The global well-posedness of the compressible fluid model of Korteweg type for the critical case
The global well-posedness of the compressible fluid model of Korteweg type for the critical case
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DOI：
10.57262/die034-0506-245
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发表时间：
2020-09
影响因子：
1.4
通讯作者：
Takayuki Kobayashi;M. Murata
中科院分区：
文献类型：
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作者：
Takayuki Kobayashi;M. Murata
本文考虑了临界情况下的Korteweg型可压缩流体模型，在此临界情况下，压力的导数等于0。证明了该方程组在最大L_p$-$L_q$正则类中存在唯一的全局强解。作为结果，我们还证明了非线性问题解的衰减估计。为了得到临界情形下的整体适定性，我们在一个附加的低频假设下证明了线性化方程解的L_p-L_q衰减性质.
In this paper, we consider the compressible fluid model of Korteweg type in a critical case where the derivative of pressure equals to $0$ at the given constant state. It is shown that the system admits a unique, global strong solution for small initial data in the maximal $L_p$-$L_q$ regularity class. As a result, we also prove the decay estimates of the solutions to the nonliner problem. In order to obtain the global well-posedness for the critical case, we show $L_p$-$L_q$ decay properties of solutions to the linearized equations under an additional assumption for a low frequencies.