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Abstract
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In this talk, I will present some recent results joint with Tao Luo and Zhouping Xin on the nonlinear asymptotic stability of the Lane-Emden solutions for spherically symmetric motions of viscous gaseous stars in the setting of vacuum free boundaries capturing the physical behavior that the sound speed is 1/2-Holder continuous across the vacuum boundary . The key issue is to establish the global-in-time regularity uniformly up to the vacuum boundary due to the difficulty caused by the degeneracy and singular behavior near the vacuum states. The main ingredients of the analysis consist of combinations of some new weighted nonlinear functionals and space-time weighted energy estimates. The constructions of these weighted nonlinear functionals and space-time weights depend crucially on the structures of the Lane-Emden solution, the balance of pressure and gravitation, and the dissipation.
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