TY - JOUR

T1 - Global well-posedness for the incompressible Navier–Stokes equations in the critical Besov space under the Lagrangian coordinates

AU - Ogawa, Takayoshi

AU - Shimizu, Senjo

N1 - Funding Information:
The authors would like to express their thank to Professor Masashi Misawa for his useful suggestion on the null-Lagrangian structure. The work of the first author is partially supported by JSPS Grant-in-aid for Scientific Research S # 19H05597 and Challenging Research (Pioneering) # 20K20284 . The work of the second author is partially supported by JSPS Grant-in-aid for Scientific Research B # 16H03945 and Fostering Joint Research B # 18KK0072 . The authors declare no conflicts of interest.
Publisher Copyright:
© 2020 Elsevier Inc.

PY - 2021/2/15

Y1 - 2021/2/15

N2 - We consider global well-posedness of the Cauchy problem of the incompressible Navier–Stokes equations under the Lagrangian coordinates in scaling critical Besov spaces. We prove the system is globally well-posed in the homogeneous Besov space B˙p,1−1+n/p(Rn) with 1≤p<∞. The former result was restricted for 1≤p<2n and the main reason why the well-posedness space is enlarged is that the quasi-linear part of the system has a special feature called a multiple divergence structure and the bilinear estimate for the nonlinear terms are improved by such a structure. Our result indicates that the Navier–Stokes equations can be transferred from the Eulerian coordinates to the Lagrangian coordinates even for the solution in the limiting critical Besov spaces.

AB - We consider global well-posedness of the Cauchy problem of the incompressible Navier–Stokes equations under the Lagrangian coordinates in scaling critical Besov spaces. We prove the system is globally well-posed in the homogeneous Besov space B˙p,1−1+n/p(Rn) with 1≤p<∞. The former result was restricted for 1≤p<2n and the main reason why the well-posedness space is enlarged is that the quasi-linear part of the system has a special feature called a multiple divergence structure and the bilinear estimate for the nonlinear terms are improved by such a structure. Our result indicates that the Navier–Stokes equations can be transferred from the Eulerian coordinates to the Lagrangian coordinates even for the solution in the limiting critical Besov spaces.

KW - Critical Besov space

KW - End-point

KW - Global well-posedness

KW - Lagrangian coordinates

KW - Maximal L regularity

KW - Navier–Stokes equations

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U2 - 10.1016/j.jde.2020.10.023

DO - 10.1016/j.jde.2020.10.023

M3 - Article

AN - SCOPUS:85096399636

SN - 0022-0396

VL - 274

SP - 613

EP - 651

JO - Journal of Differential Equations

JF - Journal of Differential Equations

ER -