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.
Publisher Copyright:
© 2018 American Institute of Mathematical Sciences. All rights reserved.
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
SN - 1534-0392
VL - 17
SP - 1511
EP - 1560
JO - Communications on Pure and Applied Analysis
JF - Communications on Pure and Applied Analysis
IS - 4
ER -