Abstract
The authors analyze continuity equations with Stratonovich stochasticity,∂ρ+divh[ρo(u(t,x)+N∑i=1ai(x)(W)i(t))]=0 defined on a smooth closed Riemannian manifold M with metric h.The velocity field u is perturbed by Gaussian noise terms W1(t),…,WN(t)driven by smooth spatially dependent vector fields a1(x),…,aN(x)on M.The velocity u belongs to L1tWx,2 with divh u bounded in Lpt,x for p＞d+2,where d is the dimension of M(they do not assume divh 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 L2 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 L2 estimate,which is obtained by a duality method,and a weak compactness argument.
基金项目
Research Council of Norway through the projects Stochastic Conservation Laws(250674)
Waves and Nonlinear Phenomena(250070)
维普
万方数据