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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.

This paper establishes a blowup criterion for the three-dimensional viscous, compressible, and heat conducting magneto-hydrodynamic (MHD) flows. It is essentially shown that for the Cauchy problem and the initial-boundary-value one of the three-dimensional compressible MHD flows with initial density allowed to vanish, the strong or smooth solution exists globally if the density is bounded from above and the velocity satisfies Serrin’s condition. Therefore, if the Serrin norm of the velocity remains bounded, it is not possible for other kinds of singularities (such as vacuum states vanishing or vacuum appearing in the non-vacuum region or even milder singularities) to form before the density becomes unbounded. This criterion is analogous to the well known Serrin’s blowup criterion for the three-dimensional incompressible Navier-Stokes equations, in particular, it is independent of the temperature and magnetic field and is just the same as that of the barotropic compressible Navier-Stokes equations. As a direct application, it is shown that the same result also holds for the strong or smooth solutions to the three-dimensional full compressible Navier-Stokes system describing the motion of a viscous, compressible, and heat conducting fluid.

In this paper, we consider the initial boundary value problem in a cylindrical domain to the three dimensional primitive equations with full eddy viscosity in the momentum equations but with only horizontal eddy diffusivity in the temperature equation. Global well-posedness of $z$-weak solution is established for any such initial datum that itself and its vertical derivative belong to $L^2$. This not only extends the results in \cite{Cao5} from the spatially periodic case to general cylindrical domains but also weakens the regularity assumptions on the initial data which are required to be $H^2$ there.

We prove a quantitative bi-Lipschitz non-embedding theorem for the Heisenberg group with its Carnot–Carathéodory metric and apply it to give a lower bound on the integrality gap of the Goemans–Linial semidefinite relaxation of the sparsest cut problem.