In this paper, self-similarity is illustrated and compared in deterministic and stochastic dissipative systems. Examples are (1) deterministic self-similarity in reaction-diffusion system and Navier-Stokes equations, where solutions eventually decay to zero due to balance of diffusion (viscosity) and nonlinearity;(2) statistical self-similarity in randomly advected passive scalar model of Kraichnan where solutions undergo turbulent decay due to roughness of advection;(3) selfsimilarity in blowup of solutions of fourth order nonlinear parabolic equations of the Cahn-Hillard type. Problems for future research are mentioned, especially those where self-similarity is conjectured based on numerical evidence or physical grounds but mathematically open.

In this paper, we will introduce a precise classification of characteristic functions in the Fourier space according to the moment constraint in the physical space of any order. Based on this, we construct measure-valued solutions to the homogeneous Boltzmann equation with the exact moment condition as the initial data.

G-equations are popular front propagation models in combustion literature and describe the front motion law of normal velocity equal to a constant plus the normal projection of fluid velocity. G-equations are Hamilton-Jacobi equations with convex but non-coercive Hamiltonians. We prove homogenization of the inviscid G-equation for space periodic incompressible flows. This extends a two space dimensional result in" Periodic homogenization of G-equations and viscosity effects," Nonlinearity, to appear. We construct approximate correctors to bypass the lack of compactness due to the non-coercive Hamiltonian. The existence of approximate correctors rely on a local reachability property of the controlled flow trajectory as well as incompressibility of the flow. Homogenization then follows from the comparison principle and the perturbed test function method. The effective Hamiltonian is convex and homogeneous of

In this paper, we study the asymptotic stability of rarefaction waves for the compressible isentropic NavierStokes equations with density-dependent viscosity. First, a weak solution around a rarefaction wave to the Cauchy problem is constructed by approximating the system and regularizing the initial values which may contain vacuum states. Then some global in time estimates on the weak solution are obtained. Based on these uniform estimates, the vacuum states are shown to vanish in finite time and the weak solution we constructed becomes a unique strong one. Consequently, the stability of the rarefaction wave is proved in a weak sense. The theory holds for large-amplitudes rarefaction waves and arbitrary initial perturbations.

We revisit the homogenization problem for the Poisson equation in periodically perforated domains with zero Neumann data at the boundary of the holes and prescribed Dirichlet data at the outer boundary. It is known that, if the periodicity of the holes goes to zero but their volume fraction remains fixed and positive, the limit problem is a Dirichlet boundary value problem posed in the domain without the holes, and the effective diffusion coefficients are non-trivially modified; if that volume fraction goes to zero instead, i.e. the holes are dilute, the effective operator remains the Laplacian (that is, unmodified). Our main results contain the study of a "continuity" in those effective models with respect to the volume fraction of the holes and some new convergence rates for homogenization in the dilute setting. Our method explores the classical two-scale expansion ansatz and relies on asymptotic analysis of the rescaled cell problems using layer potential theory.