Existence and uniqueness of weak solutions of the compressible spherically symmetric Navier-Stokes equations

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• 作者：Xiangdi Huang ^{xdhuang@amss.ac.cn}
• 关键词：35Q30 ; 76N10
• 刊名：Journal of Differential Equations
• 出版年：2017
• 出版时间：5 February 2017
• 年：2017
• 卷：262
• 期：3
• 页码：1341-1358
• 全文大小：282 K
• 卷排序：262

文摘

One of the most influential fundamental tools in harmonic analysis is the Riesz transforms. It maps LpLp functions to LpLp functions for any p∈(1,∞)p∈(1,∞) which plays an important role in singular operators. As an application in fluid dynamics, the norm equivalence between ‖∇u‖_{Lp‖∇u‖Lp and ‖divu‖Lp+‖curlu‖Lp‖divu‖Lp+‖curlu‖Lp is well established for p∈(1,∞)p∈(1,∞). However, since Riesz operators sent bounded functions only to BMO functions, there is no hope to bound ‖∇u‖L∞‖∇u‖L∞ in terms of ‖divu‖L∞+‖curlu‖L∞‖divu‖L∞+‖curlu‖L∞. As pointed out by Hoff (2006) [11], this is the main obstacle to obtain uniqueness of weak solutions for isentropic compressible flows.Fortunately, based on new observations, see Lemma 2.2, we derive an exact estimate for ‖∇u‖L∞≤(2+1/N)‖divu‖L∞‖∇u‖L∞≤(2+1/N)‖divu‖L∞ for any N-dimensional radially symmetric vector functions u. As a direct application, we give an affirmative answer to the open problem of uniqueness of some weak solutions to the compressible spherically symmetric flows in a bounded ball.}