MS14: Theoretical and computational aspects of nonlocal operators


Session A: Oct.1, 10:40am-12:00pm, Classroom Building 208
Session B: Oct.1, 3:10pm-4:30pm, Classroom Building 208
Session C: Oct.1, 5:00pm-6:20pm, Classroom Building 208

MS14-A-1 (canceled)
10:40am-11:00am (Oct 1)
CLB 208

Farhan Abedin,
Lafayette College

Hele-Shaw flow and parabolic integro-differential equations


The Hele-Shaw flow is a model for the evolution of an ideal fluid occupying the narrow gap between two parallel plates and subject to an external pressure source. Assuming the fluid interface is given by the graph of a function, it can be shown that this function must solve a parabolic integro-differential equation of 1st order. I will discuss recent joint work with Russell Schwab (Michigan State University) on the regularization properties of such nonlocal parabolic equations. Our results allow us, in certain model scenarios and for short enough times, to conclude improvement-of-regularity for the fluid interface.

11:00am-11:20am (Oct 1)
CLB 208
Fernando Charro,
Wayne State University

Asymptotic Mean Value Formulas for Nonlinear Equations
  In recent years there has been an increasing interest in whether a mean value property, known to characterize harmonic functions, can be extended in some weak form to solutions of nonlinear equations. This question has been partially motivated by a surprising connection between Random Tug-of-War games and the normalized p-Laplacian discovered some years ago by Peres et al., where a nonlinear asymptotic mean value property for solutions of a PDE is related to a dynamic programming principle for an appropriate game. In this talk we discuss asymptotic mean value formulas for a class of nonlinear second-order equations that includes the classical Monge-Ampère and k-Hessian equations among other examples.

11:20am-11:40am (Oct 1)
CLB 208
Thialita Nascimento,
University of Central Florida

New regularity estimates for a class of nonlocal operators
  In this talk, we will discuss new universal bounds for the exponent of Hessian integrability of viscosity supersolutions of fully nonlinear and uniformly elliptic equations. Such estimates produce a quantitative improvement in the decay of this exponent with respect to dimension. In particular, we solve, in the negative, the Armstrong-Silvestre-Smart Conjecture about the optimal exponent for the Hessian integrability. This is a joint work with Prof. Eduardo Teixeira (UCF-USA).

11:40am-12:00pm (Oct 1)
CLB 208
Shiping Zhou,
Missouri University of Science and Technology

Accurate and efficient spectral method for fractional wave equations
  In this talk, we will present a Fourier pseudospectral method for solving the variable-order fractional wave equation. In contrast to the constant-order case, the Fourier transform of a variable-order fractional wave equation cannot give a decoupled ODE system anymore. To deal with this, a fast algorithm is proposed by diagonalizing the variable-order fractional Laplacian. We study the accuracy and efficiency of our method, numerical results show that our method can get a spectral order of accuracy in space and a 2nd-order of accuracy in time.

3:10pm-3:30pm (Oct 1)
CLB 208
Debdeep Bhattacharya,
Louisiana State University

Load-controlled evolution of quasistatic nonlinear peridynamics
  We consider the load-controlled quasistatic evolution of a nonlinear nonlocal continuum model, which can be viewed as a peridynamic equation. The well-posedness of the solution for all loads near the local minima of the peridynamic energy is proved in a suitable space perpendicular to rigid motions. Consequently, the local existence of a stable load path originating at the local minima is established. Although the local minima belong to the strength domain of the material, the evolution of the displacement, however, is not constrained to lie in the strength domain. Our method relies on a fixed-point argument, which differs from the approaches based on the global minimizer of the energy. The load-controlled evolution is shown to exhibit energy balance. A numerical method is implemented to solve the load-controlled evolution equation.

3:30pm-3:50pm (Oct 1)
CLB 208
Mikil Foss,
University of Nebraska-Lincoln
Convergence of Solutions for Linear Peridynamic Models
  Peridynamic models have been successfully employed to predict fractures and deformations for a variety of materials. In this talk, I will present some results on the convergence of solutions to a nonlocal state-based linear elastic model to their local counterparts as the interaction horizon vanishes. The results provide explicit rates of convergence that are sensitive to the compatibility of the nonlocal boundary data and the extension of the solution for the local model.

3:50pm-4:10pm (Oct 1)
CLB 208
Qihao Ye,
University of California San Diego

A Monotone Meshfree Finite Difference Method for Linear Elliptic PDEs via Nonlocal Relaxation
  We design a monotone meshfree finite difference method for linear elliptic PDEs in non-divergence form on point clouds via a nonlocal relaxation method. Nonlocal approximations of linear elliptic PDEs are first introduced to which a meshfree finite difference method applies. Minimal positive stencils are obtained through a linear optimization procedure that automatically guarantees the stability and, therefore, the convergence of the meshfree discretization. The key to the success of the method relies on the existence of positive stencils for a given point cloud geometry. We provide sufficient conditions for the existence of positive stencils by finding neighbors within an ellipse surrounding each interior point, generalizing the study for the Poisson equation by Seibold in 2008. It is well-known that wide stencils, in general, are needed for the existence of positive weights when elliptic equations become degenerate (when the coercivity constant goes to zero). Our study allows judiciously selecting the neighboring points to reduce the computational cost for solving elliptic equations with small coercivity constants. Numerical experiments will be provided.

5:00pm-5:20pm (Oct 1)
CLB 208
Anh Vo,
University of Nebraska-Lincoln

Convergence of nonlocal nonlinear conservation laws with respect to horizon
  In recent years, conservation laws involving nonlocal terms have been extensively studied due to their applications to several fields. These nonlocal operators capture long-range interactions and often have a finite horizon which is measured through the support of the kernel. In this study, we investigate nonlinear nonlocal conservation laws and the nonlocal divergence operator. We propose a requirement for the flux density so that the nonlocal divergence will converge to the traditional divergence operator. We then apply the result to study the convergence of the conservation laws.

5:20pm-5:40pm (Oct 1)
CLB 208
Mitch Haeuser,
Iowa State University

Nonlocal Equations on the Boundary
  We will discuss regularity for a problem involving a fractional Dirichlet-to-Neumann operator associated to harmonic functions. In particular, we will define a fractional powers of the normal derivative, compatible Sobolev spaces, and consider various examples. We will further look at the extension problem characterization to obtain various estimates. This is joint work with Luis Caffarelli (UT Austin) and Pablo Raúl Stinga (Iowa State University)

5:40pm-6:00pm (Oct 1)
CLB 208
Daniel E Restrepo Montoya,
University of Texas Austin

On a semilinear nonlocal elliptic equation in the context of plasma physics
  In this talk, we will discuss regularity, qualitative properties and uniqueness of solutions to a type of semilinear equations that arises in plasma physics as an approximation to Grad equations, which were introduced by Harold Grad, to model the behavior of plasma confined in a toroidal vessel. The difficulty of this problem lies on a right-hand side which involves the measure of the superlevel sets, making the problem nonlocal. This model also develops naturally a dead core which amounts, mathematically, to a new type of free boundary problem that will be commented briefly. This is a joint work with Luis Caffarelli and Ignacio Tomasetti from UT Austin.

6:00pm-6:20pm (Oct 1)
CLB 208
Animesh Biswas,
University of Nebraska-Lincoln

Extension equation for fractional power of operator defined on Banach spaces
  In this talk, we show the extension (in spirit of Caffarelli-Silvestre) of fractional power of operators defined on Banach spaces. Starting with the Balakrishnan definition, we use semigroup method to prove the extension. This is a joint work with Pablo Raul Stinga.