| Talk_id | Date | Speaker | Title |
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17489
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Wednesday 9/11
4:10 PM
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Marcelo Disconzi, Vanderbilt University
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Rough solutions to the three-dimensional compressible Euler equations with vorticity and entropy
- Marcelo Disconzi, Vanderbilt University
- Rough solutions to the three-dimensional compressible Euler equations with vorticity and entropy
- 09/11/2019
- 4:10 PM - 5:00 PM
- C304 Wells Hall
We prove a series of intimately related results tied to the regularity and geometry of solutions to the three-dimensional compressible Euler equations.
The solutions are allowed to have nontrivial vorticity and entropy, and an arbitrary equation of state with positive sound speed. The central theme is that under low regularity assumptions on the initial data, it is possible to avoid, at least for short times, the formation of shocks. Our main result is that the time of classical existence can be controlled under low regularity assumptions on the part of the initial data associated with propagation of sound waves in the fluid. Such low regularity assumptions are in fact optimal. To implement our approach, we derive several results of independent interest: (i) sharp estimates for the acoustic geometry, which in particular capture how the vorticity and entropy interact with the sound waves; (ii) Strichartz estimates for quasilinear sound waves coupled to vorticity and entropy; (iii) Schauder estimates for the transport-div-curl-part of the systems. Compared to previous works on low regularity, the main new feature of our result is that the quasilinear PDE system under study exhibit multiple speeds of propagation. In fact, this is the first result of its kind for a system with multiple characteristic speeds. An interesting feature of our proof is the use of techniques that originated in the study of the vacuum Einstein equations in general relativity.
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19649
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Thursday 9/26
1:00 PM
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A. Volberg/P. Mozolyako
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Unexpected combinatorial properties of all planar measures
- A. Volberg/P. Mozolyako
- Unexpected combinatorial properties of all planar measures
- 09/26/2019
- 1:00 PM - 1:50 PM
- C517 Wells Hall
We will start with paraproducts--operators used in PDE to prove Leibniz rule with fractional derivatives. Then we move to bi-parameter paraproducts and prove the property from the title.
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19670
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Thursday 10/10
1:00 PM
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Pavel Mozolyako, MSU
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Embedding on bi-tree
- Pavel Mozolyako, MSU
- Embedding on bi-tree
- 10/10/2019
- 1:00 PM - 1:50 PM
- C517 Wells Hall
No abstract available.
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20694
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Friday 11/1
1:00 PM
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Victor Lie, Purdue University
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A unified approach to three themes in harmonic analysis
- Victor Lie, Purdue University
- A unified approach to three themes in harmonic analysis
- 11/01/2019
- 1:00 PM - 2:00 PM
- C304 Wells Hall
Linked Abstract
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19645
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Wednesday 11/6
4:10 PM
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Ioakeim Ampatzoglou, University of Texas, Austin
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Derivation of a ternary Boltzmann system
- Ioakeim Ampatzoglou, University of Texas, Austin
- Derivation of a ternary Boltzmann system
- 11/06/2019
- 4:10 PM - 5:00 PM
- C304 Wells Hall
In this talk work we present a rigorous derivation of a new kinetic equation describing the limiting behavior of a classical system of particles with three particle instantaneous interactions, which are modeled using
a non-symmetric version of a ternary distance. The equation, which we call ternary Boltzmann equation, can be understood as a step towards modeling a dense gas in non-equilibrium. This is a joint work with Natasa Pavlović.
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20703
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Wednesday 11/13
4:10 PM
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Farhan Abedin, Michigan State University
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Regularity results for a class of Kolmogorov-Fokker-Planck equations in non-divergence form
- Farhan Abedin, Michigan State University
- Regularity results for a class of Kolmogorov-Fokker-Planck equations in non-divergence form
- 11/13/2019
- 4:10 PM - 5:00 PM
- C304 Wells Hall
The Kolmogorov-Fokker-Planck equation is a degenerate parabolic equation arising in models of gas dynamics from kinetic theory. The operator is of the form
$$\mathcal{L}_Au := \mathrm{tr}(A(v,y,t) D^2_v u) + v \cdot \nabla_yu - \partial_tu,$$ where $$u(v,y,t): \mathbb{R}^{2d+1} \to \mathbb{R} \text{ and } 0 < \lambda \mathbb{I}_d \leq A \leq \Lambda \mathbb{I}_d.$$
It is an open problem if non-negative solutions of $\mathcal{L}_A u = 0$ in $\mathbb{R}^{2d+1}$ satisfy a scale-invariant Harnack inequality, assuming the matrix coefficient $A$ is merely bounded and measurable. I will discuss recent joint work with Giulio Tralli in which progress is made on partially solving this problem.
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20714
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Thursday 11/21
1:00 PM
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Leonardo Abbrescia, MSU
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Stability of traveling planewave solutions to Lorentzian vanishing mean curvature flow
- Leonardo Abbrescia, MSU
- Stability of traveling planewave solutions to Lorentzian vanishing mean curvature flow
- 11/21/2019
- 1:00 PM - 1:50 PM
- C517 Wells Hall
Lorentzian minimal submanifolds of Minkowski space are the dynamic analogue of minimal surfaces in the elliptic regime. They are defined by the vanishing of mean curvature, which can be expressed as a system of geometric PDEs. With the requirement that the submanifold be Lorentzian, that is, that the induced metric is Lorentzian, the equations have a hyperbolic nature. Consequently, the natural approach to study them is via the Cauchy initial value problem. In this talk we discuss stability properties of traveling planewave solutions to these equations, and highlight the difficulties introduced by the "infinite energy" planewave background. This is joint work with Willie Wong.
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