On the role of pressure in the theory of MHD equations

Abstract

We consider the system of MHD equations in Ω×(0,T), where Ω is a domain in R3 and T>0, with the no slip boundary condition for the velocity u and the Navier-type boundary condition for the magnetic induction b. We show that an associated pressure p, as a distribution with a certain structure, can be always assigned to a weak solution (u,b). The pressure is a function with some rate of integrability if the domain Ω is “smooth”, see section 3. In section 4, we study the regularity of p in a sub-domain Ω1×(t1,t2) of Ω×(0,T), where u (or, alternatively, both u and b) satisfies Serrin’s integrability conditions. Regularity criteria for weak solutions to the MHD equations in terms of π≔p+12|b|2 are studied in section 5. Finally, section 6 contains remarks on analogous results in the case of Navier’s or Navier-type boundary conditions for the velocity u.