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Nonlocality in nonlinear conservation model

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Speaker: Anh Thuong Vo (Univ. of NE Lincoln)

Abstract: In this talk, we investigate the convergence of solutions of nonlocal conservation PDE to the local counterparts in one space dimension. Nonlocal operators are integral operators that mimic differential operators but account for long-range interactions over a finite horizon. Nonlocality appears in many physical phenomena (fracture, phase separation) and has a wide range of applications (image processing). In Du et al, it was shown that nonlocal operators can be reduced into local operators in a distributional sense. The solution of the nonlocal Burgers equation is also shown to converge to the local counterpart numerically. In our research, we generalize the nonlocal advection operator. In the limit when the horizon parameter approaches zero, we are able to prove nonlocal operator convergence pointwise to its local counterpart. Then, we apply the result to show the convergence of the solution of the nonlocal conservation equation to the local counterpart.