Nakai, Eiichi; Yoneda, Tsuyoshi Applications of Campanato spaces with variable growth condition to the Navier-Stokes equation. (English) Zbl 1425.46021 Hokkaido Math. J. 48, No. 1, 99-140 (2019). Summary: 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 the 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. Cited in 4 Documents MSC: 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 35Q30 Navier-Stokes equations 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 76D05 Navier-Stokes equations for incompressible viscous fluids 46N20 Applications of functional analysis to differential and integral equations Keywords:Campanato spaces with variable growth condition; blowup criterion; 3D Navier-Stokes equation; stationary 3D Euler flow; Cauchy problem PDFBibTeX XMLCite \textit{E. Nakai} and \textit{T. Yoneda}, Hokkaido Math. J. 48, No. 1, 99--140 (2019; Zbl 1425.46021) Full Text: DOI Euclid