PDE-Applied Math Archives for Fall 2024 to Spring 2025
Recent progress on mathematical wave turbulence
When: Thu, September 7, 2023 - 3:30pmWhere: Kirwan Hall MTH1311
Speaker: Yu Deng (USC) https://sites.google.com/usc.edu/yudeng/
Abstract: The theory of wave turbulence, which started in the 1920s as the wave analog of Boltzmann's kinetic theory, has now become an active field in mathematical physics with substantial applications in science. In this talk I will review some recent works, joint with Zaher Hani, that establish the rigorous mathematical foundation of the wave turbulence theory, by justifying the derivation of the wave kinetic equation, the fundamental equation of this subject.
Invasion: robustness and universality
When: Thu, September 14, 2023 - 3:30pmWhere: EGR3102
Speaker: Cole Graham (Brown University) https://colegraham.net/
Abstract: The world teems with examples of invasion, in which one steady state spatially invades another. Invasion can even display a universal character: fine details recur in seemingly unrelated systems. Reaction-diffusion equations provide a mathematical framework for these phenomena. In this talk, I will discuss recent examples of robust and universal invasion patterns in reaction-diffusion equations, with a focus on multiple dimensions.
Formation of shocks for the Einstein-Euler system
When: Thu, September 21, 2023 - 3:30pmWhere: MTH1311
Speaker: John Anderson (Stanford) https://web.stanford.edu/~jrlander/
Abstract: In this talk, I hope to describe elements of proving a certain stable singularity formation result for the Einstein-Euler system, which is the topic of work in progress with Jonathan Luk. I'll first describe where this fits into the big picture of the study of multidimensional shocks, and why it is appropriate to call this a shock formation result. Then, I will try to describe some of the main ideas that go into proving shock formation, and the main difficulty in the case of Einstein-Euler. In a nutshell, the difficulty arises from the fact that the speed of sound is less than the speed of light. In the remaining time, I will describe how this is related to shocks for other hyperbolic PDEs arising in continuum mechanics.
Dynamics of kink clusters for scalar fields in dimension 1+1
When: Thu, September 28, 2023 - 3:30pmWhere: MTH1311
Speaker: Andrew Lawrie (MIT) https://math.mit.edu/~alawrie/
Abstract: I will present joint work with Jacek Jendrej. We consider classical scalar fields in dimension 1+1 with a symmetric double-well self-interaction potential. Examples of such equations are the phi-4 model and the sine-Gordon equation. Such equations admit non-trivial static solutions called kinks and antikinks. A kink cluster is a solution approaching, for large positive times, a superposition of alternating kinks and antikinks whose velocities converge to zero and mutual distances grow to infinity. Our main result is a determination of the leading order asymptotic behavior of any kink cluster. Our results are partially inspired by the notion of "parabolic motions" in the Newtonian n-body problem. We explain this analogy and its limitations. We also explain the role of kink clusters as universal profiles for the formation/annihilation of multikink configurations.
Effects of Network Structure on Spreading of Innovations
When: Thu, October 5, 2023 - 3:30pmWhere: MTH1311
Speaker: Gadi Fibich http://www.math.tau.ac.il/~fibich/
Abstract: Spreading (diffusion) of new products is a classical problem.Traditionally, it has been analyzed using the compartmental Bass model, which implicitly assumes that all individuals are homogeneous and connected to each other. To relax these assumptions, research has gradually shifted to the more fundamentalBass model on networks, which is an agent-based model for the stochastic adoption decision of each individual. In this talk, I will present the emerging mathematical theory for the Bass model on networks. The main focus will be on the effect of network structure. For example, which networks yield the slowest and fastest adoption? I will also discuss the effect of heterogeneity among individuals: Does it always slow down the adoption? Can it be neglected?
Convergence rates to traveling waves
When: Thu, October 12, 2023 - 3:30pmWhere: MTH1311
Speaker: Lenya Ryzhik http://math.stanford.edu/~ryzhik/
Abstract: We propose a new method for establishing the convergence rates of solutions to reaction-diffusion equations to traveling waves. The analysis is based on the study of the traveling wave shape defect function. It turns out that the convergence rate is controlled by the distance between the ``phantom front location'' for the shape defect function
and the true front location of the solution. Curiously, the convergence to a traveling wave has a pulled nature, regardless of whether the traveling wave itself is of pushed, pulled, or pushmi-pullyu type. In addition to providing new results, this approach simplifies dramatically the proof in the Fisher-KPP case and gives a unified, succinct explanation for the known algebraic rates of convergence in the Fisher-KPP case and the exponential rates in the pushed case. This is a joint work with J. An and C. Henderson.
