Quantitative Estimates for Advective Equation with Degenerate Anelastic Constraint

In these proceedings we are interested in quantitative estimates for advective equations with an anelastic constraint in presence of vacuum. More precisely, we derive a quantitative stability estimate and obtain the existence of renormalized solutions. Our main objective is to show the flexibility of the method introduced recently by the authors for the compressible Navier-Stokes’ system. This method seems to be well adapted in general to provide regularity estimates on the density of compressible transport equations with possible vacuum state and low regularity of the transport velocity field; the advective equation with degenerate anelastic constraint considered here is another good example of that. As a final application we obtain the existence of global renormalized solution to the so-called lake equation with possibly vanishing topography. AMS Classification. 35B40, 35B45, 35K35, 76Y05.