We prove almost optimal local well-posedness for the coupled Dirac–Klein–Gordon (DKG) system of equations in $1+3$ dimensions. The proof relies on the null structure of the system, combined with bilinear spacetime estimates of Klainerman–Machedon type. It has been known for some time that the Klein–Gordon part of the system has a null structure; here we uncover an additional null structure in the Dirac equation, which cannot be seen directly, but appears after a duality argument.

We consider an uncoupled, modular regularization algorithm for approximation of the Navier-Stokes equations. The method is: Step 1.1: Advance the NSE one time step, Step 1.1: Regularize to obtain the approximation at the new time level. Previous analysis of this approach has been for specific time stepping methods in Step 1.1 and simple stabilizations in Step 1.1. In this report we extend the mathematical support for uncoupled, modular stabilization to (i) the more complex and better performing...

In this paper, we make some observations on the work of Di Fazio concerning $W^{1,p}$ estimates, $1<p<\mathrm{\infty }$, for solutions of elliptic equations $\text{div}A\nabla u=\text{div}f$ , on a domain $\mathrm{\Omega }$ with Dirichlet data $0$ whenever $A\in VMO\left(\mathrm{\Omega }\right)$ and $f\in L^p\left(\mathrm{\Omega }\right)$. We weaken the assumptions allowing real and complex non-symmetric operators and $C^1$ boundary. We also consider the corresponding inhomogeneous Neumann problem for which we prove the similar result. The main tool is an appropriate representation for the Green (and Neumann) function on the upper half space. We propose...

We study the realization $A$ of the operator $\mathcal{A}=\frac{1}{2}\mathrm{△}-(DU,D\cdot )$ in $L^2\left(\mathrm{\Omega },\mu \right)$, where $\mathrm{\Omega }$ is a possibly unbounded convex open set in ${\mathbb{R}}^N$, $U$ is a convex unbounded function such that ${lim}_{x\to \partial \mathrm{\Omega },x\in \mathrm{\Omega }}U\left(x\right)=+\mathrm{\infty }$ and ${lim}_{\left|x\right|\to +\mathrm{\infty },x\in \mathrm{\Omega }}U\left(x\right)=+\mathrm{\infty }$, $DU\left(x\right)$ is the element with minimal norm in the subdifferential of $U$ at $x$, and $\mu \left(dx\right)=c\exp \left(-2U\left(x\right)\right)dx$ is a probability measure, infinitesimally invariant for $\mathcal{A}$. We show that $A$, with domain $D\left(A\right)=\left\{u\in H^2\left(\mathrm{\Omega },\mu \right):\left(DU,Du\right)\in L^2\left(\mathrm{\Omega },\mu \right)\right\}$ is a dissipative self-adjoint operator in $L^2\left(\mathrm{\Omega },\mu \right)$. Note that the functions in the domain of $A$ do not satisfy any particular boundary condition. Log-Sobolev and Poincaré inequalities allow...

We deal with a generalization of the Stokes system. Instead of the Laplace operator, we consider a general elliptic operator and a pressure gradient with small perturbations. We investigate the existence and uniqueness of a solution as well its regularity properties. Two types of regularity are provided. Aside from the classical Hilbert regularity, we also prove the Hölder regularity for coefficients in VMO space.

We first outline the procedure of averaging the incompressible Navier-Stokes equations when the flow is turbulent for various type of filters. We introduce the turbulence model called Bardina’s model, for which we are able to prove existence and uniqueness of a distributional solution. In order to reconstruct some of the flow frequencies that are underestimated by Bardina’s model, we next introduce the approximate deconvolution model (ADM). We prove existence and uniqueness of a “regular weak solution”...

I will start with a short review of the classical restriction theorem for the sphere and Strichartz estimates for the wave equation. I then plan to give a detailed presentation of their recent generalizations in the form of “bilinear estimates”. In addition to the $L^2$ theory, which is now quite well developed, I plan to discuss a more general point of view concerning the $L^p$ theory. By investigating simple examples I will derive necessary conditions for such estimates to be true. I also plan to discuss...