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].

It is well known that the nonlinear filtering problem has important applications in both military and commercial industries. The central problem of nonlinear filtering is to solve the DMZ equation in real time and memoryless manner. The purpose of this paper is to show that, under very mild conditions (which essentially say that the growth of the observation |h| is greater than the growth of the drift |f|), the DMZ equation admits a unique nonnegative weak solution u which can be approximated by a solution u_R of the DMZ equation on the ball B_R with u_R|_BR = 0. The error of this approximation is bounded by a function of R which tends to zero as R goes to infinity. The solution u_R can in turn be approximated efficiently by an algorithm depending only on solving the observation-independent Kolmogorov equation on B_R. In theory, our algorithm can solve basically all engineering problems in real time manner. Specifically

We consider a model semilinear reaction-diffusion system with cubic nonlinear reaction terms and small spatially decaying initial data on R 1. The model system is motivated by the thermal-diffusive system in combustion, and it reduces to a scalar reaction-diffusion equation with Zeldovich nonlinearity when the Lewis number is one and proper initial data are prescribed. For scalar equations of similar type it is well known that while a nonlinearity of degree greater than three (supercritical case) has no effect for large times a cubic nonlinearity qualitatively changes the long time behaviour. The latter case has been treated in the literature by a rescaling method under the additional assumption of smallness of the nonlinearity. Although for our system the cubic nonlinearity is also critical we establish large time behaviour when the nonlinearity is not necessarily small which essentially differs from the supercritical case. This

Despite its usefulness, the Kalman-Bucy filter is not perfect. One of its weaknesses is that it needs a Gaussian assumption on the initial data. Recently Yau and Yau introduced a new direct method to solve the estimation problem for linear filtering with non-Gaussian initial data. They factored the problem into two parts: (1) the on-line solution of a finite system of ordinary differential equations (ODEs), and (2) the off-line calculation of the Kolmogorov equation. Here we derive an explicit closed-form solution of the Kolmogorov equation. We also give some properties and conduct a numerical study of the solution.

G-equations are well-known front propagation models in combustion and are HamiltonJacobi type equations with convex but non-coercive Hamiltonians. Viscous G-equations arise from numerical discretization or modeling dissipative mechanisms. Although viscosity helps to overcome non-coercivity, we prove homogenization of an inviscid G-equation based on approximate correctors and attainability of controlled flow trajectories. We verify the attainability for two-dimensional mean zero incompressible flows, and demonstrate asymptotically and numerically that viscosity reduces the homogenized Hamiltonian in cellular flows. In the case of one-dimensional compressible flows, we found an explicit formula of homogenized Hamiltonians, as well as necessary and sufficient conditions for wave trapping (effective Hamiltonian vanishes identically). Viscosity restores coercivity and wave propagation.