In this paper, we study the regularity criteria for axisymmetric weak solutions to the MHD equa- tions in R3. Let we, Jo and ue be the azimuthal component of w, J and u in the cylindrical coordinates, respectively. Then the axisymmetric weak solution (u,b) is regular on (0, T) if (wo,Jo) E Lq(O,T;Lp) or (oae,V(uoeo)) e Lq(0,T;Lp) with 3 + 2 〈 2, 3 〈 p 〈 oo. In the endpoint case, one needs conditions (we, Jo) C LI(0, T;B∞∞) or (wo, V(uoeo)) C LI(0, T;B ∞∞).
We prove the global existence of weak solutions of the one-dimensional compressible Navier-stokes equations with density-dependent viscosity. In particular, we assume that the initial density belongs to L^1 and L^∞, module constant states at x = -∞ and x = +∞, which may be different. The initial vacuum is permitted in this paper and the results may apply to the one-dimensional Saint-Venant model for shallow water.
Using the fibering method introduced by Pohozaev, we prove existence of positive solution for a Diriclhlet problem with a quasilinear system involving p-Laplacian operator.