This issuePrevious Article$W^{1,p}$ regularity for the conormal derivative problem with parabolic BMO nonlinearity in reifenberg domainsNext ArticleThe thermodynamic formalism for sub-additive potentials
Stability of under-compressive waves with second and fourth order diffusions
This is a continuation of our work on the
nonlinear stability of traveling shock fronts arising in
multidimensional conservation laws with fourth order regularization
only. Our motivating example is the thin film equation for which
planar waves correspond with fluid coating a pre-wetted surface.
Under only the fourth order regularization, we established the
nonlinear stability of compressive waves for dimensions $d\geq 2$,
and of under-compressive waves for dimensions $d\geq 3$ under
general spectral conditions. The case of stability for
under-compressive waves in the thin film equations for the critical
dimensions $d=1,2$ remained open. In this paper we study the
nonlinear stability of under-compressive waves by assuming both the
second and fourth order regularization. We present a step toward the
open problem by establishing the nonlinear stability of
under-compressive waves in dimensions $d\geq 2$ under general
spectral conditions. We emphasize the above mentioned stability
question remains still open.