A Generalized Framework for Lr Convex Integration and its Application to Geophysical Models

Abstract

In this paper a general framework for convex integration is developed, in order to construct weak solutions to the Cauchy problem, by building on ideas from [C. De Lellis and L. Székelyhidi, Arch. Ration. Mech. Anal., 195 (2010)] and [S. Markfelder, Nonlinearity, 37 (2024)]. This framework may be applied to a large family of partial differential equations in order to construct weak solutions in L∞((0, T ) × Ω) (for a bounded domain Ω) which are weakly continuous in time with respect to the weak topology of Lr(Ω) for some r ∈ (1, ∞). This allows us to construct solutions which obey an energy inequality. In the second part of the paper we apply the framework to several inviscid models appearing in the field of geophysical fluid mechanics in order to show existence of (in- finitely many) weak solutions for all initial data, and to prove that there exist initial data for which there are infinitely many solutions which satisfy an energy inequality (such initial data are sometimes called “wild data”). More precisely, we first consider the incompressible and the barotropic compressible Euler equations to recover the cor- responding results from the literature. In addition, the framework allows us to prove a new result for the incompressible Euler equations, namely the global existence for the Cauchy problem in L∞. We then apply the framework to the shallow water and lake equations. Moreover, we use the framework in the context of the hydrostatic Euler equations (also known as the incompressible inviscid primitive equations), which leads to the first convex integration approach which is able to construct admissible solutions with the natural energy for this system. A crucial ingredient in the proof of this result is the computation of a large subset of the convex hull, as an explicit characterization of the convex hull does not seem to be available. Finally, we apply the framework to the compressible inviscid primitive equations and to the inviscid quasi-geostrophic equations to obtain the first results on existence of wild data for these two geophysical models