No abstract uploaded!

We study reactive front speeds in randomly perturbed cellular flows using a variational representation for the front speed. We develop this representation into a computational tool for computing the front speeds without resorting to closure approximations. We demonstrate that the front speeds depend on flow statistics and topologies in a complex and dramatic manner.

The Boltzmann equation without Grads angular cutoff assumption is believed to have a regularizing effect on the solutions because of the non-integrable angular singularity of the cross-section. However, even though this has been justified satisfactorily for the spatially homogeneous Boltzmann equation, it is still basically unsolved for the spatially inhomogeneous Boltzmann equation. In this paper, by sharpening the coercivity and upper bound estimates for the collision operator, establishing the hypo-ellipticity of the Boltzmann operator based on a generalized version of the uncertainty principle, and analyzing the commutators between the collision operator and some weighted pseudo-differential operators, we prove the regularizing effect in all (time, space and velocity) variables on the solutions when some mild regularity is imposed on these solutions. For completeness, we also show that when the initial

In this paper, we study the decomposition of the Nehari manifold via the combination of concave and convex nonlinearities. Furthermore, we use this result and the LjusternikSchnirelmann category to prove that the existence of multiple positive solutions for a Dirichlet problem involving critical Sobolev exponent.

In this paper we establish the local-in-time existence and uniqueness of strong solutions to the free boundary problem of the full compressible Navier-Stokes equations in three-dimensional space. The vanishing density and temperature condition is imposed on the free boundary, which captures the motions of the non-isentropic viscous gas surrounded by vacuum with bounded entropy. We also assume some proper decay rates of the density towards the boundary and singularities of derivatives of the temperature across the boundary on the initial data, which coincides with the physical vacuum condition for the isentropic flows. This extends the previous result of Liu [arXiv: 1612.07936] by removing the spherically symmetric assumption and considering more general initial density and temperature profiles.