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APDE Seminar

Talks earlier this year

List of speakers (in alphabetical order)
  1. Serena Dipierro, Boundary behaviour of nonlocal minimal surfaces, Monday, 22 February 2021.
  2. Rupert Frank, Blow-up of solutions of critical elliptic equations in three dimensions , Monday, 10 May 2021.
  3. François Hamel, Symmetry properties for the Euler equations and semilinear elliptic equations, Monday, 12 April 2021.
  4. In-Jee Jung, Ill-posedness for incompressible fluid models at critical Sobolev regularity, Monday, 26 April 2021.
  5. Yoshiyuki Kagei, Stability and bifurcation analysis of the compressible Navier-Stokes equations, Monday, 8 March 2021.
  6. Shuang Miao, On the free boundary hard phase fluid in Minkowski space-time, Monday, 15 February 2021.
  7. Aaron Palmer, Hidden Convexity in problem of Nonlinear Elasticity, Monday, 22 March 2021.
  8. Mitsuru Sugimoto, A constructive approach to semilinear wave equations, Monday, 25 January 2021.
  9. Emmanuel Trélat, Spectral analysis of sub-Riemannian Laplacians and Weyl measure, 8, February 2021.
  10. Rongchan Zhu, Large \(N\) Limit of the \(O(N)\) Linear Sigma Model via Stochastic Quantization, Monday, 25 January 2021.

See also the Talks in 2020 page.


Monday, 10 May 2021.
Rupert Frank
Professor @ Caltech University, United States, and @ Ludwig Maximilian University of Munich, Germany.
He defended his PhD thesis in 2007 @ the Royal Institute of Technology in Stockholm under the supervision of Ari Laptev. After a post-doctoral internship and assistant professorship @ Princeton, in 2013 he became professor @ Caltech and in 2016 at LMU Munich.

Slides to the talk

Blow-up of solutions of critical elliptic equations in three dimensions

We describe the asymptotic behavior of positive solutions \(u_\varepsilon\) of the equation \[-\Delta u + au = 3\,u^{5-\varepsilon}\qquad \textrm{in \(\Omega\subseteq\,\mathbb{R}^3\)}\] with a homogeneous Dirichlet boundary condition. The function \(a\) is assumed to be critical in the sense of Hebey and Vaugon and the functions \(u_\varepsilon\) are assumed to be an optimizing sequence for the Sobolev inequality. Under a natural nondegeneracy assumption we derive the exact rate of the blow-up and the location of the concentration point, thereby proving a conjecture of Brézis and Peletier (1989). Similar results are also obtained for solutions of the equation \[-\Delta u + (a+\varepsilon V) u = 3\,u^5\qquad\textrm{ in \(\Omega\).}\] For the variational problem corresponding to the latter problem we also obtain precise energy asymptotics and a detailed description of the blow-up behavior of almost minimizers (but not necessarily minimizers or solutions).

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Monday, 26 April 2021.
In-Jee Jung
Assistant Professor @ Seoul National University, South Korea.
Professor Jung received his PhD in 2017 from Princeton University, United States, under the supervision of Professor Yakov Sinai. He became a Research Fellow @ the Korean Institute of Advanced Study (KIAS) in 2017, where he was promoted to KIAS Fellow in 2019. Since March 2023, Jung is Assistant Professor @ Seoul National University.

Slides to the talk

Ill-posedness for incompressible fluid models at critical Sobolev regularity

We consider the incompressible fluid equations including the Euler and SQG equations in critical Sobolev spaces, which are Sobolev spaces with the same scaling as the Lipschitz norm of the velocity. We show that initial value problem for the equations are ill-posed at critial regularity.

Such an ill-posedness result can be used to prove enhanced dissipation for the dissipative counterpart. The proof relied on the Key Lemma of Kiselev and Sverak, which allows to compute the main term of the velocity gradient.

This is based on joint works with Tarek Elgindi, Tsuyoshi Yoneda, and Junha Kim.

Video of the talk not yet released.

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Monday, 22 February 2021.
François Hamel
Professor @ Aix-Marseille Université, France
Professor Hamel received his PhD in 1996 @ Sorbonne University (Ex-Paris VI) under the supervision of Henri Berestycki. Before he had the privilege to be a student @ the École Normale Supçrieure de Paris (ENS), later to be a researcher @ CNRS from 1995 until 2001. Since 2001, François Hamel is professor @ Aix-Marseille Université. He was a Visiting professor @ MIT, United States and member of the Institut Universitaire de France from 2009 until 2014.

