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

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. [ABSTRACT FROM AUTHOR]