ON BOUNDED TWO-DIMENSIONAL GLOBALLY DISSIPATIVE EULER FLOWS

We examine the two-dimensional Euler equations including the local energy (in)equality as a differential inclusion and show that the associated relaxation essentially reduces to the known relaxation for the Euler equations considered without local energy (im)balance. Concerning bounded solutions we provide a sufficient criterion for a globally dissipative subsolution to induce infinitely many globally dissipative solutions having the same initial data, pressure, and dissipation measure as the subsolution. The criterion can easily be verified in the case of a flat vortex sheet giving rise to the Kelvin--Helmholtz instability. As another application we show that there exists initial data for which associated globally dissipative solutions realize every dissipation measure from an open set in C-0(T-2 x [0, T]). In fact the set of such initial data is dense in the space of solenoidal L-2(T-2; R-2) vector fields.

Automated summary

We study the two-dimensional Euler equations with local energy balance and show that the corresponding relaxation process can be simplified compared to the relaxation process without local energy balance. For bounded solutions, we provide a criterion that allows a globally dissipative subsolution to induce infinitely many globally dissipative solutions with the same initial data, pressure, and dissipation measure. This criterion can be easily verified in the case of a flat vortex sheet causing the Kelvin-Helmholtz instability. Additionally, we show the existence of initial data such that the associated globally dissipative solutions can achieve any dissipation measure from an open set in C-0(T-2 x [0, T]). In fact, the set of such initial data is dense in the space of solenoidal L-2(T-2; R-2) vector fields.