## Ohio State University Partial Differential Equations Seminar## Year 2016-2017Time/Location: Tuesdays 1:30 - 2:30pm / MW 154 (unless otherwise noted) |
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Zhenan Sui, Ohio State (Sep 20, 2016)

Title: Complete Conformal Metrics with Prescribed Ricci Curvature Functions on a Negative ConeAbstract: The study of conformal metrics with constant scalar curvature on compact manifolds, known as the Yamabe problem, has been completely solved by Neil Trudinger, Thierry Aubin and Richard Schoen. The nonlinear Yamabe problem, concerned with the Schouten tensor, defined on compact manifolds, are also formulated by elliptic approach or flow method. Another generalization to Yamabe problem is to consider non-compact manifolds.

On the other hand, in the spirit of Lohkamp's theorem, which asserts that any smooth manifolds admit complete metrics with negative Ricci curvature. It seems an interesting quesiton whether there always exists a complete conformal metric with negative Ricci curvature. The answer is obviously negative for compact manifolds without boundary from the maximum principle. Guan has proved that the answer is yes on compact manifolds with boundary.

Motivated by the formulation of Yamabe problem as well as Guan's existence theorem regarding Ricci curvature on compact manifolds with boundary, a natural question to ask is whether such complete conformal metrics, with prescribed symmetric functions of the eigenvalues of the Ricci tensor defined on negative cones, exist on certain non-compact manifolds? The affirmative answer is obtained for Euclidean spaces, on which we are also able to detect the nonexistence of such metrics under certain decay conditions.

Blair Davey, CCNY (Oct 4, 2016)

Title: Recent progress on Landis' conjectureAbstract: In the late 1960s, E.M. Landis made the following conjecture: If $u$ and $V$ are bounded functions, and $u$ is a solution to $\Delta u = V u$ in $\mathbb{R}^n$ that decays like $|u(x)| \le c \exp(- C |x|^{1+})$, then $u$ must be identically zero. In 1992, V. Z. Meshkov disproved this conjecture by constructing bounded functions $u, V: \mathbb{R}^2 \to \mathbb{C}$ that solve $\Delta u = V u$ in $\mathbb{R}^2$ and satisfy $|u(x)| \le c \exp(- C |x|^{4/3})$. The result of Meshkov was accompanied by qualitative unique continuation estimates for solutions in $\mathbb{R}^n$. In 2005, J. Bourgain and C. Kenig quantified Meshkov's unique continuation estimates. These results, and the generalizations that followed, have led to a fairly complete understanding of the complex-valued setting. However, there are reasons to believe that Landis' conjecture may be true in the real-valued setting. We will discuss recent progress towards resolving the real-valued version of Landis' conjecture in the plane.

Peter Polacik, Minnesota (Oct 18, 2016)

Title: Propagating terraces in the dynamics of parabolic equationsAbstract: We will present recent results on the approach of front-like solutions of one-dimensional parabolic equations to propagating terraces, or, stacked families of traveling fronts. We will then show some applications of the theorem in multidimensional problems, including results on the quasiconvegence and asymptotic one-dimensional symmetry of a class of solutions.

Malcolm Brown, Cardiff Univ, UK (Oct 27, 2016)

Title : UNIQUENESS FOR AN INVERSE PROBLEM IN ELECTROMAGNETISM WITH PARTIAL DATAAbstract : A uniqueness result for the recovery of the electric and magnetic coefficients in the time-harmonic Maxwell equations from local boundary measurements is showen. No special geometrical condition are imposed on the inaccessible part of the boundary of the domain, apart from that that the boundary of the domain is C1,1. The coefficients are assumed to coincide on a neighbourhood of the boundary: a natural property in many applications.

Ming Chen (Robin), U. Pittsburgh (Nov 8, 2016)

Title : Existence of large-amplitude steady stratified water waves Abstract : We consider 2D steady water waves with heterogeneous density. The presence of stratification allows for a wide variety of traveling waves, including fronts, so-called generalized solitary waves with ripples in the far field, and even fronts with ripples! Among these many possible wave patterns, we prove that for any smooth choice of upstream velocity and monotone streamline density function, there always exists a continuous curve of solitary waves with large amplitude, which are even and decreasing monotonically on either side of a central crest. As one moves along this curve, the horizontal fluid velocity comes arbitrarily close to the wave speed.We will also discuss a number of results characterizing the qualitative features of solitary stratified waves. In part, these include bounds on the Froude number from above and below that are new even for constant density flow; an a priori bound on the velocity field and lower bound on the pressure; a proof of the nonexistence of monotone bores for stratified surface waves; and a theorem ensuring that all supercritical solitary waves of elevation have an axis of even symmetry. This is a joint work with Samuel Walsh and Miles Wheeler.

