2023-2024 Seminars | The Ohio State University Partial Differential Equations Seminar (2024)

Zoom Link for the Spring Semester 2023

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https://osu.zoom.us/j/99158238170?pwd=S1I0MCsrczcrRW1qUUF1SmRaZVV2UT09

Meeting ID: 991 5823 8170

Password: 314159

Abstracts

Fanze Kong. University of British Columbia

Title: Existence and Stability of Localized Patterns in the Population Models with Large Advection and Strong Allee Effect.

Abstract: The strong Allee effect plays an important role on the evolution of population in ecological systems. One important concept is the Allee threshold that determines the persistence or extinction of the population in a long time. In general, a small initial population size is harmful to the survival of a species since when the initial data is below the Allee threshold, the population tends to extinction, rather than persistence. Another interesting feature of population evolution is that a species whose movement strategy follows a conditional dispersal strategy is more likely to persist.

To study the interaction between Allee effect and the biased movement strategy, we mainly consider the pattern formation and local dynamics for a class of single species population models that is subject to the strong Allee effect. We first rigorously show the existence of multiple localized solutions when the directed movement is strong enough. Next, the spectrum analysis of the associated linear eigenvalue problem is established and used to investigate the stability properties of these interior spikes. This analysis proves that there exist not only unstable but also linear stable steady states. Finally, we extend results of the single equation to coupled systems for two interacting species, each with different advective terms, and competing for the same resources. We also construct several non-constant steady states and analyze their stability.

Idriss Mazari. Paris Dauphine.

Title: regarding the maximisation of the total biomass

Abstract: In this talk, we survey the qualitative conclusions of several recent studies concerning spatial heterogeneity in population dynamics. Using the heterogeneous logistic-diffusive equation as a paradigmatic model, we investigate the optimal location of resources in a domain: how should resources be spread inside a domain in order to optimise certain criteria, in particular the total biomass? Motivating our analysis by a study of the optimal survival ability, we shall present some qualitative properties recently obtained that show surprising phenomena. The talk will be mostly descriptive, and is based on collaborations with Grégoire Nadin, Yannick Privat and Domènec Ruiz-i-Balet.

Brian Reyes, Department of Mathematics, University of Notre Dame (October 17, 2023)

Title: The Cauchy problem of the modified $b$-family

Abstract: We consider the Cauchy problem of the modified $b$-family of equations and study its well-posedness in Sobolev and analytic spaces. Using bilinear estimates for estimating the nonlinearity in Bourgain spaces, we show that this equation is locally well-posed in Sobolev spaces $H^{s}$ for $s>-\frac{3}{4}$. Furthermore, we show local well-posedness for data in analytic spaces $G^{\delta, s}$, for $s>-\frac{3}{4}$ and $\delta>0$.

Finally, for $b=3$ (the Degasperis-Procesi case) we show that the local solutions are global, and study the evolution of the uniform radius of analyticity. We will talk about the relationship between the radius of analyticity $\delta$ and the critical Sobolev exponent $s_c=-\frac{3}{4}$.

Bradley Lucier, Department of Mathematics, Purdue University (October 24, 2023)

Title: High order regularity for discontinuous solutions to conservation laws

Abstract: Solutions to hyperbolic conservation laws—time dependent, first order, partial differential equations—can become discontinuous in finite time even with smooth initial data. So-called entropy weak solutions for scalar problems were formalized around 1970. It was shown that if the initial data has bounded variation—first-order smoothness in $L_1$—then the solution has bounded variation for all positive time.

Beginning in the late 1980s, a number of papers working in one space dimension showed that newly developed connections between nonlinear approximation theory and Besov function spaces lead to higher-order smoothness results, with smoothness order greater than 1. The catch is that this smoothness is measured in nonconvex spaces, like $L_q(\Bbb R)$ for $0<q<1$.

In this talk I will attempt to give relevant examples and describe the ideas behind these results. Much of the talk should be accessible to early graduate students.

Jingrui Cheng, Department of Mathematics, Stony Brook (October 31, 2023)

Title:Interior W^{2,p} estimates for complex Monge-Ampere equations

Abstract:The classical estimate by Caffarelli shows that a strictly convex solution to the real Monge-Ampere equations has W^{2,p} regularity if the right hand side is close to a constant. We partially generalize this result to the complex version, when the underlying solution is close to a smooth strictly plurisubharmonic function. The additional assumption we impose is related to the lack of Pogorelov type estimate in the complex case. The talk is based on joint work with Yulun Xu.

