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Weak and strong solutions of the complex Ginzburg-Landau equation. (English) Zbl 0810.35119
Summary: The generalized complex Ginzburg-Landau equation, $\partial_ t A= RA+ (1+ i\nu) \Delta A-(1+ i\mu)| A|^{2\sigma}A,$ has been proposed and studied as a model for “turbulent” dynamics in nonlinear partial differential equations. It is a particularly interesting model in this respect because it is a dissipative version of the Hamiltonian nonlinear Schrödinger equation possessing solutions that form localized singularities in finite time.
In this paper we investigate existence and regularity of solutions to this equation subject to periodic boundary conditions in various spatial dimensions. Appropriately defined weak solutions are established globally in time, and unique strong solutions are found locally. A new collection of a priori estimates are presented, and we discuss the relationship of our results for the complex Ginzburg-Landau equation to analogous issues for fluid turbulence described by the incompressible Navier-Stokes equations.

##### MSC:
 35Q55 NLS equations (nonlinear Schrödinger equations) 35Q35 PDEs in connection with fluid mechanics 76F05 Isotropic turbulence; homogeneous turbulence
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##### References:
 [1] Bartuccelli, M; Constantin, P; Doering, C.R; Gibbon, J.D; Gisselfält, M, On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation, Physica D, 44, 421-444, (1990) · Zbl 0702.76061 [2] Bartuccelli, M; Constantin, P; Doering, C.R; Gibbon, J.D; Gisselfält, M, Errata: hard turbulence in a finite dimensional dynamical system, Phys. lett. A, 145, 476-477, (1990) [3] Batchelor, G.K, The theory of homogeneous turbulence, (1953), Cambridge Univ. Press Cambridge · Zbl 0053.14404 [4] Bourbaki, N, Éléments de mathématique III, (), Ch. 10 · Zbl 0165.56403 [5] Bourgain, J, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part I: Schrödinger equations, Geom. funct. anal., 3, 107-156, (1993) · Zbl 0787.35097 [6] Bourgain, J, Exponential sums and nonlinear Schrödinger equations, Geom. funct. anal., 3, 157-178, (1993) · Zbl 0787.35096 [7] Cazenave, T; Weissler, F, The Cauchy problem for the critical nonlinear Schrödinger equations in H^s, Nonlinear anal. theor. meth. appl., 14, 807-836, (1990) · Zbl 0706.35127 [8] Constantin, P; Foias, C, The Navier-Stokes equations, (1988), Chicago Univ. Press Chicago [9] Doering, C.R; Gibbon, J.D; Holm, D.D; Nicolaenko, B, Low dimensional behavior in the complex Ginzburg-Landau equation, Nonlinearity, 1, 279-309, (1988) · Zbl 0655.58021 [10] Ginibre, J; Velo, G, On a class of nonlinear Schrödinger equations I: the Cauchy problem, general case, Funct. anal., 32, 1-32, (1979) · Zbl 0396.35028 [11] Glassey, R, On the blowing-up of solutions to the Cauchy problem for the nonlinear Schrödinger equation, J. math. phys., 18, 1794-1797, (1977) · Zbl 0372.35009 [12] Goldman, D, () [13] Goldman, M, Plasma turbulence, Rev. mod. phys., 56, 709-735, (1984) [14] Kavian, O, A remark on the blowing-up of solutions to the Cauchy problem for nonlinear Schrödinger equations, Trans. am. math. soc., 299, 193-203, (1987) · Zbl 0638.35043 [15] Leray, J, Sur le mouvement d’un fluide visqueux emplissant l’espace, Acta math., 53, 193-248, (1934) · JFM 60.0726.05 [16] Luce, B, () [17] Merle, F, Limit behavior of saturated approximations of nonlinear Schrödinger equation, Commun. math. phys., 146, 377-414, (1992) · Zbl 0756.35094 [18] Merle, F; Tsutsumi, Y, L^2-concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power nonlinearity, J. diff. eq., 84, 205-214, (1990) · Zbl 0722.35047 [19] Rasmussen, I; Rypdal, K, Blow-up in nonlinear Schrödinger equations I - a general review, Phys. scr., 33, 498-504, (1987) · Zbl 1063.35546 [20] Schwartz, L, Functional analysis, () [21] Sirovich, L; Rodriguez, J.D, Coherent structures and chaos: a model problem, Phys. lett. A, 120, 211-214, (1988) [22] Stark, D, () [23] Tsutsumi, Y, L^2-solutions for NLS and nonlinear groups, Funkcialaj ekvacioj, 30, 115-125, (1987) · Zbl 0638.35021 [24] Weissler, F.B, Local existence for semilinear parabolic equations in L^p, Indiana U. math J., 29, 79-102, (1980) · Zbl 0443.35034 [25] Zakharov, V.E, Collapse of Langmuir waves, Sov. phys. JETP, 35, 908-914, (1972) [26] Zakharov, V.E; Sobolev, V.V; Synach, V.S, Character of the singularity and the stochastic phenomena of self-focusing, Zh. eksp. teor. fiz. pis’ma red, 14, 390-393, (1971)
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