New Regularity Criteria for Weak Solutions to the MHD Equations in Terms of an Associated Pressure

We assume that Ω is either a smooth bounded domain in R³ or Ω = R³, and Ω ^′ is a sub-domain of Ω. We prove that if 0 ≤ T₁< T₂≤ T≤ ∞, (u, b, p) is a suitable weak solution of the initial–boundary value problem for the MHD equations in Ω × (0 , T) and either Fγ(p-)∈L∞(T1,T2;L3/2(Ω′)) or Fγ(B+)∈L∞(T1,T2;L3/2(Ω′)) for some γ> 0 , where Fγ(s)=s[ln(1+s)]1+γ, B=p+12|u|2+12|b|2 and the subscripts “−” and “+ ” denote the negative and the nonnegative part, respectively, then the solution (u, b, p) has no singular points in Ω ^′× (T₁, T₂). If b≡ 0 then our result generalizes some previous known results from the theory of the Navier–Stokes equations.