Integrability Gain for Systems in Divergence Form

The dynamical systems of interest herewith are governed by the partial differential equations of fluid mechanics and of magnetohydrodynamics. A contribution to the qualitative theory of said dynamical systems comes from a property known as ``higher integrability of determinants.'' Very roughly, given N independent variables and K >= N scalar observables, one looks for an N X N tensor, the N-divergence of which allows to rewrite N of the differential equations in the system. The entries of the tensor are functions of the observables (the unknowns). The determinant of said tensor is in an integrability class higher than that of the tensor entries. In the application the result is: (p + rho_f phi)rho_f ^{1/{(N-1)}} in L^1(Q), where Q is the time-space domain, p fluid pressure, rho_f fluid density and phi (>0) the known potential of conservative forces. Higher integrability thus affects p and rho_f, leaving out both velocity, vec u, and the magnetic field, vec H. Many problems are still open, as one shall expect from a relatively new branch of qualitative theory. AMS subject classification: 15A15, 35B45, 53A45.