Well-posedness and scattering for nonlinear Schrödinger equations with a derivative nonlinearity at the scaling critical regularity (Q2816486)
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scientific article; zbMATH DE number 6618812
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Well-posedness and scattering for nonlinear Schrödinger equations with a derivative nonlinearity at the scaling critical regularity |
scientific article; zbMATH DE number 6618812 |
Statements
23 August 2016
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Schrödinger equation
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well-posedness
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Cauchy problem
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scaling critical
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multilinear estimate
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bounded \(p\)-variation
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Well-posedness and scattering for nonlinear Schrödinger equations with a derivative nonlinearity at the scaling critical regularity (English)
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The purpose of this paper is to study the NLS equation NEWLINE\[NEWLINE\begin{cases} (i\partial_t+\triangle)u=\partial_k(\overline{u}^m),\;(t,x)\in(0,\infty)\times \mathbb{R}^d\\ u(0,x)=u_0(x),\;x\in\mathbb{R}^d.\end{cases}NEWLINE\]NEWLINE The main result states that (1) is globally well-posed, and has a unique solution in a certain Banach space. The proofs use Plancherel's theorem, Sobolev embedding, Cauchy-Schwartz and Hölder inequalities.
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