On a nonlocal dispersive equation modeling particle suspensions

We study a nonlocal, scalar conservation law ${u_t} + {\left ( \left ( {K_a} * u \right )u \right )_x} = 0$, modeling sedimentation of particles in a dilute fluid suspension, where ${K_a}\left ( x \right ) = {a^{ - 1}}K\left ( x/a \right )$ is a symmetric smoothing kernel, and $\ast$ represents convolution. We show this to be a dispersive regularization of the Hopf equation, ${u_t} + {\left ( {u^2} \right )_x} = 0$, 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.