zbMATH — the first resource for mathematics

A singular perturbation problem in a system of nonlinear Schrödinger equation occurring in Langmuir turbulence. (English) Zbl 0961.76096
Summary: We prove the existence of limit for \(\alpha\to +\infty\) in the system \(i\partial_t E+\nabla (\nabla\cdot E) -\alpha^2 \nabla\times \nabla \times E=-|E|^{2\sigma}E\), where \(E:\mathbb{R}^3 \to\mathbb{C}^3\). This corresponds to an approximation which is made in the context of Langmuir turbulence in plasma physics. The \(L^2\)-subcritical \(\sigma\) (that is \(\sigma \leq 2/3)\) and the \(H^1\)-subcritical \(\sigma\) (that is \(\sigma\leq 2)\) are studied. In the physical case \(\sigma=1\), the limit is studied in the \(H^1 (\mathbb{R}^3)\) norm.

76X05 Ionized gas flow in electromagnetic fields; plasmic flow
35Q55 NLS equations (nonlinear Schrödinger equations)
76F99 Turbulence
Full Text: DOI Link EuDML
[1] L. Bergé and T. Colin, A singular perturbation problem for an envelope equation in plasma physics. Physica D84 (1995) 437-459. · Zbl 1194.82092
[2] T. Colin, On the Cauchy problem for a nonlocal, nonlinear Schrödinger equation occurring in plasma Physics. Differential and Integral Equations6 (1993) 1431-1450. Zbl0780.35104 · Zbl 0780.35104
[3] R.O. Dendy, Plasma dynamics. Oxford University Press, New York (1990).
[4] J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. Parts I, II. J. Funct. Anal.32 (1979) 1-32, 33-71; Part III Ann. Inst. H. Poincaré A28 (1978) 287-316. · Zbl 0396.35028
[5] J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation revisited. Ann. Inst. H. Poincaré Anal Non Linéaire2 (1985) 309-402. Zbl0586.35042 · Zbl 0586.35042
[6] E.M. Stein, Singular Integrals and Differentiability properties of Functions. Princeton University Press, Princeton, New Jersey (1970). · Zbl 0207.13501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.