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We study the critical dissipative quasi-geostrophic equations in $\bR^ 2$ with arbitrary H^ 1 initial data. After showing certain decay estimate, a global well-posedness result is proved by adapting the method in [11] with a suitable modification. A decay in time estimate for higher order homogeneous Sobolev norms of solutions is also discussed.

We study the critical and super-critical dissipative quasi-geostrophic equations in $\bR^ 2$ or $\bT^ 2$. Higher regularity of mild solutions with arbitrary initial data in H^{2-} is proved. As a corollary, we obtain a global existence result for the critical 2D quasi-geostrophic equations with periodic H^{2-} data. Some decay in time estimates are also provided.

The solvability in Sobolev spaces is proved for divergence form complex-valued higher order parabolic systems in the whole space, on a half-space, and on a Reifenberg flat domain. The leading coefficients are assumed to be merely measurable in one spacial direction and have small mean oscillations in the orthogonal directions on each small cylinder. The directions in which the coefficients are only measurable vary depending on each cylinder. The corresponding elliptic problem is also considered.

We establish the solvability of second order divergence-type parabolic systems in Sobolev spaces. The leading coefficients are assumed to be merely measurable in one spatial direction on each small parabolic cylinder with the spatial direction allowed to depend on the cylinder. In the other orthogonal directions and the time variable, the coefficients have locally small mean oscillations. We also obtain the corresponding $W^1_p$-solvability of second order elliptic systems in divergence form. This type of system arises from the problems of linearly elastic laminates and composite materials. Our results are new even for scalar equations, and the proofs differ from and simplify the methods used previously in [H. Dong and D. Kim, <i>Arch. Ration. Mech. Anal.</i>, 196 (2010), pp. 2570]. As an application, we improve a result by Chipot, Kinderlehrer, and Vergara-Caffarelli [<i>Arch. Ration. Mech. Anal.</i>, 96 (1986), pp. 8196] on

We study a one-dimensional transport equation with nonlocal velocity which was recently considered in the work of Crdoba, Crdoba and Fontelos [A. Crdoba, D. Crdoba, MA Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math.(2) 162 (3)(2005) 13771389]. We show that in the subcritical and critical cases the problem is globally well-posed with arbitrary initial data in H max {3/2 , 0}. While in the supercritical case, the problem is locally well-posed with initial data in H 3/2 , and is globally well-posed under a smallness assumption. Some polynomial-in-time decay estimates are also discussed. These results improve some previous results in [A. Crdoba, D. Crdoba, MA Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math.(2) 162 (3)(2005) 13771389].

We establish several gradient estimates for second-order divergence type parabolic and elliptic systems. The coefficients and data are assumed to be Hlder or Dini continuous in the time variable and all but one spatial variable. This type of system arises from the problems of linearly elastic laminates and composite materials. For the proof, we use Campanatos approach in a novel way. Non-divergence type equations under similar conditions are also discussed.

We study regularity criteria for weak solutions of the dissipative quasi-geostrophic equation (with dissipation ()^<i></i>/2, 0 &lt; <i></i> 1). We show in this paper that if , or with is a weak solution of the 2D quasi-geostrophic equation, then <i></i> is a classical solution in . This result improves our previous result in [18].

Elliptic Bellman equations with coefficients independent of the variable x are considered. Error bounds for certain types of finite-difference schemes are obtained. These estimates are sharper than the earlier results in Krylov's article of 1997.

We establish global regularity for weak solutions to quasilinear divergence form elliptic and parabolic equations over Lipschitz domains with controlled growth conditions on low order terms. The leading coefficients belong to the class of BMO functions with small mean oscillations with respect to <i>x</i>.