We investigate Lamé systems in periodically perforated domains, and establish quantitative homogenization results in the setting where the domain is clamped at the boundary of the holes. Our method is based on layer potentials and it provides a unified proof for various regimes of hole-cell ratios (the ratio between the size of the holes and the size of the periodic cells), and, more importantly, it yields natural correctors that facilitate error estimates. A key ingredient is the asymptotic analysis for the rescaled cell problems, and this is studied by exploring the convergence of the periodic layer potentials for the Lamé system to those in the whole space when the period tends to infinity.

We consider the zero heat conductivity limit to a contact discontinuity for the mono-dimensional full compressible Navier--Stokes--Fourier system. The method is based on the relative entropy method and does not assume any smallness conditions on the discontinuity nor on the bounded variation norm of the initial data. It is proved that for any viscosity , the solution of the compressible Navier--Stokes--Fourier system (with well-prepared initial value) converges, when the heat conductivity tends to zero, to the contact discontinuity solution to the corresponding Euler system. We obtain the decay rate . It implies that the heat conductivity dominates the dissipation in the regime of the limit to a contact discontinuity. This is the first result, based on the relative entropy, of an asymptotic limit to a discontinuous solution for a system.

We study the vanishing viscosity limit of the compressible NavierStokes equations to the Riemann solution of the Euler equations that consists of the superposition of a shock wave and a rarefaction wave. In particular, it is shown that there exists a family of smooth solutions to the compressible NavierStokes equations that converges to the Riemann solution away from the initial and shock layers at a rate in terms of the viscosity and the heat conductivity coefficients. This gives the first mathematical justification of this limit for the NavierStokes equations to the Riemann solution that contains these two typical nonlinear hyperbolic waves.

We study the stability and oscillation of traveling fronts in a three-component, advection-reaction biodegradation model. The three components are pollutant, nutrient, and bacteria concentrations. Under an explicit condition on the biomass growth and decay coefficients, we derive reduced, two-component, semilinear hyperbolic models through a relaxation procedure, during which biomass is slaved to pollutant and nutrient concentration variables. The reduced two-component models resemble the Broadwell model of the discrete velocity gas. The traveling fronts of the reduced system are explicit and are expressed in terms of hyperbolic tangent function in the nutrient-deficient regime. We perform energy estimates to prove the asymptotic stability of these fronts under explicit conditions on the coefficients in the system. In the small damping limit, we carry out Wentzel--Kramers--Brillouin (WKB) analysis on front

Let G=(V,E) be a locally finite connected weighted graph, and Ω be an unbounded subset of V. Using Rothe's method, we study the existence of solutions for the semilinear heat equation ∂_t u+|u|^{p−1}⋅u=Δu (p≥1) and the parabolic variational inequality ∫_{Ω^∘} ∂_t u⋅(v−u)dμ ≥ ∫_{Ω^∘} (Δu+f)⋅(v−u)dμ for any v∈H, where H={u∈W^{1,2}(V):u=0 on V∖Ω^∘}.