The complex Ginzburg-Landau equation as a model problem.

*(English)*Zbl 0845.35003
Deift, Percy (ed.) et al., Dynamical systems and probabilistic methods in partial differential equations. 1994 summer seminar on dynamical systems and probabilistic methods for nonlinear waves, June 20-July 1, 1994, MSRI, Berkeley, CA, USA. Providence, RI: American Mathematical Society. Lect. Appl. Math. 31, 141-190 (1996).

The generalized complex Ginzburg-Landau equation
\[
\partial_t u= Ru+ (1+ i\nu) \Delta u- (1+ i\mu)|u|^{2\sigma}u\tag{CGL}
\]
is considered as a model problem of a variety of possible phenomena in nonlinear partial differential equations. This equation describes the evolution of a complex-valued field \(u= u(x, t)\) over a \(d\)-dimensional spatial domain. The positive parameter \(R\) is the coefficient of the linear driving term, without which all solutions would decay to zero, and \(\sigma> 0\) sets the degree of nonlinearity. The real constants \(\nu\) and \(\mu\) are the coefficients of the linear and nonlinear dispersive terms, respectively. The authors’ aim is to present to nonspecialists an overview of how the generalized CGL equation can be viewed as a dynamical system, and this goal is excellently reached. The style is expository and the paper is very well written.

The authors start their exposition with painting the CGL phase portrait by identifying some special solutions, examining their stability and describing how chaotic behavior might occur. Later on, the relationship is discussed of the well-posedness and regularity results for (CGL) to the analogous questions for Navier-Stokes fluid turbulence, in particular, for the nonlinear Schrödinger equation. The authors prove existence of global in time weak solutions to (CGL) for \(L^2\) initial data in any spatial dimension \(d\) and any degree of nonlinearity \(\sigma\). Existence and uniqueness of local (in time) classical (even \(C^\infty\)) solutions is studied provided \(\sigma\) is a positive integer, and taking \(L^p\) or \(H^q\) initial data. Making use of a characterization of Gevrey classes in terms of decay of Fourier coefficients, the authors derive local estimates which show that if \(\sigma\) is an integer then the solution to (CGL) are real analytic functions. A priori estimates in \(L^p\) are also proved for each positive \(\sigma\) and suitable values of \(\nu\) and \(\mu\), depending on \(d\) and \(\sigma\). In particular, these estimates ensure existence of global classical solutions to (CGL). In the case of positive and integer \(\sigma\), direct global estimates are derived on each Sobolev space \(H^n\) for the solutions of (CGL) with suitable \(L^p\) initial data. Finally, the authors discuss the significance of the results presented in applying dynamical systems methods when studying long-time behavior of solutions to (CGL).

For the entire collection see [Zbl 0831.00019].

The authors start their exposition with painting the CGL phase portrait by identifying some special solutions, examining their stability and describing how chaotic behavior might occur. Later on, the relationship is discussed of the well-posedness and regularity results for (CGL) to the analogous questions for Navier-Stokes fluid turbulence, in particular, for the nonlinear Schrödinger equation. The authors prove existence of global in time weak solutions to (CGL) for \(L^2\) initial data in any spatial dimension \(d\) and any degree of nonlinearity \(\sigma\). Existence and uniqueness of local (in time) classical (even \(C^\infty\)) solutions is studied provided \(\sigma\) is a positive integer, and taking \(L^p\) or \(H^q\) initial data. Making use of a characterization of Gevrey classes in terms of decay of Fourier coefficients, the authors derive local estimates which show that if \(\sigma\) is an integer then the solution to (CGL) are real analytic functions. A priori estimates in \(L^p\) are also proved for each positive \(\sigma\) and suitable values of \(\nu\) and \(\mu\), depending on \(d\) and \(\sigma\). In particular, these estimates ensure existence of global classical solutions to (CGL). In the case of positive and integer \(\sigma\), direct global estimates are derived on each Sobolev space \(H^n\) for the solutions of (CGL) with suitable \(L^p\) initial data. Finally, the authors discuss the significance of the results presented in applying dynamical systems methods when studying long-time behavior of solutions to (CGL).

For the entire collection see [Zbl 0831.00019].

Reviewer: D.Palagachev (Sofia)

##### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35G25 | Initial value problems for nonlinear higher-order PDEs |

35Q55 | NLS equations (nonlinear Schrödinger equations) |

35Q30 | Navier-Stokes equations |

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\textit{C. D. Levermore} and \textit{M. Oliver}, in: Dynamical systems and probabilistic methods in partial differential equations. 1994 summer seminar on dynamical systems and probabilistic methods for nonlinear waves, June 20-July 1, 1994, MSRI, Berkeley, CA, USA. Providence, RI: American Mathematical Society. 141--190 (1996; Zbl 0845.35003)