Uniqueness for cross-diffusion systems issuing from seawater intrusion problems
Abstract
We consider a model mixing sharp and diffuse interface approaches
for seawater intrusion phenomenons in confined and unconfined aquifers. More
precisely, a phase field model is introduced in the boundary conditions on
the virtual sharp interfaces. We thus include in the model the existence of
diffuse transition zones but we preserve the simplified structure allowing front
tracking. The three-dimensional problem then reduces to a two-dimensional
model involving a strongly coupled system of partial differential equations
of parabolic and elliptic type describing the evolution of the depth of the
interface between salt- and freshwater and the evolution of the freshwater
hydraulic head. Assuming a low hydraulic conductivity inside the aquifer, we
prove the uniqueness of a weak solution for the model completed with initial
and boundary conditions. Thanks to a generalization of a Meyer’s regularity
result, we establish that the gradient of the solution belongs to the space Lr
,
r > 2. This additional regularity combined with the Gagliardo-Nirenberg
inequality for r = 4 allows to handle the nonlinearity of the system in the
proof of uniqueness.