We consider a stochastic system of $N$ particles, usually called vortices in that setting, approximating the 2D Navier-Stokes equation written in vorticity. Assuming that the initial distribution of the position and circulation of the vortices has finite (partial) entropy and a finite moment of positive order, we show that the empirical measure of the particle system converges in law to the unique (under suitable a priori estimates) solution of the 2D Navier-Stokes equation. We actually prove a slightly...

The propagation of weak discontinuities for quasilinear systems with coefficients functionally dependent on the solution is studied. We demonstrate that, similarly to the case of usual quasilinear systems, the transport equation for the intensity of weak discontinuity is quadratic in this intensity. However, the contribution from the (nonlocal) functional dependence appears to be in principle linear in the jump intensity (with some exceptions). For illustration, several examples, including two hyperbolic...

Let ${\ell }_j:=-d²/dx²+k²q_j\left(x\right)$, k = const > 0, j = 1,2, $0<essinf{q}_j\left(x\right)\le esssup{q}_j\left(x\right)<\infty$. Suppose that (*) ${\int }_0^{1}p\left(x\right)u₁(x,k)u₂(x,k)dx=0$ for all k > 0, where p is an arbitrary fixed bounded piecewise-analytic function on [0,1], which changes sign finitely many times, and $u_j$ solves the problem ${\ell }_j{u}_j=0$, 0 ≤ x ≤ 1, $u_j^{\text{'}}(0,k)=0$, $u_j(0,k)=1$. It is proved that (*) implies p = 0. This result is applied to an inverse problem for a heat equation.

In this paper new methods for solving elliptic variational inequalities with weakly coercive operators are considered. The use of the iterative prox-regularization coupled with a successive discretization of the variational inequality by means of a finite element method ensures well-posedness of the auxiliary problems and strong convergence of their approximate solutions to a solution of the original problem. In particular, regularization on the kernel of the differential operator and regularization...