In this paper, we review several recent results dealing with elliptic equations with non local diffusion. More precisely, we investigate several problems involving the fractional laplacian. Finally, we present a conformally covariant operator and the associated singular and regular Yamabe problem.

Nous rappelons tout d’abord l’approche maintenant classique de renormalisation pour établir l’unicité des solutions faibles des équations de transport linéaires, en mentionnant les résultats récents qui s’y rattachent. Ensuite, nous montrons comment l’approche alternative introduite par Crippa et DeLellis estimant directement le flot lagrangien permet d’obtenir des résultats nouveaux. Nous établissons l’existence et l’unicité du flot associé à une équation de transport dont le coefficient a un gradient...

This paper deals with a mixed boundary-value problem of Ventcel type in two variables. The peculiarity of the Ventcel problem lies in the fact that one of the boundary conditions involves second order differentiation along the boundary. Under suitable assumptions on the data, we first give the definition of a weak solution, and then we prove that the problem is uniquely solvable. We also consider a particular case arising in real-world applications and discuss the resulting model.

In this article we are interested in the existence and uniqueness of solutions for the Dirichlet problem associated with the degenerate nonlinear elliptic equations $$\begin{array}{cc}\hfill \Delta \left(v\left(x\right){\left|\Delta u\right|}^{p-2}\Delta u\right)& -\sum _{j=1}^n{D}_j\left[\omega \left(x\right){𝒜}_j(x,u,\nabla u)\right]\hfill \\ \hfill =& f_0\left(x\right)-\sum _{j=1}^n{D}_j{f}_j\left(x\right),in\Omega \hfill \end{array}$$ in the setting of the weighted Sobolev spaces.

We prove the existence and uniqueness of a renormalized solution for a class of nonlinear parabolic equations with no growth assumption on the nonlinearities.

We consider the problem of influencing the motion of an electrically conducting fluid with an applied steady magnetic field. Since the flow is originating from buoyancy, heat transfer has to be included in the model. The stationary system of magnetohydrodynamics is considered, and an approximation of Boussinesq type is used to describe the buoyancy. The heat sources given by the dissipation of current and the viscous friction are not neglected in the fluid. The vessel containing the fluid is embedded...

We study a general class of nonlinear elliptic problems associated with the differential inclusion $\beta \left(u\right)-div\left(a\right(x,Du)+F(u\left)\right)\ni f$ in Ω where $f\in L^{\infty }\left(\Omega \right)$. The vector field a(·,·) is a Carathéodory function. Using truncation techniques and the generalized monotonicity method in function spaces we prove existence of renormalized solutions for general $L^{\infty }$-data.

In this article, we prove the existence of entropy solutions for the Dirichlet problem $$\left(P\right)\left\{\begin{array}{cc}-\mathrm{div}\left[\omega \right(x\left)𝒜\right(x,u,\nabla u\left)\right]=f\left(x\right)-\mathrm{div}\left(G\right),\hfill & \text{in}\Omega \hfill \\ u\left(x\right)=0,\hfill & \text{on}\partial \Omega \hfill \end{array}\right.$$ where $\Omega$ is a bounded open set of ${}^N$, $N\ge 2$, $f\in L^1\left(\Omega \right)$ and $G/\omega \in {\left[L^{p^{\text{'}}}(\Omega ,\omega )\right]}^N$.

We study the problem ∂b(x,u)/∂t - div(a(x,t,u,Du)) + H(x,t,u,Du) = μ in Q = Ω×(0,T), ${b(x,u)|}_{t=0}=b(x,u₀)$ in Ω, u = 0 in ∂Ω × (0,T). The main contribution of our work is to prove the existence of a renormalized solution without the sign condition or the coercivity condition on H(x,t,u,Du). The critical growth condition on H is only with respect to Du and not with respect to u. The datum μ is assumed to be in $L¹\left(Q\right)+L^{p^{\text{'}}}(0,T;W^{-1,p^{\text{'}}}\left(\Omega \right))$ and b(x,u₀) ∈ L¹(Ω).

We show existence of solutions to two types of generalized anisotropic Cahn-Hilliard problems: In the first case, we assume the mobility to be dependent on the concentration and its gradient, where the system is supplied with dynamic boundary conditions. In the second case, we deal with classical no-flux boundary conditions where the mobility depends on concentration $u$, gradient of concentration $\nabla u$ and the chemical potential $\Delta u-s^{\text{'}}\left(u\right)$. The existence is shown using a newly developed generalization of gradient...

This paper has two objectives. First, we prove the existence of solutions to the general advection-diffusion equation subject to a reasonably smooth initial condition. We investigate the behavior of the solution of these problems for large values of time. Secondly, a numerical scheme using the Sinc-Galerkin method is developed to approximate the solution of a simple model of turbulence, which is a special case of the advection-diffusion equation, known as Burgers' equation. The approximate solution...