Isometric decomposition operators, function spaces \(E_{p,q}^{\lambda}\) and applications to nonlinear evolution equations. (English) Zbl 1099.46023
The main goal of this paper is to study the basic estimates for the semigroup
\[
U(t):=\exp\bigl((a+i\alpha)t\Delta\bigr),\;a\geq 0,\;\alpha \in\mathbb{R},\;a+|\alpha|\neq 0,\tag{1}
\]
in certain function spaces. The authors using the isometric decomposition to the frequency spaces, introduce a new class of function spaces \(\mathbb{E}^\lambda_{p,q}\), which is a subspace of the Gevrey 1-class \(G_1(\mathbb{R}^n)\subset C^\infty (\mathbb{R}^n)\) for \(\lambda>0\), and study the Cauchy problem for the nonlinear Schrödinger equation, the complex Ginzburg–Landau equation and the Navier–Stokes equation. Some well-posed results are obtained as well as the regularity behaviour.
Reviewer: Messoud A. Efendiev (Berlin)
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 35K55 | Nonlinear parabolic equations |
| 35Q30 | Navier-Stokes equations |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 47D06 | One-parameter semigroups and linear evolution equations |
Keywords:
nonlinear Schrödinger equation; complex Ginzburg-Landau equation; Navier-Stokes equation; Cauchy problem; local well posedness; regularity behaviorReferences:
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