In this paper, we give a priori estimation of derivatives of solution of Duncan-Mortensen-Zakai (DMZ) equation up to 2nd order. We also prove the existence and uniqueness of weak solution for DMZ equation, which is a fundamental equation in nonlinear filtering.

In this paper, we investigate the initial-boundary value problem to the heat conductive compressible Navier-Stokes equations. Local existence and uniqueness of strong solutions is established with any such initial data that the initial density $\rho_0$, velocity $u_0$, and temperature $\theta_0$ satisfy $\rho_0\in W^{1,q}$, with $q\in(3,6)$, $u_0\in H^1$, and $\sqrt{\rho_0}\theta_0\in L^2$. The initial density is assumed to be only nonnegative and thus the initial vacuum is allowed. In addition to the necessary regularity assumptions, we do not require any initial compatibility conditions such as those proposed in (Y. Cho and H. Kim, \emph{Existence results for viscous polytropic fluids with vacuum}, J. Differential Equations {\bf 228} (2006), no.~2, 377--411.), which although are widely used in many previous works but put some inconvenient constraints on the initial data. Due to the weaker regularities of the initial data and the absence of the initial compatibility conditions, leading to weaker regularities of the solutions compared with those in the previous works, the uniqueness of solutions obtained in the current paper does not follow from the arguments used in the existing literatures. Our proof of the uniqueness of solutions is based on the following new idea of two-stages argument: (i) showing that the difference of two solutions (or part of their components) with the same initial data is controlled by some power function of the time variable; (ii) carrying out some singular-in-time weighted energy differential inequalities fulfilling the structure of the Gr\"onwall inequality. The existence is established in the Euler coordinates, while the uniqueness is proved in the Lagrangian coordinates first and then transformed back to the Euler coordinates.

No abstract uploaded!

In this paper, we consider the Cauchy problem to the planar non-resistive magnetohydrodynamic equations without heat conductivity, and establish the global well-posedness of strong solutions with large initial data. The key ingredient of the proof is to establish the a priori estimates on the effective viscous flux and a newly introduced ``transverse effective viscous flux" vector field inducted by the transverse magnetic field. The initial density is assumed only to be uniformly bounded and of finite mass and, in particular, the vacuum and discontinuities of the density are allowed.

A review is presented of recent results on front propagation in reaction-diffusion-advection equations in homogeneous and heterogeneous media. Formal asymptotic expansions and heuristic ideas are used to motivate the results wherever possible. The fronts include constant-speed monotone traveling fronts in homogeneous media, periodically varying traveling fronts in periodic media, and fluctuating and fractal fronts in random media. These fronts arise in a wide range of applications such as chemical kinetics, combustion, biology, transport in porous media, and industrial deposition processes. Open problems are briefly discussed along the way.