On the Stochastic Strichartz Estimates and the Stochastic Nonlinear Schrödinger Equation on a Compact Riemannian Manifold
On the Stochastic Strichartz Estimates and the Stochastic Nonlinear Schrödinger Equation on a Compact Riemannian Manifold
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DOI：
10.1007/s11118-013-9369-2
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发表时间：
2012-09
影响因子：
1.1
通讯作者：
Z. Brzeźniak;A. Millet
中科院分区：
文献类型：
--
作者：
Z. Brzeźniak;A. Millet
证明了二维紧致黎曼流形上随机非线性方程组解的存在唯一性。因此，我们推广（并改进）了Burq等人最近的工作（J Nonlinear Math Phys 10（1）：12-27，2003）以及de Bouard和Debussche的一系列论文，例如de Bouard和Debussche（Commun Math Phys 205（1）：161-181，1999和Stoch Anal Appl 21（1）：97-126，2003），他们在平坦欧氏空间的情况下研究了类似的问题。证明了紧致维黎曼流形上带乘性噪声的随机非线性薛定谔方程局部极大解的存在唯一性。在噪声更规则的情况下，证明了当非线性项为散焦型或聚焦型，d= 2，初始数据属于有限能量空间时，解是全局的。我们的证明是基于改进的随机Eschenhartz不等式。
We prove the existence and the uniqueness of a solution to the stochastic NSLEs on a two-dimensional compact riemannian manifold. Thus we generalize (and improve) a recent work by Burq et al. (J Nonlinear Math Phys 10(1):12–27, 2003) and a series of papers by de Bouard and Debussche, see e.g. de Bouard and Debussche (Commun Math Phys 205(1):161–181, 1999 and Stoch Anal Appl 21(1):97–126, 2003) who have examined similar questions in the case of the flat euclidean space. We prove the existence and the uniqueness of a local maximal solution to stochastic nonlinear Schrödinger equations with multiplicative noise on a compactd-dimensional riemannian manifold. Under more regularity on the noise, we prove that the solution is global when the nonlinearity is of defocusing or of focusing type,d= 2 and the initial data belongs to the finite energy space. Our proof is based on improved stochastic Strichartz inequalities.