A kinetic decomposition for singular limits of non-local conservation laws

Abstract

We consider a non-local regularization of nonlinear hyperbolic conservation laws in several space variables. The regularization is motivated by the theory of phase dynamics and is based on a convolution operator. We formulate the initial value problem and begin by deriving a priori estimates which are independent of the regularization parameter. Following Hwang and Tzavaras we establish a kinetic decomposition associated with the problem under consideration, and we conclude that the sequence of solutions generated by the non-local model converges to a weak solution of the corresponding hyperbolic problem. Depending on the scaling introduced in the non-local dispersive term, this weak limit is either a classical Kruzkov solution satisfying all entropy inequalities or, more interestingly, a nonclassical entropy solution in the sense defined by LeFloch, that is, a weak solution satisfying a single entropy inequality and containing undercompressive shock waves possibly selected by a kinetic relation. Finally, we illustrate our analytical conclusions with numerical experiments in one spatial variable.

Nonlocal Conservation Laws. A New Class of Monotonicity-Preserving Models

“…More general nonlocal regularization terms can be found in spectral viscosity methods introduced by Tadmor [35,8]. Other nonlocal regularizations can be found in [41], Liu [33], Chmaj [9], Duan,Feller and Zhu [19], Rohde [39], Kissling and Rohde [26], and Kissling, LeFloch and Rohde [27]. Nonlocal convection may also be introduced through a nonlocal regularization of the convective velocity, that is, for a transport equation of the form u t + (vu) x = 0, we may have v being an integral average of some function of u, see for instance Zumbrun [52], Logan [34] and Amorim, Colombo and Teixeira [4].…”

Peridynamics as a rigorous coarse-graining of atomistics for multiscale materials design.

“…Assuming that the time derivative is treated locally (i.e., as the usual partial derivative ∂/∂t), there are two main aspects of the general advection equation for which nonlocal extensions have been examined: (i) nonlocal f , nonlocal f x (u) in (2.0.2a), or nonlocal f (u) in (2.0.2b), and (ii) nonlocal regularizations, i.e., "small" nonlocal terms replacing the zero on the RHS of the equality in both equations in (2.0.2). The first of these approaches was considered, e.g., in [6,9,14,38,56,66], while the second category of nonlocalizations includes some of the above cited works, as well as, e.g., [1,2,4,7,11,15,16,19,28,27,36,37,56,49,63,64]. A focus of many of these works is on nonlocal generalizations of Burgers equation.…”