We study a class of bistable reaction-diffusion systems used to model two competing species. Systems in this class possess two uniform stable steady states representing semi-trivial solutions. Principally, we are interested in the case where the ratio of the diffusion coefficients is small, i.e. in the near-degenerate case. First, limiting arguments are presented to relate solutions to such systems to those of the degenerate case where one species...

We consider complex-valued solutions $u_{\epsilon }$ of the Ginzburg–Landau equation on a smooth bounded simply connected domain $\Omega$ of ${ℝ}^N$, $N\ge 2$, where $\epsilon \>0$ is a small parameter. We assume that the Ginzburg–Landau energy $E_{\epsilon }\left(u_{\epsilon }\right)$ verifies the bound (natural in the context) $E_{\epsilon }\left(u_{\epsilon }\right)\le M_0|log\epsilon |$, where $M_0$ is some given constant. We also make several assumptions on the boundary data. An important step in the asymptotic analysis of $u_{\epsilon }$, as $\epsilon \to 0$, is to establish uniform $L^p$ bounds for the gradient, for some $p\>1$. We review some recent techniques developed in...

We consider complex-valued solutions uE of the Ginzburg–Landau equation on a smooth bounded simply connected domain Ω of ${ℝ}^N$, N ≥ 2, where ε > 0 is a small parameter. We assume that the Ginzburg–Landau energy $E_{\epsilon }\left(u_{\epsilon }\right)$ verifies the bound (natural in the context) $E_{\epsilon }\left(u_{\epsilon }\right)\le M_0|log\epsilon |$, where M0 is some given constant. We also make several assumptions on the boundary data. An important step in the asymptotic analysis of uE, as ε → 0, is to establish uniform Lp bounds for the gradient, for some p>1. We review some...

It is well known that people can derive the radiation MHD model from an MHD-$P1$ approximate model. As pointed out by F. Xie and C. Klingenberg (2018), the uniform regularity estimates play an important role in the convergence from an MHD-$P1$ approximate model to the radiation MHD model. The aim of this paper is to prove the uniform regularity of strong solutions to an isentropic compressible MHD-$P1$ approximate model arising in radiation hydrodynamics. Here we use the bilinear commutator and product estimates...

In this paper we construct upper bounds for families of functionals of the form$$E_{\epsilon }\left(\phi \right):={\int }_{\Omega }\left({\epsilon \left|\nabla \phi \right|}^2+\frac{1}{\epsilon }W\left(\phi \right)\right)\mathrm{d}x+\frac{1}{\epsilon }{\int }_{{ℝ}^N}{\left|\nabla {\overline{H}}_{F\left(\phi \right)}\right|}^2\mathrm{d}x$$where Δ${\overline{H}}_u$ = div {${\chi }_{\Omega }$u}. Particular cases of such functionals arise in Micromagnetics. We also use our technique to construct upper bounds for functionals that appear in a variational formulation of the method of vanishing viscosity for conservation laws.

We prove an upper bound for the Aviles–Giga problem, which involves the minimization of the energy $E_{\epsilon }\left(v\right)=\epsilon {\int }_{\Omega }|{\nabla }^2{v|}^{2}dx+{\epsilon }^{-1}{\int }_{\Omega }{\left(1-\right|\nabla v|}^2{)}^{2}dx$ over $v\in H^2\left(\Omega \right)$, where $\epsilon >0$ is a small parameter. Given $v\in W^{1,\infty }\left(\Omega \right)$ such that $\nabla v\in BV$ and $\left|\nabla v\right|=1$ a.e., we construct a family $\left\{v_{\epsilon }\right\}$ satisfying: $v_{\epsilon }\to v$ in $W^{1,p}\left(\Omega \right)$ and $E_{\epsilon }\left(v_{\epsilon }\right)\to \frac{1}{3}{\int }_{J_{\nabla v}}{\left|{\nabla }^{+}v-{\nabla }^{-}v\right|}^{3}d{\mathscr{H}}^{N-1}$ as $\epsilon$ goes to 0.

We consider variational problems of P. D. E. depending on a small parameter $\epsilon$ when the limit process $\epsilon \downarrow 0$ implies vanishing of the higher order terms. The perturbation problem is said to be sensitive when the energy space of the limit problem is out of the distribution space, so that the limit problem is out of classical theory of P. D. E. We present here a review of the subject, including abstract convergence theorems and two very different model problems (the second one is presented for the first...

We study solutions of the Gross-Pitaevsky equation and similar equations in $m\ge 3$ space dimensions in a certain scaling limit, with initial data $u_0^{ϵ}$ for which the jacobian $J{u}_0^{ϵ}$ concentrates around an (oriented) rectifiable $m-2$ dimensional set, say ${\Gamma }_0$, of finite measure. It is widely conjectured that under these conditions, the jacobian at later times $t\>0$ continues to concentrate around some codimension $2$ submanifold, say ${\Gamma }_t$, and that the family $\left\{{\Gamma }_t\right\}$ of submanifolds evolves by binormal mean curvature flow. We prove...

We discuss the asymptotics of the parabolic Ginzburg-Landau equation in dimension $N\ge 2.$ Our only asumption on the initial datum is a natural energy bound. Compared to the case of “well-prepared” initial datum, this induces possible new energy modes which we analyze, and in particular their mutual interaction. The two dimensional case is qualitatively different and requires a separate treatment.