Hokkaido Mathematical Journal

Applications of Campanato spaces with variable growth condition to the Navier-Stokes equation

Eiichi NAKAI and Tsuyoshi YONEDA

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Abstract

We give new viewpoints of Campanato spaces with variable growth condition for applications to the Navier-Stokes equation. Namely, we formulate a blowup criteria along maximum points of the 3D-Navier-Stokes flow in terms of stationary Euler flows and show that the properties of Campanato spaces with variable growth condition are very useful for this formulation, since variable growth condition can control the continuity and integrability of functions on the neighborhood at each point. Our criterion is different from the Beale-Kato-Majda type and Constantin-Fefferman type criterion. If geometric behavior of the velocity vector field near the maximum point has a kind of stationary Euler flow configuration up to a possible blowup time, then the solution can be extended to be the strong solution beyond the possible blowup time. As another application we also mention the Cauchy problem for the Navier-Stokes equation.

Article information

Source
Hokkaido Math. J., Volume 48, Number 1 (2019), 99-140.

Dates
First available in Project Euclid: 18 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.hokmj/1550480646

Digital Object Identifier
doi:10.14492/hokmj/1550480646

Mathematical Reviews number (MathSciNet)
MR3914171

Subjects
Primary: 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10] 76D03: Existence, uniqueness, and regularity theory [See also 35Q30] 76D05: Navier-Stokes equations [See also 35Q30]

Keywords
Campanato spaces with variable growth condition blowup criterion 3D Navier-Stokes equation stationary 3D Euler flow Cauchy problem

Citation

NAKAI, Eiichi; YONEDA, Tsuyoshi. Applications of Campanato spaces with variable growth condition to the Navier-Stokes equation. Hokkaido Math. J. 48 (2019), no. 1, 99--140. doi:10.14492/hokmj/1550480646. https://projecteuclid.org/euclid.hokmj/1550480646


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