Philip Korman, U. Cincinnatti (Nov 15, 2016)

Title : Exact multiplicity of solutions for some semilinear Dirichlet problems.Abstract : By considering generalized harmonics, we give a generalization of the classical results of A. Ambrosetti and G. Prodi, and of M.S. Berger and E. Podolak, and present an application to a population model involving fishing.

Sebastien Picard, Columbia Univ. (Jan 24, 2017)

Title : Geometric flows and Strominger systemsAbstract : The Anomaly flow is a geometric flow which implements the Green-Schwarz anomaly cancellation mechanism originating from superstring theory, while preserving the conformally balanced condition of Hermitian metrics. I will discuss criteria for long time existence and convergence of the flow on toric fibrations with the Fu-Yau ansatz. In this setting, the flow reduces to a scalar fully-nonlinear parabolic PDE with non-concave elliptic terms. This is joint work with D.H. Phong and X.W. Zhang.

Kesh Govinder, MBI. (Feb 14, 2017)

Title : Applications of symmetry methods to differential equationsAbstract : Group analysis (which exploits the invariance of equations under transfor- mations) has been very successfully applied in the classification and explicit solution of differential equations. We will show how the theory has been used to great effect in a number of applications in Mathematical Physics and Math- ematical Biology. We will also show how some of these applications have re- sulted in refinement of the theory itself. In particular, there are fundamentally different approaches for ordinary differential equations as opposed to partial differential equations.

Qiliang Wu, Michigan State. (Feb 21, 2017)

Title : Periodic Patterns on Unbounded DomainsâStability, Persistence and DefectsAbstract : Time-independent, spatially periodic patterns, or, stripe patterns, are ubiquitous in nature. Their manifestation in nature, nevertheless, is usually deformed, accompanied with various defects. We investigate typical nonlinear pattern forming systems which admit spatially periodic solutions, with an emphasis on the nonlinear stability of such patterns, their persistence in the presence of spatial inhomogeneity, and the existence of grain boundariesâa particular form of defects. Our study employs various analytic tools, mainly from dynamical system, functional analysis, and harmonic analysis. We focus on the most interesting case when the linearized operator at the periodic pattern admits a diffusively stable spectrum, that is, the spectrum lies in the left half of the complex plane and continues up to the imaginary axis only at the origin. More specifically, we show that, 1)for a reaction diffusion system on the real line, any sufficiently small initial perturbation to the periodic pattern decays to zero at the rate t^{-1/2} as time t goes to infinity; 2)In the presence of any sufficiently small localized inhomogeneity, every Eckhaus stable periodic solution in the 1D Swift- Hohenberg equation persists, yielding a deformed solution with selected wave number and phase; 3)in 2D Swift-Hohenberg equation, for any sufficiently small systematic parameter \mu>0 and spanning angle Î± between 0 and Ï, there always exists a symmetric grain boundary with angle Î±, admitting its limiting periodic pattern with a selected wavenumber determined by the parameter \mu and the angle \alpha.

Arnd Scheel, Minnesota. (Feb 28, 2017)

Title : Growing stripesAbstract : I will give an overview of work that explores how spatial growth acts as a selection mechanism, focusing on the creation of "striped phases". A prototypical example is the Allen-Cahn equation with heterogeneity that renders the system bistable in a half plane and monostable in the complement. Expanding the bistable region leads to creation of patterns at the boundary of the bistable region. Beyond Allen-Cahn, I will comment on effects in Cahn-Hilliard, reaction-diffusion, and Swift-Hohenberg equations.

Anna Ghazaryan, Miami University. (Mar 7, 2017)

Title : Existence of fronts a population model for mussel-algae interaction.Abstract : I will discuss some new results about a known model that describes formation of mussel beds on soft sediments. The model consists of nonlinearly coupled pdes that capture evolution of mussel biomass on the sediment and algae in the water layer overlying the mussel bed. The system accounts for the diffusive spread of mussel, while the diffusion of algae is neglected and at the same time the tidal flow of the water is considered to be the main source of transport for algae, but does not affect mussels. Therefore both the diffusion and the advection matrices in the system are singular. We use Geometric Singular Perturbation theory to analytically study wave formation mechanisms in this system. This is joint work with V. Manukian.

