On nonlocal conservation laws modelling sedimentation

Abstract

The well-known kinematic sedimentation model by Kynch states that the settling velocity of small equal-sized particles in a viscous fluid is a function of the local solids volume fraction. This assumption converts the one-dimensional solids continuity equation into a scalar, nonlinear conservation law with a nonconvex and local flux. This work deals with a modification of this model, and is based on the assumption that either the solids phase velocity or the solid-fluid relative velocity at a given position and time depends on the concentration in a neighbourhood via convolution with a symmetric kernel function with finite support. This assumption is justified by theoretical arguments arising from stochastic sedimentation models, and leads to a conservation law with a nonlocal flux. The alternatives of velocities for which the nonlocality assumption can be stated lead to different algebraic expressions for the factor that multiplies the nonlocal flux term. In all cases, solutions are in general discontinuous and need to be defined as entropy solutions. An entropy solution concept is introduced, jump conditions are derived and uniqueness of entropy solutions is shown. Existence of entropy solutions is established by proving convergence of a difference-quadrature scheme. It turns out that only for the assumption of nonlocality for the relative velocity it is ensured that solutions of the nonlocal equation assume physically relevant solution values between zero and one. Numerical examples illustrate the behaviour of entropy solutions of the nonlocal equation.

Nonlocal Systems of Conservation Laws in Several Space Dimensions

“…However, a bound on the L ∞ norm of the solution is provided below, as well as the L 1 Lipschitz continuous dependence of the solution on time. As a consequence, the present result also extends to a wider class of equations the existence part of the results in [3,4,8,9,10]. More precisely, in the scalar 1D case, a first convergence proof for a Lax-Friedrichs type algorithm was obtained in [4], see also [3,5].…”

“…As a consequence, the present result also extends to a wider class of equations the existence part of the results in [3,4,8,9,10]. More precisely, in the scalar 1D case, a first convergence proof for a Lax-Friedrichs type algorithm was obtained in [4], see also [3,5]. Analytical well posedness results for nonlocal conservation laws in the scalar case were obtained in [8,9] and in [10] in the case of systems.…”

“…As usual, t is time, x the space variable, U the vector of the unknown densities, F the matrix valued flow and η is a matrix of smooth convolution kernels. Systems of the type (1.1), with the particular coupling considered below, comprise, for instance, sedimentation models [4] and various crowd dynamics models [8,9,10]. The nonlocal nature of (1.1) is particularly suitable in describing the behavior of crowds, where each member moves according to her/his evaluation of the crowd density and its variations within her/his horizon.…”

Well-posedness of a conservation law with non-local flux arising in traffic flow modeling

“…Space-integral terms are considered for example in models for granular flows [1], sedimentation [6], crowd motion [9], or more general problems like gradient constrained equations [2]. Also, non-local in time terms arise in conservation laws with memory, starting from [13].…”

“…Following [17,Definition 1], [6,Definition 4.1] and [10, Definition 2.1], we can also give an entropy criterion to select admissible solutions.…”