Global classical large solutions to 1D compressible Navier–Stokes equations with density-dependent viscosity and vacuum
Global classical large solutions to 1D compressible Navier–Stokes equations with density-dependent viscosity and vacuum
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DOI：
10.1016/j.jde.2011.05.025
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发表时间：
2011-09
影响因子：
2.4
通讯作者：
S. Ding;Huanyao Wen;Changjiang Zhu
中科院分区：
文献类型：
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作者：
S. Ding;Huanyao Wen;Changjiang Zhu
本文研究了具有大初始数据、密度相关粘性、外力和真空的一维可压缩等熵Navier-Stokes方程的初边值问题。充分利用Cho和Kim（2006）[3]中解的局部估计、方程的一维性质和Sobolev不等式，我们得到了唯一的整体古典解（ρ，u），其中ρ∈ C1（[0，T]; H1（[0，1]）），u∈ H1（[0，T]; H2（[0，1]）），对任意T> 0.辛（1998）[31]指出，当初始密度为非平凡紧支集时，Cauchy问题的光滑解（ρ，u）∈ C1（[0，T]; H3（R1））（T足够大）必然在有限时间爆破.我们得到的解的性质似乎可以改进，这促使我们借助于一个新的检验函数ρ 2 ut t得到一些新的估计，如引理3.2-3.6.这导致了（ρ，u）的进一步扩张，其中ρ∈ C1（[0，T]; H3（[0，1]）），u∈ H1（[0，T]; H3（[0，1]））.由于初边值问题在C1（[0，T]; H3（[0，1]））中的解在有限时间内爆破并不明显，因此u的正则性是否能随着真空的出现而改进为C1（[0，T]; H3（[0，1]））仍然是一个未知数.
In this paper, we investigate an initial boundary value problem for 1D compressible isentropic Navier–Stokes equations with large initial data, density-dependent viscosity, external force, and vacuum. Making full use of the local estimates of the solutions in Cho and Kim (2006)[3] and the one-dimensional properties of the equations and the Sobolev inequalities, we get a unique global classical solution (ρ, u) where ρ∈ C 1 ([0, T]; H 1 ([0, 1])) and u∈ H 1 ([0, T]; H 2 ([0, 1])) for any T> 0. As it is pointed out in Xin (1998)[31] that the smooth solution (ρ, u)∈ C 1 ([0, T]; H 3 (R 1))(T is large enough) of the Cauchy problem must blow up in finite time when the initial density is of nontrivial compact support. It seems that the regularities of the solutions we obtained can be improved, which motivates us to obtain some new estimates with the help of a new test function ρ 2 u t t, such as Lemmas 3.2–3.6. This leads to further regularities of (ρ, u) where ρ∈ C 1 ([0, T]; H 3 ([0, 1])), u∈ H 1 ([0, T]; H 3 ([0, 1])). It is still open whether the regularity of u could be improved to C 1 ([0, T]; H 3 ([0, 1])) with the appearance of vacuum, since it is not obvious that the solutions in C 1 ([0, T]; H 3 ([0, 1])) to the initial boundary value problem must blow up in finite time.