Abstract
Time-dependent free surface problem for the incompressible Navier–Stokes equations which describes the motion of viscous incompressible fluid nearly half-space are considered. We obtain global well-posedness of the problem for a small initial data in scale invariant critical Besov spaces. Our proof is based on maximal $L^{1}$-regularity of the corresponding Stokes problem in the half-space and special structures of the quasi-linear term appearing from the Lagrangian transform of the coordinate.
Funding Statement
The first author is partially supported by JSPS grant-in-aid for Scientific Research (S) #19H05597 and Challenging Research (Pioneering) #20K20284. The second author is partially supported by JSPS grant-in-aid for Scientific Research (B) #16H03945 (B) #21H00992 and by JSPS Fostering Joint Research Program (B) #18KK0072.
Citation
Takayoshi OGAWA. Senjo SHIMIZU. "Maximal $L^{1}$-regularity and free boundary problems for the incompressible Navier–Stokes equations in critical spaces." J. Math. Soc. Japan 76 (2) 593 - 672, April, 2024. https://doi.org/10.2969/jmsj/88288828
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