TY - JOUR

T1 - Dynamical behavior for the solutions of the Navier-Stokes equation

AU - Li, Kuijie

AU - Ozawa, Tohru

AU - Wang, Baoxiang

N1 - Funding Information:
by the National Science Foundation of China, grants 11271023 and 11771024. The second and third named authors were supported in part by Mathematics and Physics Unit “Multiscale Analysis Moddelling and Simulation”, Top Global University Project, Waseda University and the second author was also supported in part by JSPS Japanese-German Graduate Externship. Part of the paper was carried out when the first named author was visiting Laboratoire J. A. Dieudonnéand he is grateful to Professor Fabrice Planchon for his valuable suggestions, comments and warm hospitality and also the support from the China Scholarship Council. The authors are grateful to the reviewers for their careful reading to the manuscript and for their enlightening suggestions to the paper.

PY - 2018/7

Y1 - 2018/7

N2 - We study several quantitative properties of solutions to the incompressible Navier-Stokes equation in three and higher dimensions: (Equation Presented) More precisely, for the blow up mild solutions with initial data in L∞(ℝRd) and Hd/2-1(ℝd), we obtain a concentration phenomenon and blowup profile decomposition respectively. On the other hand, if the Fourier support has the form supp u0 ⊂ {ξ ∈ ℝn : ξ1 ≥ L} and ||u0||∞ ≪ L for some L > 0, then (1) has a unique global solution u ∈ C(ℝR+;L∞). In 3D, we show the compactness of the set consisting of minimal-Lp singularity-generating initial data with 3 < p < 1, furthermore, if the mild solution with data in Lp(ℝ3) blows up in a Type-I manner, we prove the existence of a blowup solution which is uniformly bounded in critical Besov spaces B-1+6/pp/2,∞ (ℝ3).

AB - We study several quantitative properties of solutions to the incompressible Navier-Stokes equation in three and higher dimensions: (Equation Presented) More precisely, for the blow up mild solutions with initial data in L∞(ℝRd) and Hd/2-1(ℝd), we obtain a concentration phenomenon and blowup profile decomposition respectively. On the other hand, if the Fourier support has the form supp u0 ⊂ {ξ ∈ ℝn : ξ1 ≥ L} and ||u0||∞ ≪ L for some L > 0, then (1) has a unique global solution u ∈ C(ℝR+;L∞). In 3D, we show the compactness of the set consisting of minimal-Lp singularity-generating initial data with 3 < p < 1, furthermore, if the mild solution with data in Lp(ℝ3) blows up in a Type-I manner, we prove the existence of a blowup solution which is uniformly bounded in critical Besov spaces B-1+6/pp/2,∞ (ℝ3).

KW - Blowup profile

KW - Concentration phenomena

KW - L-minimal singularity-generating data

KW - Navier-stokes equation

KW - Type-I blowup solution

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U2 - 10.3934/cpaa.2018073

DO - 10.3934/cpaa.2018073

M3 - Article

AN - SCOPUS:85045313237

VL - 17

SP - 1511

EP - 1560

JO - Communications on Pure and Applied Analysis

JF - Communications on Pure and Applied Analysis

SN - 1534-0392

IS - 4

ER -