Norm Inflation for Generalized Navier-Stokes Equations

被引:0
作者
Cheskidov, Alexey [1 ]
Dai, Mimi [2 ]
机构
[1] Univ Illinois, Dept Math Stat & Comp Sci, Chicago, IL 60607 USA
[2] Univ Colorado, Dept Appl Math, Boulder, CO 80303 USA
关键词
Fractional Navier-Stokes equation; norm inflation; Besov spaces; interactions of plane waves;
D O I
暂无
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We consider the incompressible Navier-Stokes equation with a fractional power alpha is an element of E [1, infinity) of the Laplacian in the three-dimensional case. We prove the existence of a smooth solution with arbitrarily small initial data in (B) over dot (-alpha)(infinity,p) (2 < p <= infinity) that becomes arbitrarily large in (B) over dot (-s)(infinity,infinity) for all s > 0 in arbitrarily small time. This extends the result of Bourgain and Pavlovic [1] for the classical Navier-Stokes equation, a result which uses the fact that the energy transfer to low modes increases norms with negative smoothness indexes. It is remarkable that the space (B) over dot (-alpha)(infinity,infinity) is supercritical for alpha > 1. Moreover, the norm inflation occurs even in the case alpha >= 5/4 where the global regularity is known.
引用
收藏
页码:869 / 884
页数:16
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