The nonlinear theory of sound
When: Thu, October 19, 2023 - 3:30pmWhere: MTH1311
Speaker: Robin Yong (UMass Amherst)
Abstract: We prove the existence of nonlinear sound waves, which are smooth, time periodic, oscillatory solutions to the compressible Euler equations, in one space dimension. In the mid-19th century, Riemann proved that compressions always form shocks in the simpler isentropic system, which is inconsistent with sound wave solutions of the (linear) wave equation. We prove that for generic entropy profiles, the fully nonlinear compressible Euler equations support perturbations of the linearized solutions for every frequency. This shows that Riemann's result is a highly degenerate special case, and brings the mathematics of the compressible Euler equations back into line with two centuries of verified Acoustics technology.
This is joint work with Blake Temple.
Regularity of Bubbles in Porous Media
When: Thu, October 26, 2023 - 3:30pmWhere: MTH1311
Speaker: Neel Patel
Abstract: The free boundary problem for fluids in porous media is known as the Muskat problem. The well-posedness of this PDE system for an infinite free boundary is dependent on the Rayleigh-Taylor condition. When the fluid velocity is changed by gravity, then this condition implies stability when the denser fluid is below the boundary. However, in the case of a closed curve boundary, or a bubble, this condition always fails to hold. In this talk, we will study the effect of surface tension and gravity on the stability and regularity of fluid bubbles in porous media.
Parabolic equations and diffusion processes with degeneration: boundary problems, metastability, and homogenization
When: Thu, November 2, 2023 - 3:15amWhere: MTH1311
Speaker: Leonid Koralov (UMD)
Abstract: Stated in PDE terms, the problems concern the asymptotic behavior of solutions to
parabolic equations whose coefficients degenerate at the boundary of a domain.
The operator may be regularized by adding a small diffusion term. Metastability effects arise in this case: the
asymptotics of solutions, as the size of the perturbation tends to zero, depends on the time scale.
Initial-boundary value problems with both the Dirichlet and the Neumann boundary conditions are considered.
We also consider periodic homogenization for operators with degeneration.
The talk is based on joint work with M. Freidlin.
To Flutter or Not: Mathematical Aeroelasticity
When: Thu, November 16, 2023 - 3:30pmWhere: MTH1311
Speaker: Justin Webster (UMBC) http://webster.math.umbc.edu/
Abstract: When a thin elastic structure is immersed in a fluid flow, certain conditions may bring about excitations in the structure. That is, the dynamic loading of the fluid feeds back with the natural oscillatory modes of the structure. In this case we have a bounded-response instability, and the oscillatory behavior may persist until the flow velocity changes or energy is dissipated from the structure. This interactive phenomenon is referred to as flutter. Beyond the obvious applications in aeroscience (projectile paneling and flaps, flags, and airfoils), the flutter phenomenon arises in: (i) the biomedical realm (in treating sleep apnea), and (ii) sustainable energies (in providing a low-cost power generating mechanisms). Modeling, predicting, and controlling flutter have been foremost problems in engineering for nearly 70 years.In this talk we describe the basics of modeling flutter in the simplest configuration (an aircraft panel) using differential equations and dynamical systems. After discussing the partial differential equation model, we will discuss theorems that can be proved about solutions to these equations using modern analysis (e.g., monotone operator theory, the theory of global attractors, and PDE control). Specifically, we will remark on a recent result concerning the strong stabilization to equilibria for a hyperbolic system with damping on a portion of the boundary. We will relate these results back to experimental results in engineering.
Nonlocal Boundary Value Problems with Local Boundary Conditions
When: Thu, November 30, 2023 - 3:30pmWhere:
Speaker: James Scott (Columbia) https://sites.google.com/view/jamesmichaelscott/home
Abstract: We state and analyze nonlocal problems with classically-defined, local boundary conditions. The model takes its horizon parameter to be spatially dependent, vanishing near the boundary of the domain. We establish a Green's identity for the nonlocal operator that recovers the classical boundary integral, which permits the use of variational techniques. We show the existence of solutions, as well as their variational convergence to classical counterparts as the horizon uniformly converges to zero. In certain circumstances, global regularity of solutions can be established, resulting in improved modes and rates of variational convergence. We also show that Galerkin discretization schemes for the nonlocal problems converge unconditionally with respect to the nonlocal parameter, i.e. that the schemes are asymptotically compatible.
Matrix generalization of the cubic Szegő equation
When: Thu, February 8, 2024 - 3:30pmWhere: MTH1311
Speaker: Ruoci Sun https://sites.google.com/view/sun-ruoci/home
Abstract: This presentation is devoted to studying matrix solutions of the cubic Szegő equation, leading to the following matrix Szegő equation on the 1-d torus and on the real line. The matrix Szegő equation still enjoys a two-Lax-pair structure, which is slightly different from the Lax pair structure of the cubic scalar Szegő equation introduced in Gérard-Grellier [arXiv:0906.4540]. We can establish an explicit formula for general solutions both on the torus and on the real line of the matrix Szegő equation. This presentation is based on the works Sun [arXiv:2309.12136, arXiv:2310.13693].