Junping Shi, College of William and Mary. (Mar 8, 2017) (Joint with Applied Math)

Title : Existence, uniqueness and stability of positive steady state solution to nonlocal Fisher type equationsAbstract : Nonlocal Fisher type equation is a reaction-diffusion equation with logistic non-linearity which has an integral form carrying capacity. In general such models do not satisfy maximal principle. We will discuss the existence, uniqueness and stability of positive steady state solution for such equations under different boundary conditions and different kernel functions. Local and global stability of the steady state in time evolution model with or without delay effect will also be considered. Finally we will also discuss some results for two-species competition models with nonlocal carrying capacity. In particular we show that the co-existence steady state may not be unique.

Irfan Glogic, Ohio State University. (Mar 21, 2017)

Title : TBDAbstract : TBD

Changyou Wang, Purdue University. (Mar 28, 2017)

Title : High dimensional Ginzburg-Landau equations under weak anchoring boundary conditionsAbstract : In this talk, I will introduce a modified Ginzburg-Landau functional that includes a boundary surface energy term, i.e. $$E_\epsilon(u):=\int_\Omega (1/2|Du|^2+1/4\epsilon^2 (1-|u|^2)^2)+\lambda_\epsilon \int_{\partial\Omega}|u-g|^2,$$ where $\epsilon>0$ is Ginzburg-Landau parameter, and $\lambda_\epsilon=\epsilon^{a}$, $0

Irfan Glogic, Ohio State University. (Mar 21, 2017)

Title : Existence and stability of blowup for wave maps into a negatively curved targetAbstract : Wave maps are stationary points of the geometric action functional for smooth maps between Lorenzian and Riemannian manifolds. In addition to being used as models in particle physics they also serve as simplified versions of Einstein's equations of general relativity. We consider wave maps from $(1+d)$-dimensional Minkowski space into negatively curved target manifolds. The underlying Euler-Lagrange equations then become a system of semilinear wave equations. While their local well-posedness is standard, global existence is known only for $d=2$ from the work of Shatah and Tahvildar-Zadeh during â90s. Furthermore, they showed that for $d=7$ singularities can form. In this talk we show the existence of singularity formation for dimensions $d \geq 8$ and furthermore prove that the blowup process is stable for the lowest odd dimension $d=9$. Moreover, we conjecture that stability of blowup is true for all dimensions $d \geq 8$ and show how the conjecture reduces to a spectral problem for a second order ordinary differential operator with four regular singularities. Whether blowup for this model exists when $3 \leq d \leq 6$ still remains open. This is joint work with Roland Donninger, University of Bonn, Germany.

Renhao Cui, Harbin Normal University (April 18, 2017)

Title : A Spatial SIS Model in Advective Heterogeneous EnvironmentsAbstract : We study the effects of diffusion and advection for a susceptible-infected-susceptible epidemic reaction-diffusion model in heterogeneous environments. The definition of the basic reproduction number $\mathcal{R}_0$ is given. The persistence of infected and susceptible populations and the global stability of the disease free equilibrium are established when the basic reproduction number is greater than or less than or equal to one, respectively. We futher consider the effects of diffusion and advection on asymptotic profiles of endemic equilibrium: When the advection rate is relatively large comparing to the diffusion rates of both populations, then two population persist and concentrate at the downstream end. As the diffusion rate of the susceptible population tends to zero, the density of the infected population decays exponentially for positive advection rate but linearly when there is no advection. Our results suggest that advection can speed up the elimination of disease. This is joint work with King-Yeung Lam and Yuan Lou.

Jiaping Wang, University of Minnesota. (April 20, 2017)

Title : Geometry of shrinking Ricci solitonsAbstract : Introduced by Hamilton about thirty-five years ago, Ricci flow has led to spectacular successes including the resolution of the Poincare conjecture for three manifolds and the complete classification of quarter pinched Riemannian manifolds. Ricci solitons, as self-similar solutions to Ricci flow, play an important role in understanding the singularity formation and long time dynamics of the flow. The talk will focus on the so-called shrinking solitons. We will review their classification in dimension two and three, and explain some recent progress made jointly with Ovidiu Munteanu concerning their geometry in dimension four.

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