Time global existence and finite time blow-up criterion for solutions to the Keller-Segel system coupled with the Navier-Stokes fluid
Time global existence and finite time blow-up criterion for solutions to the Keller-Segel system coupled with the Navier-Stokes fluid
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DOI：
10.1016/j.jde.2019.05.035
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发表时间：
2019-10
影响因子：
2.4
通讯作者：
H. Kozono;M. Miura;Y. Sugiyama
中科院分区：
文献类型：
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作者：
H. Kozono;M. Miura;Y. Sugiyama
我们将处理Navier-Stokes流体（即不可压缩粘性流体）作用下的趋化性模型。我们将证明在尺度不变类中存在一个对于大初始数据的局部温和解和一个对于小初始数据的全局温和解，证明n 0∈l1 （r2）和u 0∈L σ 2 （r2）。我们的方法是基于线性化的扰动和热半群的L p−L q估计。作为我们方法的副产品，我们将证明我们温和解的平滑效果和唯一性。此外，我们将给出一个爆破判据，该判据几乎覆盖了流体运动静止状态下尺寸为‖n 0‖l1 （r2）的众所周知的阈值数8π。此外，还将讨论爆破率。
We will deal with the chemotaxis model under the effect of the Navier-Stokes fluid, ie, the incompressible viscous fluid. We shall show the existence of a local mild solution for large initial data and a global mild solution for small initial data in the scale invariant class demonstrating that n 0∈ L 1 (R 2) and u 0∈ L σ 2 (R 2). Our method is based on the perturbation of linearization together with the L p− L q-estimates of the heat semigroup. As a by-product of our method, we shall prove the smoothing effect and uniqueness of our mild solution. In addition, we shall show a blow-up criterion which almost covers the well-known threshold number 8π of the size‖ n 0‖ L 1 (R 2) under the rest state of the fluid motion. Furthermore, the blow-up rate will be also discussed.