Speaker: Jiqiang Zheng(Institute for Applied Physics and Computational Mathematics)
Time：14:00-15:00 June 25 ，2020
Title：Laplacian operator with Hardy potential and applications to nonlinear dispersive equations
Abstract： In this talk, we first discuss the Sobolev space theory and harmonic analysis tools(such as Littlewood-Paley theory) for the Laplacian operator associated with Hardy potential. And then we consider the energy-critical nonlinear wave equation in the presence of an inverse-square potential in dimensions three and four. In the defocusing case, we prove that arbitrary initial data in the energy space lead to global solutions that scatter. In the focusing case, we prove scattering below the ground state threshold. This talk is based on a series of joint works with Rowan Killip, Changxing Miao, Jason Murphy, Monica Visan and Junyong Zhang.
Speaker: Yong Wang (CAS)
Time：15:15-16:15 June 25 ，2020
Title：Global Solutions of the Compressible Euler Equations with Large Initial Data of Spherical Symmetry and Positive Far-Field Density
Abstract：We are concerned with the global existence theory for spherically symmetric solutions of the multidimensional compressible Euler equations with large initial data of positive far-field density so that the total initial-energy is unbounded. The central feature of the solutions is the strengthening of waves as they move radially inward toward the origin. For the large initial data of positive far-field density, various examples have shown that the spherically symmetric solutions of the Euler equations blow up near the origin at certain time. A fundamental unsolved problem is whether the density of the global solution would form concentration to become a measure near the origin for the case when the total initial-energy is unbounded. Another longstanding problem is whether a rigorous proof could be provided for the inviscid limit of the multidimensional compressible Navier-Stokes to Euler equations with large initial data. In this paper, we establish a global existence theory for spherically symmetric solutions of the compressible Euler equations with large initial data of positive far-field density and relative finite-energy. This is achieved by developing a new approach via adapting a class of degenerate density-dependent viscosity terms, so that a rigorous proof of the vanishing viscosity limit of global weak solutions of the Navier-Stokes equations with the density-dependent viscosity terms to the corresponding global solution of the Euler equations with large initial data of spherical symmetry and positive far-field density can be obtained. One of our main observations is that the adapted class of degenerate density-dependent viscosity terms not only includes the viscosity terms for the Navier-Stokes equations for shallow water (Saint Venant) flows but also, more importantly, is suitable to achieve our key objective of this paper. These results indicate that concentration is not formed in the vanishing viscosity limit even when the total initial-energy is unbounded, though the density may blow up near the origin at certain time. The talk is based on a joint work with Gui-Qiang Chen.
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