally, we extend the existence...