Abstract:Â 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 Â the wave turbulence theory, by justifying the derivation of the wave kinetic equation, the fundamental equation of this subject.
Abstract:Â 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.
Abstract:Â In this talk, I hope to describe elements of proving a certain stable singularity formation result for the Einstein-Euler system,Â 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.
Abstract:Â 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.
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: Â Does it always slow down the adoption? Can it be neglected?Â
Abstract:Â We propose a new method for establishing the convergence rates of solutions to reaction-diffusion equations to traveling waves.Â 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'' Â for the shape defect functionÂ and the true front location of the solution.Â Curiously, the convergence to a traveling wave has a pulled nature, regardless of whether the travelingÂ wave itself is of pushed, pulled, or pushmi-pullyu type.Â 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.
Abstract:Â We prove the existence of nonlinear sound waves,Â which are smooth, time periodic, oscillatory solutions to theÂ compressible Euler equations, in one space dimension.Â In the mid-19thÂ 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.Â We prove that for generic entropy profiles,Â the fully nonlinear compressible Euler equations support perturbations of the linearized solutions for every frequency.Â This shows thatÂ Riemann's result is aÂ 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.
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 curveÂ 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.
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. Â 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.
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.
Abstract:Â 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.
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