Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn approximate solutions of the equations of incompressible magnetohydrodynamics
101
0
0.0
(
0
)
Authors
Livio Pizzocchero
Ask ChatGPT about the research
No Arabic abstract
Inspired by an approach proposed previously for the incompressible Navier-Stokes (NS) equations, we present a general framework for the a posteriori analysis of the equations of incompressible magnetohydrodynamics (MHD) on a torus of arbitrary dimension d; this setting involves a Sobolev space of infinite order, made of C^infinity vector fields (with vanishing divergence and mean) on the torus. Given any approximate solution of the MHD Cauchy problem, its a posteriori analysis with the method of the present work allows to infer a lower bound on the time of existence of the exact solution, and to bound from above the Sobolev distance of any order between the exact and the approximate solution. In certain cases the above mentioned lower bound on the time of existence is found to be infinite, so one infers the global existence of the exact MHD solution. We present some applications of this general scheme; the most sophisticated one lives in dimension d=3, with the ABC flow (perturbed magnetically) as an initial datum, and uses for the Cauchy problem a Galerkin approximate solution in 124 Fourier modes. We illustrate the conclusions arising in this case from the a posteriori analysis of the Galerkin approximant; these include the derivation of global existence of the exact MHD solution with the ABC datum, when the dimensionless viscosity and resistivity are equal and stay above an explicitly given threshold value.
rate research
Read More
It is known that the Maxwell-Klein-Gordon equations in $mathbb{R}^{3+1}$ admit global solutions with finite energy data. In this paper, we present a new approach to study the asymptotic behavior of these global solutions. We show the quantitative energy flux decay of the solutions with data merely bounded in some weighted energy space. We also establish an integrated local energy decay and a hierarchy of $r$-weighted energy decay. The results in particular hold in the presence of large total charge. This is the first result to give a complete and precise description of the global behavior of large nonlinear charged scalar fields.
Consider Yudovich solutions to the incompressible Euler equations with bounded initial vorticity in bounded planar domains or in $mathbb{R}^2$. We present a purely Lagrangian proof that the solution map is strongly continuous in $L^p$ for all $pin [1, infty)$ and is weakly-$*$ continuous in $L^infty$.
We prove that the energy equality holds for weak solutions of the 3D Navier-Stokes equations in the functional class $L^3([0,T);V^{5/6})$, where $V^{5/6}$ is the domain of the fractional power of the Stokes operator $A^{5/12}$.
We study the well-posedness theory for the linearized free boundary problem of incompressible ideal magnetohydrodynamics equations in a bounded domain. We express the magnetic field in terms of the velocity field and the deformation tensors in the Lagrangian coordinates, and substitute the magnetic field into the momentum equation to get an equation of the velocity in which the initial magnetic field serves only as a parameter. Then, we linearize this equation with respect to the position vector field whose time derivative is the velocity, and obtain the local-in-time well-posedness of the solution by using energy estimates of the tangential derivatives and the curl with the help of Lie derivatives and the smooth-out approximation.
We prove, by a shooting method, the existence of infinitely many solutions of the form $psi(x^0,x) = e^{-iOmega x^0}chi(x)$ of the nonlinear Dirac equation {equation*} iunderset{mu=0}{overset{3}{sum}} gamma^mu partial_mu psi- mpsi - F(bar{psi}psi)psi = 0 {equation*} where $Omega>m>0,$ $chi$ is compactly supported and [F(x) = {{array}{ll} p|x|^{p-1} & text{if} |x|>0 0 & text{if} x=0 {array}.] with $pin(0,1),$ under some restrictions on the parameters $p$ and $Omega.$ We study also the behavior of the solutions as $p$ tends to zero to establish the link between these equations and the M.I.T. bag model ones.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
