GLOBAL WELL-POSEDNESS OF THE VISCOUS CAMASSA-HOLM EQUATION WITH GRADIENT NOISE

被引:10
|
作者
Holden, Helge [1 ]
Karlsen, Kenneth H. [2 ]
Pang, Peter H. C. [1 ,2 ]
机构
[1] NTNU Norwegian Univ Sci & Technol, Dept Math Sci, NO-7491 Trondheim, Norway
[2] Univ Oslo, Dept Math, NO-0316 Oslo, Norway
关键词
Shallow water equation; viscous Camassa-Holm equation; stochastic perturbation; convective noise; existence; Faedo-Galerkin method; compactness; tightness; Skorokhod-Jakubowski representation; uniqueness; commutator estimate; NAVIER-STOKES EQUATIONS; DIFFERENTIAL-EQUATIONS; WAVE-EQUATIONS; WEAK SOLUTIONS; CONSERVATIVE SOLUTIONS; PATHWISE SOLUTIONS; MARTINGALE; EXISTENCE; BREAKING; VALUES;
D O I
10.3934/dcds.2022163
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
We analyse a nonlinear stochastic partial differential equation that corresponds to a viscous shallow water equation (of the Camassa-Holm type) perturbed by a convective, position-dependent noise term. We establish the existence of weak solutions in Hm (m is an element of N) using Galerkin approximations and the stochastic compactness method. We derive a series of a priori esti-mates that combine a model-specific energy law with non-standard regularity estimates. We make systematic use of a stochastic Gronwall inequality and also stopping time techniques. The proof of convergence to a solution argues via tightness of the laws of the Galerkin solutions, and Skorokhod-Jakubowski a.s. representations of random variables in quasi-Polish spaces. The spatially dependent noise function constitutes a complication throughout the analysis, repeatedly giving rise to nonlinear terms that "balance" the martingale part of the equation against the second-order Stratonovich-to-Ito correction term. Fi-nally, via pathwise uniqueness, we conclude that the constructed solutions are probabilistically strong. The uniqueness proof is based on a finite-dimensional Ito formula and a DiPerna-Lions type regularisation procedure, where the reg-ularisation errors are controlled by first and second order commutators.
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页码:568 / 618
页数:51
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