Nearly self-similar blowup of the slightly perturbed homogeneous Landau equation with very soft potentials
When: Thu, February 22, 2024 - 3:30pmWhere: MTH1311
Speaker: Jiajie Chen (Courant) https://jiajiechen94.github.io/
Abstract: Whether the Landau equation can develop a finite time singularity is an important open problem in kinetic equations. In this talk, we will first discuss several similarities between the Landau equation and some incompressible fluids equations. Then we will focus on the slightly perturbed homogeneous Landau equation with very soft potentials, where we increase the nonlinearity from $ c(f) f$ in the Landau equation to $\alpha c(f) f$ with $\alpha>1$. For $\alpha >1$ and close to $1$, we establish finite time nearly self-similar blowup from some smooth non-negative initial data, which can be radially symmetric or non-radially symmetric. The blowup results are sharp as the homogeneous Landau equation $(\alpha=1)$ is globally well-posed, which was recently established by Guillen and Silvestre. The proof builds on our previous framework on sharp blowup results of the De Gregorio model with nearly self-similar singularity to overcome the diffusion. Our results shed light on potential singularity formation in the inhomogeneous setting.
One example of Residual Diffusivity
When: Thu, April 4, 2024 - 3:30pmWhere: MTH1311
Speaker: Gautam Iyer https://www.math.cmu.edu/~gautam/sj/index.html
Abstract: Consider a diffusive passive scalar that is advected by a periodic
incompressible flow. On long time scales it is known that the effect of
the periodic drift averages and the behavior is the same as that of a
purely diffusive scalar with an effective diffusion coefficient. In all
examples where the effective diffusivity is known it vanishes with the
molecular diffusivity.
Residual diffusivity is the remarkable pheonomenon where the effective
diffusivity does not vanish with the molecular diffusivity. It is
conjectured to happen in situations where the advecting drift is
chaotic. In this talk we provide one (time discrete) example that
exhibits residual diffusivity. Our proof is probablistic and provides a
Doeblin minorization condition that can be used to obtain a lower bound
for the effective diffusivity. We are only able to verify this condition
theoretically in one case, but believe it holds for a larger class of
examples. This is joint work with Jim Nolen.
Global regularity for critical SQG in bounded domains
When: Thu, April 18, 2024 - 3:30pmWhere: MTH3206
Speaker: Peter Constantin https://web.math.princeton.edu/~const/
Abstract: Critical SQG (surface quasi-geostrophic) equations are aequations originating from atmospheric science which are widely studied in
relation to rapid formation of small scales in fluids. In the whole
space or on the torus, the equations have been proved to have global
smooth solutions some fifteen years ago by Caffarelli-Vasseur and,
independently, by Kiselev-Nazarov-Volberg.
The problem of existence and uniqueness of global smooth solution in
bounded domains was open until now. I will present a proof of global
regularity obtained recently with Ignatova and Q-H. Nguyen. We
introduce a new methodology of transforming the single nonlocal
nonlinear evolution equation in a bounded domain into an interacting
system of extended nonlocal nonlinear evolution equations in the whole
space. The proof then uses the method of the nonlinear maximum
principle for nonlocal operators in the extended system.
Where does friction come from? Hamiltonian systems for classical and quantum particles
When: Thu, May 2, 2024 - 3:30pmWhere: MTH1311
Speaker: Thierry Goudon (INRIA, France)
Abstract: A very intuitive idea, formalized in particular by Caldeira and Leggett, consists of explaining friction as resulting from interactions with the surrounding environment: if energy is globally conserved, it is a question of understanding how it is evacuated by the environment, leading to velocity damping for the particle.
This point of view was developed by Stephan de Bièvre and his collaborators by adopting a Hamiltonian description of the particle/medium assembly.
A research program aims to extend this modeling by considering several classical or quantum particles interacting with the environment, involving couplings with Vlasov or Schrödinger type PDEs. This subject uses a wide range of mathematical techniques: asymptotic analysis, stability of dynamic systems in infinite dimension… We will present the modeling issues and some results obtained recently.
Local well-posedness and smoothing of MMT kinetic wave equation
When: Thu, May 9, 2024 - 3:30pmWhere: MTH1311
Speaker: Joonhyn La (KIAS, Korea)
Abstract: In this talk, I will prove local well-posedness of kinetic wave equation arising from MMT equation, which is introduced by Majda, Mclaughlin, and Tabak and is one of the standard toy models to study wave turbulence. Surprisingly, our result reveals a regularization effect of the collision operator, which resembles the situation of non-cutoff Boltzmann. This talk is based on a joint work with Pierre Germain (Imperial College London) and Katherine Zhiyuan Zhang (Northeastern).