2015/07/14 by Serfaty, Sylvia
#35K55 #35K99 #35Q31 #35Q35 #35Q55 #35Q56 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.1507.03821
We prove that in a certain asymptotic regime, solutions of the Gross-Pitaevskii equation converge to solutions of the incompressible Euler equation, and solutions to the parabolic Ginzburg-Landau equations converge to solutions of a limiting equation which we identify. We work in the setting of the whole plane (and possibly the whole three-dimensional space in the Gross-Pitaevskii case), in the asymptotic limit where ε, the characteristic lengthscale of the vortices, tends to 0, and in a situation where the number of vortices N blows up as ε→ 0. The requirements are that N should blow up faster than |log ε| in the Gross-Pitaevskii case, and at most like |log ε| in the parabolic case. Both results assume a well-prepared initial condition and regularity of the limiting initial data, and use the regularity of the solution to the limiting equation. In the case of the parabolic Ginzburg-Landau equation, the limiting mean-field dynamical law that we identify coincides with the one proposed by Chapman-Rubinstein-Schatzman and E in the regime N≪ |log ε|, but not if N grows faster.