### Spring 2015

### Wednesday, February 18 3:30pm - 5:30pm at BU

Spatially localized states in pattern forming PDEs often result from bistability between a spatially homogeneous state and a spatially periodic state, and can be understood from the dynamical systems perspective. This talk focuses on the 1:1 forced complex Ginzburg-Landau equation, which is the normal form PDE for harmonically forced Hopf bifurcations in spatially extended systems.

In the 1990s it was discovered that Burgers' equation $\rho_t + \rho \rho_x = 0$ interacts nicely with stochastic initial data. This observation is due to several authors, beginning with special cases and alternative notions of solution, culminating with the 1998 work of Jean Bertoin. This paper showed that if $\rho(x,0)$ is a Levy process without positive jumps, then for fixed $t \gt 0$ the solution $\rho(x,t)$ has the same property, and gave a description for the evolution of the Laplace exponent.

### Wednesday, March 18 3:30pm - 5:30pm at Brown

It has long been observed that multi-scale systems, particularly those in climatology, exhibit behavior typical of stochastic models, most notably in the unpredictability and statistical variability of events. This is often in spite of the fact that the underlying physical model is completely deterministic.

One possible explanation for this stochastic behavior is deterministic chaotic effects. In fact, it has been well established that the statistical properties of chaotic systems can be well approximated by stochastic differential equations.

I will present some recent progress in stochastic homogenization of Hamilton-Jacobi equations. In particular, I will describe a new approach to establish homogenization via an inverse problem and describe some new results for the non-convex case. Joint work with Scott Armstrong and Yifeng Yu.

### Wednesday, April 8 3:30pm - 5:30pm at BU

We consider scalar, bistable lattice differential equations on rectangular lattices in two space dimensions. We show (under mild conditions) that front-like solutions persist when holes are introduced into the domain by removing one or more lattice points. This is joint work with Hupkes and Van Vleck.

We consider a heterogeneous interacting particle system, where each particle is subject to failure. Particles interact through mean field interactions and exogenous effects. We study the limiting behavior of the empirical measure. It turns out that the limiting empirical measure solves an SDPE. Next we study deviations of the empirical measure from its limit. We prove a weak convergence result for the fluctuation process. The limiting fluctuation process lives in a distribution space, also satisfying an SDPE.

### Wednesday, May 6 3:30pm - 5:30pm at Brown

$p$-ground states are functions that minimize a quantity known as the $p$-Rayleigh quotient. These functions satisfy a nonlinear PDE that has an eigenvalue problem interpretation. We present two schemes that provide approximations of these functions. One is based on inverse iteration for square matrices and the other is a flow that is an appropriate generalization of the heat equation. In both cases, property scaled solutions converge to $p$-ground states.

We consider solutions to the infinite depth water wave problem in 2D and 3D neglecting surface tension which are to leading order wave packets with small $O(\epsilon)$ amplitude and slow spatial decay that are balanced. Multiscale calculations formally suggest that such solutions have modulations that evolve on $O(\epsilon^{-2})$ time scales according to a version of a cubic NLS equation depending on dimension. Justifying this rigorously is a real problem, since standard existence results do not yield solutions to the water wave problem that exist for long enough to see the NLS dynamics.

### Fall 2014

### Wednesday, September 24 3:30pm at Brown

Zebrafish (Danio rerio), a small fish with black and yellow stripes, has the ability to regenerate its stripes in response to growth or artificial disturbance. We simulate past laser ablation experiments using discrete (agent-based) and continuum (non-local conservation law) models. Both models consider two cell types diffusing and reacting based on rules of short range activation and long range inhibition. Our results suggest that the radius of interaction is a key determinant of stripe formation.

Taylor dispersion, first described by G.I. Taylor in 1953, occurs when the diffusion of a solute in a pipe or channel is enhanced by a background flow, in that the solute asymptotically approaches a form that solves a different diffusion equation (with larger diffusion coefficient, in an appropriate moving frame). By introducing scaling variables, I'll show how one can obtain this asymptotic form in a very natural way using a center manifold reduction. This is joint work with Margaret Beck and Gene Wayne.

It is well known that the FitzHugh-Nagumo system exhibits stable, spatially monotone traveling pulses. Also, there is numerical evidence for the existence of spatially oscillatory pulses, which would allow for the construction of multi-pulses. Here we show the existence of oscillatory pulses rigorously, using geometric blow-up techniques and singular perturbation theory.