Rossana Capuani, Department of Mathematics, University of Arizona (November 7, 2023)

Title:First orderMean Field Games with State Constraints

Abstract:This talk will address deterministic mean field games for which agents are restricted in a closed domain with smooth boundary.In this case, the existence and uniqueness of Nash equilibria cannot be deduced as for unrestricted state space because, for a large set of initial conditions, the uniqueness of solutions to the minimization problem which is solved by each agent is no longer guaranteed. Therefore we attack the problem by considering a relaxed version of it, for which the existence of equilibria can be proved by set-valued fixed point arguments. Finally, by analyzing the regularity and sensitivity with respect to space variables of the relaxed solution, we will show that it satisfies the Mean Field Games system in a suitable point-wise sense.

Jiaxin Jin, Department of Mathematics, The Ohio State University (November 14, 2023)

Title: Nonlinear Asymptotic Stability of 3D Relativistic Vlasov-Poisson systems

Abstract: Motivated by solar wind models in the low altitude, we explore a boundary problem of the nonlinear relativistic Vlasov-Poisson systems in the 3D half space in the presence of a constant vertical magnetic field and strong background gravity with the inflow boundary condition. As the main result, we construct stationary solutions and establish their nonlinear dynamical asymptotic stability.

Qiliang Wu, Department of Mathematics, Ohio University (Mar 7, 2024)

Title: Weak Diffusive Stability Induced by High-order Spectral Degeneracies

Abstract: The Lyapunov stability of equilibria in dynamical systems is determined by the interplay between the linearization and nonlinear terms. In this talk, we present our recent results on the case when the spectrum of the linearization is diffusively stable with high-order spectral degeneracy at the origin. Roll solutions at the zigzag boundary of the Swift-Hohenberg equation are shown to be nonlinearly stable, serving as examples that linear decays weaker than the classical diffusive decay, together with quadratic nonlinearity, still give nonlinear stability of spatially periodic patterns. The study is conducted on two physical domains: the 2D plane and the infinite 2D torus. Linear analysis reveals that, instead of the classical $t^{-1}$ diffusive decay rate, small perturbations of zigzag stable roll solutions decay with slower algebraic rates ($t^{-3/4}$ for the 2D plane; $t^{-1/4}$ for the infinite 2D torus) due to the high-order degeneracy of the translational mode at the origin in the Bloch-Fourier spaces. The nonlinear stability proofs are based on decompositions of the neutral translational mode and the faster decaying modes, and fixed-point arguments, demonstrating the irrelevancy of the nonlinear terms.

Arnd Scheel, Department of Mathematics, University of Minnesota (Mar 7, 2024)

Title: Patterning and self-organization beyond Turing: from myxobacteria to flatworms

Abstract: Turing’s idea that diffusion differences between chemical species can drive pattern formation and select wavelengths has been a central building block for the modeling of patterns arising in chemistry and biology, from simple tabletop chemistry such as the CIMA reaction to morphogenesis and the formation of presomites.
I will report on two studies of pattern formation that invoke pattern selection mechanisms quite different from Turing’s. In the first example, I will show how simple nonlinear run-and-tumble dynamics can reproduce complex functionality from equidistribution over rippling to fruiting body formation in myxobacteria colonies [arXiv:1805.11903,arXiv:1609.05741]. In the second example, I will describe simple models for the astounding ability of planarian flatworms to regenerate completely from small fragments of body tissue, preserving polarity (that is, position of head versus tail) in the recovery [arXiv:1908.04253].

Son Tu, Department of Mathematics, Michigan State University (April 9, 2024)

Title: Properties of the effective Hamiltonian and hom*ogenization of the Hamilton-Jacobi equation

Abstract: The hom*ogenization of Hamilton-Jacobi equations in the periodic setting was pioneered by Lions, Papanicolaou, and Varadhan in the 1980s. However, understanding the intricate properties of the effective Hamiltonian and its impact on the hom*ogenization limit across various settings remains a substantial open problem. In this presentation, we will discuss recent developments that highlight a connection between this phenomenon and weak KAM theory on a qualitative level, as well as present some quantitative results.