| My thesis consists five results related to the regularity theory for second-order elliptic-parabolic equations and their numerical approximations. (1) We consider the probabilistic solutions of the heat equation ux2=ux1 x1 + f in D, where D is a bounded domain in R2 = {lcub}(x1, x 2){rcub} of class C2k. We give sufficient conditions for u to have the kth order continuous derivatives with respect to ( x1, x2) in D¯, for integers k ≥ 2. The equation is supplemented with C2k boundary data and we assume that f ∈ C 2(k-1). We also prove that our conditions are sharp by given examples in the border cases. (2) We consider the Dirichlet problem for two types of degenerate elliptic Hessian equations. New results about solvability of the equations in the C1,1 space are provided. (3) Elliptic Bellman equations with coefficients independent of variable x are considered. Error bounds for certain types of finite-difference schemes are obtained. These estimates are sharper than those earlier results in [48]. (4) Degenerate parabolic and elliptic equations of second order with C1 and C2 coefficients are considered. Error bounds for certain types of finite-difference schemes are obtained. (5) We consider the Cauchy problem for incompressible Navier-Stokes equations ut + u∇ xu - Deltaxu + ∇ xp = 0, div u = 0 in Rd x R+ with initial data in Ld( Rd ), and study in some detail the smoothing effect of the equation. We prove that for T < infinity and for any positive integers n and m we have t m+n /2 DmtDnx u ∈ Ld +2( Rd x (0,T)), as long as the uL d+2x,t&parl0;Rdx&parl0;0,T&parr0; &parr0; stays finite. |