The Dynamic Neural Simulator (DNSim) is a tool for rapid prototyping of large neural models, batch simulation management, and efficient model sharing. It is designed to speed up and simplify the process of generating, sharing, and exploring network models of neurons. It requires no user programming and interfaces a crowd-sourced model database (DNSim DB) to facilitate model sharing and modular model building.

### Wednesday, October 15 3:30pm - 5:30pm at BU

For dispersive PDE on spatially periodic domains, we formulate the time-periodic solutions problem as a fixed point problem. The operator in question is the composition of a linear operator and a nonlinear operator. The linear operator can be bounded with small divisor estimates, losing derivatives in the process. If the nonlinear operator can be shown to have smoothing properties, then the composition can be shown to be a local contraction. Thus, the nonexistence of nontrivial small-amplitude doubly periodic waves follows from dispersive smoothing estimates.

The planar Euler equations describe the motion of a 2-D inviscid incompressible fluid, and also arise as a model problem for the study of the barotropic mode (to put it simply, the vertical average) of the Primitive equations of the ocean. It is a result by Yudovich that, in the space-periodic case, there exist a unique weak solution to the Euler system whenever the initial data has bounded vorticity.

### Wednesday, November 12 3:30pm - 5:30pm at Brown

The problem of finite-time singularity versus global regularity for active scalar equations with nonlocal velocities (for instance, the vorticity equation of the Euler equations, the 2D quasi-geostrophic equations with or without dissipation, and the α-patch model) has attracted much attention in recent years. In this talk, I will discuss some recent results in this direction.

In this talk we describe a new class of optimal continuity results for the action of the Hessian determinant on spaces of Besov type into the space of distributions on $\mathbb{R}^N$. In particular, inspired by recent work of Brezis and Nguyen on the distributional Jacobian determinant, we show that the action is continuous on the Besov space of fractional order $B(2-\frac{2}{N},N)$, and that all continuity results in this scale of Besov spaces are consequences of this result.

### Wednesday, December 10 3:30pm - 5:30pm at BU

We consider the focusing 3D quantum many-body dynamic which models a dilute bose gas strongly confined in two spatial directions. We assume that the microscopic pair interaction is focusing and matches the Gross-Pitaevskii scaling condition. We carefully examine the effects of the fine interplay between the strength of the confining potential and the number of particles on the 3D N-body dynamic. We overcome the difficulties generated by the attractive interaction in 3D and establish new focusing energy estimates.

Oscillatory electrical activity is ubiquitous in the brain at a wide range of frequencies, each with different behavioral correlates in different brain regions. The "communication though coherence" hypothesis states that coordination between oscillating populations can dynamically facilitate or prevent information flow between them. This hypothesis links two often separate fields of mathematics: dynamical systems and information theory, and raises questions in both fields.

### Fall 2015

### Tuesday, November 17 2:00pm - 6:00pm at BU

Abstract: Many biological rhythms evolve over multiple timescales, such as the beating of the heart or the electrical activity of neurons. Mathematical analysis of such multi-timescale problems has revealed special types of solutions, so-called canards, that play a crucial role in organising and shaping the dynamics. In this talk, we first give a brief overview of the current state of canard theory and its implications in mathematical neuroscience. We pay particular attention to two special classes of canard solutions: folded singularity canards and torus canards.

Abstract: We present recent results of a numerical bifurcation study of multidimensional

planar viscous shock waves in Magnetohydrodynamics using the Evans function.

We describe the numerical methods used and the bifurcation properties observed.

Abstract: Consider the angle of inclination of the profile of a steady 2D inviscid

symmetric periodic or solitary water wave subject to gravity. Although the

angle may surpass 30 degrees for some irrotational waves close to the

extreme wave, Amick proved in 1987 that the angle must be less than 31.15

degrees if the wave is irrotational. For any wave that is not

irrotational, the question of whether there is a bound on the angle has

been completely open. An example is the extreme Gerstner wave, which has

adverse vorticity and vertical cusps. Moreover, numerical calculations

Stochastic Reaction Diffusion Master Equations (RDMEs) have been widely used in biochemistry, computational biology, and biophysics to understand the reaction and diffusion of molecules within cells. However, the lattice RDME does not converge to any spatially-continuous model incorporating bimolecular reactions as the lattice spacing approaches zero. In order to contribute to a more accurate solution to this problem, we derive a convergent RDME (CRDME) via finite volume discretization of a spatially-continuous model and provide efficient numerical methods for approximating its solutions.

