Convex integration for a class of active scalar equations

Abstract

We show that a general class of active scalar equations, including porous media and certain magnetostrophic turbulence models, admits non-unique weak solutions in the class of bounded functions. The proof is based upon the method of convex integration recently implemented for equations of fluid dynamics.

The ℎ-principle and the equations of fluid dynamics

“…It is however possible to build up such a sequence if instead of requiring that its support be compact, we require that z j converges uniformly to 0 in the complement of B 1 (0) (cp. Lemma 2.1 in [71]). …”

“…In particular these weak solutions are extendable by zero to R d ⊃ D. In applications to evolution equations D is a space-time domain, say D = R n × (0, T ), and thus this argument yields weak solutions with compact time support, as in [66,26,71]. For the construction of weak solutions with arbitrary initial data, in particular for the construction of admissible weak solutions, refinements of this argument are necessary.…”

“…Assume m is given by (38). Then there exist infinitely many weak solutions of (33) and ( [26,71] use some refined tools which were developed in the theory of laminates and differential inclusions and they present some substantial differences with the proofs in [30,31]. In order to address these differences we start by recalling some of the standard notions in the theory of differential inclusions.…”

“…In [26] and [71] the authors avoid calculating the full hull and instead restrict themselves to exhibiting a nontrivial (but possibly much smaller) open set U satisfying (P). However, in exchange they are forced to use much more complicated sequences z j .…”

“…However, in exchange they are forced to use much more complicated sequences z j . Indeed, the z j 's used in [30] are localizations of simple plane waves, whereas the ones used in [26] and [71] arise as an infinite nested sequence of repeated plane waves.…”