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Abstract
We prove the global existence of weak solutions to the Navier-Stokesequations of compressible heat-conducting fluids in two spatialdimensions with initial data and external forces which are large andspherically symmetric. The solutions will be obtainedas the limit of the approximate solutions in an annular domain.We first derive a number of regularity results on theapproximate physical quantities in the ``fluid region'', as well asthe new uniform integrability of the velocity and temperature in the entirespace-time domain by exploiting the theory of the Orlicz spaces.By virtue of these a priori estimates we then argue in amanner similar to that in [Arch. Rational Mech. Anal. 173 (2004),297-343] to pass to the limit and show that the limiting functions areindeed a weak solution which satisfies the mass and momentumequations in the entire space-time domain in the sense ofdistributions, and the energy equation in any compactsubset of the ``fluid region''.
Highlights
The two-dimensional Navier-Stokes equations of compressible heat-conducting fluids express the conservation of mass, and the balance of momentum and energy, which can be written as follows in Eulerian coordinates
The main purpose of this paper is to prove the global existence of weak solutions to the problem (1.1)–(1.5) when the initial data and external forces are large and spherically symmetric
The analysis shows that r(t) may be positive, and that, if it is, a vacuum state of radius r(t) centered at the origin
Summary
The two-dimensional Navier-Stokes equations of compressible heat-conducting fluids express the conservation of mass, and the balance of momentum and energy, which can be written as follows in Eulerian coordinates. By exploiting the theory of the Orlicz spaces we are able to derive some new uniform global integrability of the approximate solutions (cf Lemma 2.4) to show that (1.2) holds in the entire space-time domain in the weak sense (i.e., (d) of Theorem 1.1) This method can be applied to the screw pinch model with positive constant heat-conduction coefficient in [17] and the cylindrically symmetric rotating model of (1.1)–(1.3) (that is, in the symmetric equations (6)–(10) in [9], we take u = v, w = 0, f1 = f2 and f3 = 0) to obtain similar results. The global existence of a solution for large data in the non-isentropic case needs further study
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