Well-Posedness of Stochastic Continuity Equations on Riemannian Manifolds  

Well-Posedness of Stochastic Continuity Equations on Riemannian

作　　者：Luca GALIMBERTI Kenneth H.KARLSEN 

机构地区：[1]Department of Mathematical Sciences,NTNU Norwegian University of Science and Technology,N-7491 Trondheim,Norway [2]Department of mathematics,University of Oslo,P.O.Box 1053,Blindern,N-0316 Oslo,Norway

出　　处：《Chinese Annals of Mathematics,Series B》2024年第1期81-122,共42页数学年刊（B辑英文版）

基　　金：supported by the Research Council of Norway through the projects Stochastic Conservation Laws (No. 250674);(in part) Waves and Nonlinear Phenomena (No. 250070)

摘　　要：The authors analyze continuity equations with Stratonovich stochasticity,■ρ+divh[ρo(u(t,x)+∑_(i=1)^(N)a_(i)(x)w_(i)(t))]=0defined on a smooth closed Riemannian manifold M with metric h.The velocity field u is perturbed by Gaussian noise terms Wi(t),:WN(t)driven by smooth spatially dependent vector fields a1(x),...,aN(x)on M.The velocity u belongs to L_(t)^(1)W_(x)^(1,2)with divh u bounded in Lf,for p>d+2,where d is the dimension of M(they do not assume div_(h) u∈L_(t,x)^(∞)).For carefully chosen noise vector fields ai(and the number N of them),they show that the initial-value problem is well-posed in the class of weak L^(2) solutions,although the problem can be ill-posed in the deterministic case because of concentration effects.The proof of this“regularization by noise”result is based on a L^(2) estimate,which is obtained by a duality method,and a weak compactness argument.

关 键 词：Stochastic continuity equation Riemannian manifold Hyperbolic equa-tion Non-smooth velocity field Weak solution EXISTENCE UNIQUENESS 

分 类 号：O186.12[理学—数学]