Uniqueness and inviscid limits of solutions for the complex Ginzburg-Landau equation in a two-dimensional domain
The authors discuss the uniqueness and inviscid limit of weak solutions to the initial-boundary value problem (I-BVP) for the complex Ginzburg-Landau equation \[ \begin{matrix} \partial_tu-(\lambda+i\alpha)\Delta u+(a+i\beta)|u|^{p-1}u-\gamma u=0,\quad t>0,\quad x\in\Omega;\\ u(t,x)=0,\quad t>0,\quad x\in\partial\Omega;\qquad u(0,x)=u_0(x), \quad t=0,\quad x\in\Omega, \end{matrix} \] \(\Omega\subset \mathbb R^2\) is not necessary having compact smooth boundaries, \(p>1\), \(\lambda\geq 0\), \(a\), \(\alpha\), \(\beta\), \(\gamma\) are real constants. The first aim of the article is to prove the uniqueness of weak solutions of the problem without the restrictive condition \(|\beta|\leq\frac{2\sqrt{p}}{p-1}a\) of previous authors, provided \(p<3\), which includes the physically important situation under \(n=2\). Two kinds of uniqueness theorem are proved for different classes of weak solutions. The first is for a class with \(L^2\) initial data, the second is for a class with \(H^1\) initial data. The other purpose of the article is to discuss the inviscid limit for the I-BVP in a general domain \(\Omega\subset \mathbb R^2\). The case \(\lambda=0\) gives the I-BVP for the nonlinear Schrödinger equation.
scientific article; zbMATH DE number 2044318