Existence of infinite-energy and discretely self-similar global weak solutions for 3D MHD equations

This paper deals with the existence of global weak solutions for 3D MHD equations when the initial data belong to the weighted spaces $L^2_{w_\gamma}$, with $w_\gamma(x)=(1+\vert x\vert)^{-\gamma}$ and $0 \leq \gamma \leq 2$. Moreover, we prove the existence of discretely self-similar solutions for 3D MHD equations for discretely self-similar initial data which are locally square integrable. Our methods are inspired of a recent work of P. Fern\'aidez-Dalgo and P.G. Lemarié-Rieusset for the 3D Navier-Stokes equations

Blow-up of dynamically restricted critical norms near a potential Navier–Stokes singularity

In this paper we develop new methods to obtain regularity criteria for the three-dimensional Navier–Stokes equations in terms of dynamically restricted endpoint critical norms: the critical Lebesgue norm in general or the critical weak Lebesgue norm in the axisymmetric case. This type of results is inspired in particular by a work of Neustupa (Arch Ration Mech Anal 214(2):525–544, 2014), which handles certain non endpoint critical norms. Our work enables to have a better understanding of the nonlocal effect of the pressure on the regularity of the solutions.</p

Blow-up of dynamically restricted critical norms near a potential Navier–Stokes singularity

In this paper we develop new methods to obtain regularity criteria for the three-dimensional Navier–Stokes equations in terms of dynamically restricted endpoint critical norms: the critical Lebesgue norm in general or the critical weak Lebesgue norm in the axisymmetric case. This type of results is inspired in particular by a work of Neustupa (Arch Ration Mech Anal 214(2):525–544, 2014), which handles certain non endpoint critical norms. Our work enables to have a better understanding of the nonlocal effect of the pressure on the regularity of the solutions.</p

Weak solutions for Navier--Stokes equations with initial data in weighted L2L^2 spaces.

We show the existence of global weak solutions of the 3D Navier-Stokes equations with initial velocity in the weighted spaces L 2 wγ , where w γ (x) = (1 + |x|) −γ and 0 < γ ≤ 2, using new energy controls. As application we give a new proof of the existence of global weak discretely self-similar solutions of the 3D Navier-Stokes equations for discretely self-similar initial velocities which are locally square inte-grable

Weighted energy estimates for the incompressible Navier-Stokes equations and applications to axisymmetric solutions without swirl

We consider a family of weights which permit to generalize the Leray procedure to obtain weak suitable solutions of the 3D incom-pressible Navier-Stokes equations with initial data in weighted L 2 spaces. Our principal result concerns the existence of regular global solutions when the initial velocity is an axisymmetric vector field without swirl such that both the initial velocity and its vorticity belong to L 2 ((1 + r 2) -- $\gamma$ 2 dx), with r = x 2 1 + x 2 2 and $\gamma$ $\in$ (0, 2)

Existence of infinite-energy and discretely self-similar global weak solutions for 3D MHD equations

This paper deals with the existence of global weak solutions for 3D MHD equations when the initial data belong to the weighted spaces $L^2_{w_\gamma}$, with $w_\gamma(x)=(1+\vert x\vert)^{-\gamma}$ and $0 \leq \gamma \leq 2$. Moreover, we prove the existence of discretely self-similar solutions for 3D MHD equations for discretely self-similar initial data which are locally square integrable. Our methods are inspired of a recent work of P. Fern\'aidez-Dalgo and P.G. Lemarié-Rieusset for the 3D Navier-Stokes equations

Weak suitable solutions for 3D MHD equations for intermittent initial data

In this note, we extend some recent results on the local and global existence of solutions for 3D magneto-hydrodynamics equations to the more general setting of the intermittent initial data, which is characterized through a local Morrey space. This large initial data space was also exhibit in a contemporary work [3] in the context of 3D Navier-Stokes equations