On a nonlocal dispersive equation modeling particle suspensions

Source abstract

We study a nonlocal, scalar conservation law u t + ( ( K a ∗ u ) u ) x = 0 {u_t} + {\left ( \left ( {K_a} * u \right )u \right )_x} = 0 , modeling sedimentation of particles in a dilute fluid suspension, where K a ( x ) = a − 1 K ( x / a ) {K_a}\left ( x \right ) = {a^{ - 1}}K\left ( x/a \right ) is a symmetric smoothing kernel, and ∗ \ast represents convolution. We show this to be a dispersive regularization of the Hopf equation, u t + ( u 2 ) x = 0 {u_t} + {\left ( {u^2} \right )_x} = 0 , analogous to KdV and certain dispersive difference schemes. Using the smoothing property of convolution and the physical principle of conservation of mass, we establish the global existence of smooth solutions.