By introducing a new approximate Green function, we obtain the pointwise estimates on the solutions of Euler equations with linear frictional damping, from which we can deduce the optimal L p (1 p+) convergence rates to the nonlinear diffusion waves. The pointwise estimates and L p (1 p< 2) convergence rates given in this paper are new.

In the paper, we study the Prandtl system with initial data admitting non-degenerate critical points. For any index \sigma\in [3/2, 2], we obtain the local in time well-posedness in the space of Gevrey class \sigma\in [3/2, 2], in the tangential variable and Sobolev class in the normal variable so that the monotonicity condition on the tangential velocity is not needed to overcome the loss of tangential derivative. This answers the open question raised in the paper of D. Grard-Varet and N. Masmoudi [{\it Ann. Sci. c. Norm. Supr}.(4) 48 (2015), no. 6, 1273-1325], in which the case \sigma\in [3/2, 2], is solved.

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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 consider the optimization approach to the acousto-electric imaging problem. Assuming that the electric conductivity distribution is a small perturbation of a constant, we investigate the least-squares estimate analytically using (multiple) Fourier series, and confirm the widely observed fact that acousto-electric imaging has high resolution and is statistically stable. We also analyze the case of partial data and the case of limited-view data, in which some singularities of the conductivity can still be imaged.