Colloquia

Ahmed Bou-Rabee, Courant Institute

Monday, January 13th, 3:30 – 4:30 pm, KAP 414

Title: Homogenization with critical disorder

Abstract:

Homogenization is the approximation of a complex, “disordered” system by a simpler, “ordered” one. Picture a walker on a grid. In each step, the walker chooses to walk along a neighboring edge with equal probability. At large scales, the walker approximates Brownian motion. But what if some edges are more likely to be traversed than others?

I will discuss recent advances in the theory of quantitative homogenization which make it possible to analyze the random walk with drift and other models in statistical physics.

Joint work with Scott Armstrong and Tuomo Kuusi.

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Wojciech Ozanski, Florida State University

Monday, February 24th, 3:30 – 4:30 pm, KAP 414

Title: Ill-posedness of PDEs arising in incompressible inviscid fluid mechanics

Abstract: The 3D incompressible Euler equations are the fundamental model of inviscid incompressible fluids. The issue of regularity of solutions to the Euler equations and its relation to the emergence of turbulence remains a major open problem of fluid dynamics. In the talk we will explore these concepts from the viewpoint of recent analytic results which are concerned with ill-posedness of unique solutions at time t=0. We will discuss possible mechanisms of growth of solutions to the equations, and we will demonstrate that there exists an initial condition such that the unique solution to the 2D Euler equations admits a dramatic instantaneous loss of regularity; namely a gap loss of the order of Sobolev regularity. Moreover, we will discuss another result, where Sobolev regularity of a unique solution decreases continuously in time, and is also localized in space. Both results are the first results of these kinds in mathematical fluid mechanics, and develop new analytical tools capturing the phenomenology of inviscid and viscous fluids.

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Alexis Vasseur, University of Texas, Austin

Monday, March 3rd, 3:30 – 4:30 pm, KAP 414

Title: From Navier-Stokes to discontinuous solutions of compressible Euler

Abstract: The compressible Euler equation can lead to the emergence of shock discontinuities in finite time, notably observed behind supersonic planes. A very natural way to justify these singularities involves studying solutions as inviscid limits of Navier-Stokes solutions with evanescent viscosities. The mathematical study of this problem is however very difficult because of the destabilization effect of the viscosities. Bianchini and Bressan proved the inviscid limit to small BV solutions using the so-called artificial viscosities in 2004. However, until very recently, achieving this limit with physical viscosities remained an open question. In this presentation, we will provide the basic ideas of classical mathematical theories to compressible fluid mechanics and introduce the recent method of a-contraction with shifts. This method is employed to describe the physical inviscid limit in the context of the barotropic Euler equation, and to solve the Bianchini and Bressan conjecture in this special case. This is a joint work with Geng Chen and Moon-Jin Kang.

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Neil Siegel, USC Viterbi

CAMS Distinguished Lecturer

Monday, March 10th, 3:30 – 4:30 pm, KAP 414

Title: A theory of development for complex engineered systems

Abstract: Society depends in an essential way on modern engineered systems, but the development of such systems remains problematic: a significant majority of such efforts fail. Dr. Neil Siegel spent many years in industry “fixing” such problem engineering projects, and through that experience developed insight regarding what is a common recurring cause of such failures, and also developed a theoretical method for decreasing the probability of that recurring failure mode. In addition, he had the opportunity to apply this method to a large number of actual engineering development projects. In today’s talk, he explains the causes of such failures, his corrective theory, and some of the results of its actual application. Potential implications for future development efforts are also discussed.

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Natalia Komarova, UC San Diego

Monday, March 31st, 3:30 – 4:30 pm, KAP 414

Title: TBA

Abstract: TBA

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Dayung Koh, JPL

Monday, April 14th, 3:30 – 4:30 pm, KAP 414

Title: TBA

Abstract: TBA

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Franziska Weber, UC Berkeley

Monday, April 21st, 3:30 – 4:30 pm, KAP 414

Title: TBA

Abstract: TBA

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Bjoern Bringmann, Princeton University

Monday, April 28th, 3:30 – 4:30 pm, KAP 414

Title: TBA

Abstract: TBA

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