We study boundary regularity of weak solutions of the Navier-Stokes equations in the half-space in dimension . We prove that a weak solution which is locally in the class with near boundary is Hölder continuous up to the boundary. Our main tool is a point-wise estimate for the fundamental solution of the Stokes system, which is of independent interest.

In present paper we elaborated the numerical schemes of iterative methods for an approximate solution of weak singular integral equations with logarithmic Kernel. The equation is examined in a pair of spaces. The results obtained could be used for any pair of the functional spaces where the problem of finding the solution of weak singular integral equations is correctly formulated problem by Ti...

Using a weak convergence approach, we prove a LPD for the solution of 2D stochastic Navier Stokes equations when the viscosity converges to 0 and the noise intensity is multiplied by the square root of the viscosity. Unlike previous results on LDP for hydrodynamical models, the weak convergence is proven by tightness properties of the distribution of the solution in appropriate functional spaces.

Consider a nonlinear diffusion equation related to the p-Laplacian. Different from the usual evolutionary p-Laplacian equation, the equation is degenerate on the boundary due to the fact that the diffusion coefficient is dependent on the distance function. Not only the existence of the weak solution is established, but also the uniqueness of the weak solution is proved.

We consider a circulation system arising in turbulence modelling in fluid dynamics with unbounded eddy viscosities. Various notions of weak solution are considered and compared. We establish existence and regularity results. In particular we study the boundedness of weak solutions. We also establish an existence result for a classical solution.

We consider the equation modeling the compressible hydrodynamic flow of liquid crystals in one dimension. As mentioned in [12], the weak solution was obtained with the initial density having a positive lower bound and Hintegrable. In this paper, we get a weak solution with initial density nonnegative and L-integrable.

We present a regularity result for weak solutions of the 2D quasi-geostrophic equation with supercritical (α < 1/2) dissipation (−∆) : If a Leray-Hopf weak solution is Hölder continuous θ ∈ C(R) with δ > 1 − 2α on the time interval [t0, t], then it is actually a classical solution on (t0, t]. AMS (MOS) Numbers: 76D03, 35Q35

This paper presents a new algorithm that arrives at a generalized solution for the generation of restricted weak compositions of n-parts. In particular, this generalized algorithm covers many commonly sought compositions such as bounded compositions, restricted compositions, weak compositions, and restricted part compositions. Introduced is an algorithm for generating generalized types of restr...

Choice functions on tournaments always select the maximal element (Condorcet winner), provided they exist, but this property does not hold in the more general case of weak tournaments. In this paper we analyze the relationship between the usual choice functions and the set of maximal elements in weak tournaments. We introduce choice functions selecting maximal elements, whenever they exist. Mor...

This paper is concerned with Sobolev weak solution of Hamilton-Jacobi-Bellman (HJB) equation. This equation is derived from the dynamic programming principle in the study of the stochastic optimal control problem. Adopting Doob-Meyer decomposition theorem as one of main tool, we prove that the optimal value function is the unique Sobolev weak solution of the corresponding HJB equation. For the ...