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 [Formula presented] 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.
Bibliographical noteFunding Information:
The authors would like to thank the anonymous reviewer for his/her valuable comments and suggestions. The first author acknowledges the support of the Grant Agency of the Czech Republic , Grant No. GA19-04243S , and the Academy of Sciences of the Czech Republic ( RVO 67985840 ). The second author has been supported by the National Research Foundation of Korea No. 2016R1C1B2015731 , No. 2015R1A5A1009350 and the Yonsei University, Republic of Korea No. 2019-22-0034 .
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All Science Journal Classification (ASJC) codes
- Economics, Econometrics and Finance(all)
- Computational Mathematics
- Applied Mathematics