Existence and uniqueness of weak solutions of the compressible spherically symmetric Navier–Stokes equations
Existence and uniqueness of weak solutions of the compressible spherically symmetric Navier–Stokes equations
复制标题
可压缩球对称纳维斯托克斯方程弱解的存在唯一性
DOI：
10.1016/j.jde.2016.10.013
复制
发表时间：
2017
影响因子：
2.4
通讯作者：
Xiangdi Huang
中科院分区：
文献类型：
--
作者：
Xiangdi Huang
Riesz变换是谐波分析中最有影响力的基本工具之一。它将Lp函数映射为Lp函数，对任意p∈（1，∞），它在奇异算子中起着重要作用.作为流体力学中的一个应用，当p∈（1，∞）时，证明了<$div u <$Lp与<$div u <$Lp +<$curl u <$Lp之间的范数等价性.然而，由于Riesz算子只将有界函数发送到BMO函数，因此没有希望用<$div u <$L∞+<$curl u <$L∞来绑定<$<$u <$L∞。正如霍夫（2006）[11]所指出的，这是获得等熵可压缩流弱解唯一性的主要障碍。幸运的是，基于新的观测，见引理2.2，我们对任何N维径向对称向量函数u导出了一个精确的估计：作为一个直接应用，我们对有界球中可压缩球对称流的某些弱解的唯一性公开问题给出了肯定的回答。
One of the most influential fundamental tools in harmonic analysis is the Riesz transforms. It maps L p functions to L p functions for any p∈(1,∞) which plays an important role in singular operators. As an application in fluid dynamics, the norm equivalence between‖∇ u‖ L p and‖ div u‖ L p+‖ curl u‖ L p is well established for p∈(1,∞). However, since Riesz operators sent bounded functions only to BMO functions, there is no hope to bound‖∇ u‖ L∞ in terms of‖ div u‖ L∞+‖ curl u‖ 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)‖ div u‖ 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.