31 research outputs found
On the vacuum free boundary problem of the viscous Saint-Venant system for shallow water in two dimensions
In this paper, we establish the local-in-time well-posedness of classical
solutions to the vacuum free boundary problem of the viscous Saint-Venant
system for shallow water in two dimensions. The solutions are shown to possess
higher-order regularities uniformly up to the vacuum free boundary, although
the depth degenerates as a singularity of the distance to the vacuum boundary.
Since the momentum equations degenerate in both the dissipation and time
evolution, there are difficulties in constructing approximate solutions by the
Galerkin's scheme and gaining higher-order regularities uniformly up to the
vacuum boundary for the weak solution. To construct the approximate solutions,
we introduce some degenerate-singular elliptic operator, whose eigenfunctions
form an orthogonal basis of the projection space. Then the high-order
regularities on the weak solution are obtained by using some carefully designed
higher-order weighted energy functional
Wave breaking for the generalized Fornberg-Whitham equation
This paper aims to show that the Cauchy problem of the Burgers equation with
a weakly dispersive perturbation involving the Bessel potential (generalization
of the Fornberg-Whitham equation) can exhibit wave breaking for initial data
with large slope. We also comment on the dispersive properties of the equation