In this talk we introduce the Maslov index as a tool to analyze stability of steady state solutions to a reaction-diffusion equation in one spatial dimension. We show that the path of unstable subspaces associated to this equation is governed by a matrix Riccati equation whose solution $S$ develops singularities when changes in the Maslov index occur. We then prove that at these singularities the change in Maslov index equals the number of eigenvalues of $S$ that increase to $+\infty$ minus the number of eigenvalues that decrease to $-\infty$.

Abstract: Xenopus is a species of frog and a model organism for many biological studies. During early development of Xenopus laevis oocytes, mRNA moves from the nucleus to the periphery. The accumulation of messenger RNA at the bottom of the cell is called localization and is believed to depend on bidirectional movement by molecular motor proteins. This accumulation gives the cell an axis that is crucial for embryo development. We use discrete and continuous (partial differential equations) modeling approaches for mRNA populations that diffuse, move or are paused in the cell.

### Spring 2016

### Monday, April 25 1:30pm - 6:15pm at Brown

The motion of a vibrating material that is exposed to both deterministic and random forcing can be described by the damped semi-linear stochastic wave equation. As the mass density of the material converges to zero, the solutions of the damped semi-linear stochastic wave equations converge to the solution of a semi-linear stochastic heat equation uniformly on finite time intervals. This is called the Smoluchowski-Kramers approximation.

Zebrafish (Danio rerio) is a small fish with distinctive black and yellow stripes that form due to the interaction of different pigment cells. Working closely with the biological data, we present an agent-based model for these stripes that accounts for the migration, differentiation, and death of three types of pigment cells. The development of both wild-type and mutated patterns will be discussed, as well as the non-local continuum limit associated with the model.

(work is with Bjorn Sandstede)

We investigate the relationship between the nonlinear partial differential equations (PDEs) of mathematical physics and their linearizations around stationary localized solutions. It turns out that for the PDEs of sine-Gordon type, it is possible to solve the Inverse Linearization Problem, which is as follows: given linear stability analysis equations (LSAE) for an unknown PDE around an unknown stationary solution, restore the PDE. Of a particular interest are the instances of transparency of the LSAEs that may hint on the integrability of the corresponding PDEs.

A problem of great interest regarding conservation laws is the propagation of certain special initial data, and in particular, the behavior and evolution of the solution. For example, if we start with initial data in the form of a Markov process (in $x$), will this property continue to be preserved in time? What about other more general classes of initial data? I aim to address this and other questions, as well as provide some insight into the so-called $n$-point equations, the relationships between positions and velocities at certain points.

I will introduce Prandtl's boundary layer hypothesis, which was formulated in 1904 to describe the inviscid convergence of Navier-Stokes flows towards an Euler flow. I will then describe the setup and outline the proof of a recent result validating Prandtl's hypothesis in the case of Navier-Stokes flows over a rotating disk.

### Fall 2016

### Wednesday, December 14 2:00pm - 6:00pm at BU

We present a new result, joint with Andrea Nahmod, Natasa Pavlovic, and Gigliola Staffilani, in which extremely low regularity flows are constructed globally in time almost surely for a family of modified SQG equations using Gibbs measures techniques; we also construct a Gibbs measure with respect to which these flows are invariant.

Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and algorithms to solve and analyze problems that involve fluid flows governed by classical Navier-Stokes (NS) equations.

In this seminar I will be introducing a selection of well-posedness results in which randomness is exploited. I will briefly present the work of Burq-Tzvetkov for the NLW and the work of Bourgain for the NLS equation and then I will move to more recent results and open questions.

I will outline the proof that steady, incompressible Navier-Stokes flows posed over the moving boundary, $y = 0$, can be decomposed into Euler and Prandtl flows globally in the tangential variable, assuming a sufficiently small velocity mismatch. The main obstacles in the analysis center around obtaining sharp decay rates for the linearized profiles and the remainders. The remainders are controlled via a high-order energy method, supplemented with appropriate embedding theorems, which I will present.