Solvability of the Dirichlet problem for elliptic equations in weighted Sobolev spaces on unbounded domains.
Some boundary value problems for systems of linear partial differential equations of hyperbolic type.
Some linear parabolic system in Besov spaces
We study the solvability in anisotropic Besov spaces , σ ∈ ℝ₊, p,q ∈ (1,∞) of an initial-boundary value problem for the linear parabolic system which arises in the study of the compressible Navier-Stokes system with boundary slip conditions. The proof of existence of a unique solution in is divided into three steps: 1° First the existence of solutions to the problem with vanishing initial conditions is proved by applying the Paley-Littlewood decomposition and some ideas of Triebel. All considerations...
Some new regularity criteria for the Navier-Stokes equations containing gradient of the velocity
We study the nonstationary Navier-Stokes equations in the entire three-dimensional space and give some criteria on certain components of gradient of the velocity which ensure its global-in-time smoothness.
Some properties of solutions for a class of metaparabolic equations.
Some properties of solutions for the generalized thin film equation in one space dimension.
Some regularity results for quasilinear parabolic systems
Some regularity theorems in riemannian geometry
Some regularizing methods for transport equations and the regularity of solutions to scalar conservation laws
We study several regularizing methods, stationary phase or averaging lemmas for instance. Depending on the regularity assumptions that are made, we show that they can either be derived one from the other or that they lead to different results. Those are applied to Scalar Conservation Laws to precise and better explain the regularity of their solutions.
Some remarks on the Navier-Stokes equations with regularity in one direction
We review the developments of the regularity criteria for the Navier-Stokes equations, and make some further improvements.
Some remarks on the regularity of weak solutions of degenerate elliptic systems.
Some results of Gevrey and analytic regularity for semilinear weakly hyperbolic equations of Oleinik type
Spectral analysis of strongly propagative systems.
Stability results for Harnack inequalities
We develop new techniques for proving uniform elliptic and parabolic Harnack inequalities on weighted Riemannian manifolds. In particular, we prove the stability of the Harnack inequalities under certain non-uniform changes of the weight. We also prove necessary and sufficient conditions for the Harnack inequalities to hold on complete non-compact manifolds having non-negative Ricci curvature outside a compact set and a finite first Betti number or just having asymptotically...
Stationary p-harmonic maps into spheres
Stochastic calculus and degenerate boundary value problems
Consider the boundary value problem (L.P): in , on where is written as , and is a general Venttsel’s condition (including the oblique derivative condition). We prove existence, uniqueness and smoothness of the solution of (L.P) under the Hörmander’s condition on the Lie brackets of the vector fields (), for regular open sets with a non-characteristic boundary.Our study lies on the stochastic representation of and uses the stochastic calculus of variations for the -diffusion process...
Stochastic flow for SDEs with jumps and irregular drift term
We consider non-degenerate SDEs with a β-Hölder continuous and bounded drift term and driven by a Lévy noise L which is of α-stable type. If β > 1 - α/2 and α ∈ [1,2), we show pathwise uniqueness and existence of a stochastic flow. We follow the approach of [Priola, Osaka J. Math. 2012] improving the assumptions on the noise L. In our previous paper L was assumed to be non-degenerate, α-stable and symmetric. Here we can also recover relativistic and truncated stable processes and some classes...
Strichartz inequalities for the Schrödinger equation with the full Laplacian on the Heisenberg group
We prove Strichartz inequalities for the solution of the Schrödinger equation related to the full Laplacian on the Heisenberg group. A key point consists in estimating the decay in time of the norm of the free solution; this requires a careful analysis due also to the non-homogeneous nature of the full Laplacian.
Strict positivity of the solution to a 2-dimensional spatially homogeneous Boltzmann equation without cutoff
Structure of entropy solutions to the eikonal equation