Abstract
The authors study the compressible limit of the nonlinear Schrödinger equation with different-degree small parameter nonlinearities in small time for initial data with Sobolev regularity before the formation of singularities in the limit system. On the one hand, the existence and uniqueness of the classical solution are proved for the dispersive perturbation of the quasi-linear symmetric system corresponding to the initial value problem of the above nonlinear Schrödinger equation. On the other hand, in the limit system, it is shown that the density converges to the solution of the compressible Euler equation and the validity of the WKB expansion is justified.
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Project supported by the National Natural Science Foundation of China (Nos. 10801102, 10771151), the Sichuan Youth Sciences and Technology Foundation (No. 07ZQ026-009) and the China Postdoctoral Science Foundation.
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Gan, Z., Guo, B. Compressible limit of the nonlinear Schrödinger equation with different-degree small parameter nonlinearities. Chin. Ann. Math. Ser. B 32, 105–122 (2011). https://doi.org/10.1007/s11401-010-0620-5
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DOI: https://doi.org/10.1007/s11401-010-0620-5