English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this paper, we establish the local-in-time well-posedness of classicalsolutions to the vacuum free boundary problem of the viscous Saint-Venantsystem for shallow water in two dimensions. The solutions are shown to possesshigher-order regularities uniformly up to the vacuum free boundary, althoughthe depth degenerates as a singularity of the distance to the vacuum boundary.  Since the momentum equations degenerate in both the dissipation and timeevolution, there are difficulties in constructing approximate solutions by theGalerkin's scheme and gaining higher-order regularities uniformly up to thevacuum boundary for the weak solution. To construct the approximate solutions,we introduce some degenerate-singular elliptic operator, whose eigenfunctionsform an orthogonal basis of the projection space. Then the high-orderregularities on the weak solution are obtained by using some carefully designedhigher-order weighted energy functional.

On the vacuum free boundary problem of the viscous Saint-Venant system for shallow water in two dimensions