Slides to the talk

Symmetry properties for the Euler equations and semilinear elliptic equations

In this talk, I will discuss radial and one-dimensional symmetry properties for the stationary incompressible Euler equations in dimension 2 and some related semilinear elliptic equations. I will show that a steady flow of an ideal incompressible fluid with no stagnation point and tangential boundary conditions in an annulus is a circular flow. The same conclusion holds in complements of disks as well as in punctured disks and in the punctured plane, with some suitable conditions at infinity or at the origin. I will also discuss the case of parallel flows in two-dimensional strips, in the half-plane and in the whole plane.

The proofs are based on the study of the geometric properties of the streamlines of the flow and on radial and one-dimensional symmetry results for the solutions of some elliptic equations satisfied by the stream function.

The talk in based on joint work with N. Nadirashvili.

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Serena Dipierro
Professor and Head of the Department of Mathematics and Statistics @ the University of Western Australia.
Serena Dipierro took her PhD in Mathematical Analysis @ the International School for Advanced Studies (SISSA, Trieste) in 2012. After PostDoc positions @ the Universidad de Chile and University of Edinburgh, and a Humboldt Fellowship, she held permanent positions @ the University of Melbourne and the Università di Milano. In August 2018 she moved @ the University of Western Australia, where she is now Professor and Head of the Department of Mathematics and Statistics.

Slides to the talk

Boundary behaviour of nonlocal minimal surfaces

In this talk we present a peculiar behaviour of nonlocal minimal surfaces (i.e. local minimisers of a nonlocal perimeter functional), namely the capacity, and the strong tendency, of adhering to the boundary of the reference domain. This characteristic is in contrast not only with the boundary behaviour of classical minimal surfaces but also with the pattern produced by solutions of linear equations. We will discuss this phenomenon and present some recent results.

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Monday, 8 March 2021.
Yoshiyuki Kagei
Professor @ Tokyo Institute of Technology.
Professor Kagei received his Doctor of Science from Hiroshima University in 1994. He was research fellow of the Alexander von Humboldt Foundation from 1997 until 1998 @ the University of Bayreuth, Germany. In 1998, Kagei becam Associate Professor @ Kyushu University in Japan, where he got promoted to Professor in 2006. Since 2019, Kagei is Professor @ the Tokyo Institute of Technology. He was awarded in 1998 with the prestigeous MSJ Takebe Katahiro Prize of the Mathematical Society of Japan and in 2012 with the Analysis Prize of the Mathematical Society of Japan.
Stability and bifurcation analysis of the compressible Navier-Stokes equations

The compressible Navier-Stokes equation, which is the basic equation for compressible viscous fluids, is classified as a quasi-linear hyperbolic-parabolic system. Due to the hyperbolic and parabolic aspects of the sysytem, solutions exhbit iinteresting behavior. In this talk, I will review the stability and bifurcation analysis of the compressible Navier-Stokes equation, and state some recent results on a related bifurcation phenomenon and singular limit problem.

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Monday, 15 February 2021.
Shuang Miao
Professor @ Wuhan University, China.
Professor Miao received his PhD in July 2013 @ the University of Chinese Academy of Sciences under the supervision of Prof. Demetrios Christodoulou and Prof. Ping Zhang. Before he accepted the professorship @ Wuhan University, he held several postdoc positions @ University of Michigan, @ ETH Zurich, and @ EPFL in Switzland.

Slides to the talk

On the free boundary hard phase fluid in Minkowski space-time

I will present a recent work on the free boundary hard phase fluid model with Minkowski background. The hard phase model is an idealized model for a relativistic fluid where the sound speed approaches the speed of light. This work consists of two results: First, we prove the well-posedness of this model in Sobolev spaces. Second, we give a rigorous justification of the non-relativistic limit for this model as the speed of light approaches infinity.

This is joint work with Sohrab Shahshahani and Sijue Wu.

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Monday, 22 March 2021.
Aaron Zeff Palmer
Postdoctoral Fellow @ University of British Columbia, Canada.
Palmer received his PhD in mathematics @ Cornell University in 2016 under the supervision of Tim Healey on a topic of nonlinear elasticity. Currently, he is working as a postdoctoral fellow @ the University of British Columbia jointly with Nassif Ghoussoub and Young-Heon Kim on topics related to optimal transport.

