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Well-Posedness of Stochastic Continuity Equations on Riemannian Manifolds

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Abstract

The authors analyze continuity equations with Stratonovich stochasticity,

$$\partial \rho + {\rm{di}}{{\rm{v}}_h}\left[ {\rho \circ \left( {u(t,x) + \sum\limits_{i = 1}^N {{a_i}(x){{\dot W}_i}(t)} } \right)} \right] = 0$$

defined on a smooth closed Riemannian manifold M with metric h. The velocity field u is perturbed by Gaussian noise terms Ẇ1(t), …, ẆN(t) driven by smooth spatially dependent vector fields a1(x), …, aN(x) on M. The velocity u belongs to L 1t W 1,2x with divhu bounded in L pt,x for p > d + 2, where d is the dimension of M (they do not assume divhu ∈ 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.

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Authors and Affiliations

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

    Luca Galimberti

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

    Kenneth H. Karlsen

Authors
  1. Luca Galimberti
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  2. Kenneth H. Karlsen
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Correspondence to Luca Galimberti or Kenneth H. Karlsen.

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Conflicts of interest The authors declare no conflicts of interest.

Additional information

This work was supported by the Research Council of Norway through the projects Stochastic Conservation Laws (No. 250674) and (in part) Waves and Nonlinear Phenomena (No. 250070).

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Galimberti, L., Karlsen, K.H. Well-Posedness of Stochastic Continuity Equations on Riemannian Manifolds. Chin. Ann. Math. Ser. B 45, 81–122 (2024). https://doi.org/10.1007/s11401-024-0005-9

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  • DOI: https://doi.org/10.1007/s11401-024-0005-9

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