Asymptotic Analysis

Let p>1 and Ω be a smoothly bounded domain in R^N not necessarily convex. This paper is concerned with a Cauchy–Dirichlet problem $\def\theequation{(P)}\begin{equation}\left\{\begin{array}{l@{\quad}l}u_{t}=\Delta u+u^{p}&\mbox{in }\varOmega \times (0,\infty),\\[3pt]u(x,t)=0&\mbox{on}\ \curpartial \varOmega \times (0,\infty),\\[3pt]u(x,0)=\phi(x)\geq 0&\mbox{in }\varOmega.\end{array}\right.\end{equation}$ We show that when u blows up at t=T, it holds

|u(t)|_∞≤C(T−t)^−1/(p−1) in (0,T)

with some C>0 under some condition on the Cauchy problem for (P).

We define an abstract nonlinear elliptic system, admitting a variational structure, and including the vortex equations for some Maxwell–Chern–Simons gauge theories as special cases. We analyze the asymptotic behavior of its solutions, and we provide a general simplified framework for the asymptotics previously derived in those special cases. As a byproduct of our abstract formulation, we also find some new qualitative properties of solutions.

In this paper, we study the behaviour, in terms of compressibility, of steady flows of weakly compressible viscoelastic fluids having a differential constitutive law. The models considered here are Jeffreys' and Maxwell's. In both cases, we establish the existence of an asymptotic expansion in the neighbourhood of the steady incompressible fluid flow, with respect to the vanishing isothermal compressibility coefficient.

The paper is devoted to an elliptic perturbation of a parabolic problem. The solution v of the heat equation with a nonlinear boundary condition is compared with the solution v_ε of an elliptic regularization. An asymptotic expansion of v_ε is constructed and the order of accuracy for the difference v_ε−v is obtained. The boundary function method by Vishik and Lyusternik is used.

In this article, we rigorously investigate the diffusive limit of a velocity‐discrete Boltzmann equation which is used in the lattice Boltzmann method (LBM) to construct approximate solutions of the incompressible Navier–Stokes equation. Our results apply to LBM collision operators with multiple collision frequencies (generalized lattice Boltzmann) which include the widely used BGK operators.