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On the Local Pressure Expansion for the Navier–Stokes Equations

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Abstract

We show that the pressure associated with a distributional solution of the Navier–Stokes equations on the whole space satisfies a local expansion defined as a distribution if and only if the solution is mild. This gives a new perspective on Lemarié-Rieusset’s “equivalence theorem.” Here, the Leray projection operator composed with a gradient is defined without using the Littlewood-Paley decomposition. Prior sufficient conditions for the local expansion assumed spatial decay or estimates on the gradient and imply the considered solution is mild. An important tool is an explicit description of the bounded mean oscillation solution to a Poisson equation, which we examine in detail. As applications we include an improvement of a uniqueness criteria by the authors in Morrey spaces and revisit a proof of a regularity criteria in dynamically restricted local Morrey spaces due to Grujić and Xu.

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Notes

  1. It is in fact sufficient to have \(u\rightarrow u_0\) in an \(L^1_\mathrm {loc}\) sense if we also have \(\limsup _{t\rightarrow 0^+} \Vert u(t)\Vert _{L^2(K)}<\infty \) for every compact K.

  2. This conclusion also appears in [28, Lemma 11.3]. We include the details for completeness.

  3. These assumptions are satisfied if \(u\in L^\infty ([0,T_0);L^\infty )\) is a strong, mild solution as in [15].

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Acknowledgements

The research of Bradshaw was partially supported by the Simons Foundation. The research of Tsai was partially supported by NSERC grant RGPIN-2018-04137.

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Authors and Affiliations

  1. Department of Mathematics, University of Arkansas, Fayetteville, AR, 72701, USA

    Zachary Bradshaw

  2. Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

    Tai-Peng Tsai

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  1. Zachary Bradshaw
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  2. Tai-Peng Tsai
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Correspondence to Zachary Bradshaw.

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Communicated by G. P. Galdi

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Bradshaw, Z., Tsai, TP. On the Local Pressure Expansion for the Navier–Stokes Equations. J. Math. Fluid Mech. 24, 3 (2022). https://doi.org/10.1007/s00021-021-00637-4

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