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We will discuss recent progress on the vacuum free boundary problems arising in the dynamics of isolated gases with or without gravity. We give an overview of the well-posedness and stability theory, and present some new results on waiting time solutions.
Vacuum Free Boundary Problems in Gas Dynamics Sponsored by the Meyer Fund
Fri, Apr. 11 3:35pm (MATH 2…
Willie Wong (Michigan State University)
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Our understanding of cosmological processes, like many other predictions of physical theories, are based on studying regimes where the equations of motion reduce to a finite dimensional dynamical system. An example of a conclusion derived from such reductions is the idea of a big bang cosmology in general relativity. Such reductions are physically justified by the working assumption that when viewed from the largest scales, the inhomogeneities average out and the matter content can be approximated by a homogeneous compressible fluid. Jointly with Shih-Fang Yeh, we probe whether this working assumption is justified mathematically. Our results show that on the cosmological timescale, some big bang solutions are susceptible to instabilities generated through nonlinear self-interactions of the constituent matter when inhomogeneities are present. The goal of this talk is to present the mathematical context of this result and briefly describe the mechanism driving the instability, focusing on the relevance of the conformal (or causal) geometry of the big bang solutions. (No prior familiarity with mathematical relativity is assumed.)
Some Big Bangs are Unstable Sponsored by the Meyer Fund
Fri, Apr. 25 3:35pm (MATH 2…
Maja Taskovic (Emory University)
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The wave kinetic equation is one of the fundamental models in the theory of wave turbulence, and provides a statistical description of weakly nonlinear interacting waves.
This talk will address the global in time well-posedness of the spatially inhomogeneous wave kinetic equation by applying techniques inspired by the analysis of the Boltzmann equation – another model of statistical mechanics that describes evolution of rarefied gases in which particles undergo predominantly binary interactions.
We will also discuss the well-posedness of the wave kinetic hierarchy – an infinite system of coupled equations closely related to the wave kinetic equation. Two essential tools for obtaining these results are the Hewitt-Savage theorem, which allows us to lift the existence result for the equation to the hierarchy, and the Klainerman-Machedon board game argument, which allows us to control the factorial growth of the Dyson series and consequently prove uniqueness of solutions.
This is a joint work with Ioakeim Ampatzoglou, Joseph K. Miller and Natasa Pavlovic.
On the inhomogeneous wave kinetic equation and its associated hierarchy Sponsored by the Meyer Fund
Vacuum Free Boundary Problems in Gas Dynamics