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Global-in-time probabilistically strong solutions to stochastic power-law equations: Existence and non-uniqueness

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  • Lü, Huaxiang
  • Zhu, Xiangchan

Abstract

We are concerned with the power-law fluids driven by an additive stochastic forcing in dimension d⩾3. For the power index r∈(1,3d+2d+2), we establish existence of infinitely many global-in-time probabilistically strong and analytically weak solutions in Llocp([0,∞);L2)∩C([0,∞);W1,max{1,r−1}),p⩾1 for every divergence free initial condition in L2∩W1,max{1,r−1}. This result in particular implies non-uniqueness in law. Our result is sharp in the three dimensional case in the sense that the solution is unique if r⩾3d+2d+2.

Suggested Citation

  • Lü, Huaxiang & Zhu, Xiangchan, 2023. "Global-in-time probabilistically strong solutions to stochastic power-law equations: Existence and non-uniqueness," Stochastic Processes and their Applications, Elsevier, vol. 164(C), pages 62-98.
    • Handle: RePEc:eee:spapps:v:164:y:2023:i:c:p:62-98
    DOI: 10.1016/j.spa.2023.06.014
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    References listed on IDEAS

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    1. Yamazaki, Kazuo, 2022. "Remarks on the non-uniqueness in law of the Navier–Stokes equations up to the J.-L. Lions’ exponent," Stochastic Processes and their Applications, Elsevier, vol. 147(C), pages 226-269.
    2. Röckner, Michael & Zhu, Rongchan & Zhu, Xiangchan, 2014. "Local existence and non-explosion of solutions for stochastic fractional partial differential equations driven by multiplicative noise," Stochastic Processes and their Applications, Elsevier, vol. 124(5), pages 1974-2002.
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