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Well-Posedness of Stochastic Continuity Equations on Riemannian Manifolds

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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.

Stochastic continuity equationRiemannian manifoldHyperbolic equa-tionNon-smooth velocity fieldWeak solutionExistenceUniqueness

Luca GALIMBERTI、Kenneth H.KARLSEN

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Department of Mathematical Sciences,NTNU Norwegian University of Science and Technology,N-7491 Trondheim,Norway

Department of mathematics,University of Oslo,P.O.Box 1053,Blindern,N-0316 Oslo,Norway

Research Council of Norway through the projects Stochastic Conservation LawsWaves and Nonlinear Phenomena

250674250070

2024

10.1007/s11401-024-0005-9

数学年刊B辑(英文版)
国家教育部委托复旦大学主办

数学年刊B辑(英文版)

CSTPCD
影响因子：0.129
ISSN：0252-9599
年,卷(期)：2024.45(1)
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