Inviscid limit for the energy-critical complex Ginzburg–Landau equation

In this paper, we consider the limit behavior for the solution of the Cauchy problem of the energy-critical complex Ginzburg–Landau equation in Rn, n⩾3. In lower dimension case (3⩽n⩽6), we show that its solution converges to that of the energy-critical nonlinear Schrödinger equation in , T>0, s=0,1, as a by-product, we get the regularity of solutions in H3 for the nonlinear Schrödinger equation. In higher dimension case (n>6), we get the similar convergent behavior in C(0,T,L2(Rn)). In both cases we obtain the optimal convergent rate.