Slides to the talk

Hidden Convexity in problem of Nonlinear Elasticity

Problems in elasticity / solid mechanics have been essential to the development of the theory of partial differential equations and Calculus of Variations. However, fundamental questions regarding the regularity of equilibria in nonlinear elasticity remain largely unanswered. The theory of optimal transport has had a lot of recent success in the analysis of nonlinear PDE, ranging from the Monge-Ampere equation to kinetic gas equations.

In a recent work, we introduced techniques from optimal transport to problems of nonlinear elasticity and uncover a hidden convexity. I will discuss the connection between optimal transport and nonlinear elasticity, and our results on when equilibria in nonlinear elasticity correspond to minimizers of an optimal transport problem.

This is joint work with Nassif Ghoussoub and Hugo Lavenant.

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Monday, 25 January 2021.
Mitsuru Sugimoto
Professor @ Nagoya University, Japan
Professor Sugimoto was a Research Associate @ University of Tsukuba in 1987-90 and received his Ph.D. there in 1992. He worked @ Osaka University from 1990 until 2008, and is now a Professor @ Nagoya University since 2008.

Slides to the talk

A constructive approach to semilinear wave equations

In this talk, I will explain a new attempt to construct self-similar solutions to semilinear wave equations with power nonlinearity. The existence of self-similar solutions to the same equations has been already established by Pecher (2000), Kato-Ozawa (2003), etc. based on the standard fixed point theorem. We will rediscuss it by a constructive approach using the theory of hypergeometric differential equations.

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Monday, 8 February 2021.
Emmanuel Trélat
Head of the Laboratoire Jacques-Louis Lions and Professor @ Sarbonne Université, CNRS, Université de Paris, France
Emmanuel Trélat got his PhD in 2000 in optimal control and sub-Riemannian geometry, under the supervision of Bernard Bonnard, at Dijon University, France. He became an assistant professor at Paris-Sud University (Orsay) in 2001, then professor at Orléans University in 2006, and he moved to Pierre et Marie Curie University (now named Sorbonne University) at Paris in 2011. He is the head of Laboratoire Jacques-Louis Lions.

Slides to the talk

Spectral analysis of sub-Riemannian Laplacians and Weyl measure

In collaboraton with Yves Colin de Verdière and Luc Hillairet, we study spectral properties of sub-Riemannian Laplacians, which are selfadjoint hypoelliptic operators satisfying the Hörmander condition.

Thanks to the knowledge of the small-time asymptotics of heat kernels in a neighborhood of the diagonal, we establish the local and microlocal Weyl law.

When the Lie bracket configuration is regular enough (equiregular case), the Weyl law resembles that of the Riemannian case. But in the singular case (e.g., Baouendi-Grushin, MarVnet) the Wey law reveals much more complexity. In turn, we derive quantum ergodicity properties in some sub-Riemannian cases.

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Monday, 15 March 2021.
Rongchan Zhu
Professor @ the Beijing Institute of Technology.
Rongchan Zhu got her Phd in 2021 @ Chinese Academy of Science and Bielefeld University. Now, she is professor @ the Beijing Institute of Technology.

Slides to the talk

Large \(N\) Limit of the \(O(N)\) Linear Sigma Model via Stochastic Quantization

In this talk, I will discuss large \(N\) limits of a coupled system of \(N\) interacting \(\Phi^4\) equations posed over \(\mathbb{T}^{d}\) for \(d=1,2,3\), known as the \(O(N)\) linear sigma model. Uniform in \(N\) bounds on the dynamics are established, allowing us to show convergence to a mean-field singular SPDE, also proved to be globally well-posed. Moreover, I show tightness of the invariant measures in the large \(N\) limit. For large enough mass, they converge to the (massive) Gaussian free field, the unique invariant measure of the mean-field dynamics, at a rate of order \(1/\sqrt{N}\) with respect to the Wasserstein distance.

I will also consider fluctuations and obtain tightness results for certain \(O(N)\) invariant observables, along with an exact description of the limiting correlations in \(d=1,2\).

This talk is based on joint work with Hao Shen, Scott Smith and Xiangchan Zhu.

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