Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userEnergy conservation for the non-resistive MHD equations with physical boundaries
80
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper, we study the energy equality for weak solutions to the non-resistive MHD equations with physical boundaries. Although the equations of magnetic field $b$ are of hyperbolic type, and the boundary effects are considered, we still prove the global energy equality provided that $u in L^{q}_{loc}left(0, T ; L^{p}(Omega)right) text { for any } frac{1}{q}+frac{1}{p} leq frac{1}{2}, text { with } p geq 4,text{ and } b in L^{r}_{loc}left(0, T ; L^{s}(Omega)right) text { for any } frac{1}{r}+frac{1}{s} leq frac{1}{2}, text { with } s geq 4 $. In particular, compared with the existed results, we do not require any boundary layer assumptions and additional conditions on the pressure $P$. Our result requires the regularity of boundary $partialOmega$ is only Lipschitz which is the minimum requirement to make the boundary condition $bcdot n$ sense. To approach our result, we first separate the mollification of weak solutions from the boundary effect by considering a non-standard local energy equality and transform the boundary effects into the estimates of the gradient of cut-off functions. Then, by establishing a sharp $L^2L^2$ estimate for pressure $P$, we use zero boundary conditions of $u$ to inhibit the boundary effect and obtain global energy equality by choosing suitable cut-off functions.
rate research
Read More
This paper establishes the local-in-time existence and uniqueness of strong solutions in $H^{s}$ for $s > n/2$ to the viscous, non-resistive magnetohydrodynamics (MHD) equations in $mathbb{R}^{n}$, $n=2, 3$, as well as for a related model where the advection terms are removed from the velocity equation. The uniform bounds required for proving existence are established by means of a new estimate, which is a partial generalisation of the commutator estimate of Kato & Ponce (Comm. Pure Appl. Math. 41(7), 891-907, 1988).
This paper is concerned with the asymptotic behaviors of global strong solutions to the incompressible non-resistive viscous magnetohydrodynamic (MHD) equations with large initial perturbations in two-dimensional periodic domains in Lagrangian coordinates. First, motivated by the odevity conditions imposed in [Arch. Ration. Mech. Anal. 227 (2018), 637--662], we prove the existence and uniqueness of strong solutions under some class of large initial perturbations, where the strength of impressive magnetic fields depends increasingly on the $H^2$-norm of the initial perturbation values of both velocity and magnetic field. Then, we establish time-decay rates of strong solutions. Moreover, we find that $H^2$-norm of the velocity decays faster than the perturbed magnetic field. Finally, by developing some new analysis techniques, we show that the strong solution convergence in a rate of the field strength to the solution of the corresponding linearized problem as the strength of the impressive magnetic field goes to infinity. In addition, an extension of similar results to the corresponding inviscid case with damping is presented.
In this paper, we address the space-time decay properties for strong solutions to the incompressible viscous resistive Hall-MHD equations. We obtained the same space-time decay rates as those of the heat equation. Based on the temporal decay results in cite{cs}, we find that one can obtain weighted estimates of the magnetic field $B$ by direct weighted energy estimate, and then by regarding the magnetic convection term as a forcing term in the velocity equations, we can obtain the weighted estimates for the vorticity, which yields the corresponding estimates for the velocity field. The higher order derivative estimates will be obtained by using a parabolic interpolation inequality proved in cite{k01}. It should be emphasized that the the magnetic field has stronger decay properties than the velocity field in the sense that there is no restriction on the exponent of the weight. The same arguments also yield the sharp space-time decay rates for strong solutions to the usual MHD equations.
In this paper, we mainly investigate the Cauchy problem of the non-resistive MHD equation. We first establish the local existence in the homogeneous Besov space $dot{B}^{frac{d}{p}-1}_{p,1}times dot{B}^{frac{d}{p}}_{p,1}$ with $p<infty$, and give a lifespan $T$ of the solution which depends on the norm of the Littlewood-Paley decomposition of the initial data. Then, we prove that if the initial data $(u^n_0,b^n_0)rightarrow (u_0,b_0)$ in $dot{B}^{frac{d}{p}-1}_{p,1}times dot{B}^{frac{d}{p}}_{p,1}$, then the corresponding existence times $T_nrightarrow T$, which implies that they have a common lower bound of the lifespan. Finally, we prove that the data-to-solutions map depends continuously on the initial data when $pleq 2d$. Therefore the non-resistive MHD equation is local well-posedness in the homogeneous Besov space in the Hadamard sense. Our obtained result improves considerably the recent results in cite{Li1,chemin1,Feffer2}.
Onsager conjectured that weak solutions of the Euler equations for incompressible fluids in 3D conserve energy only if they have a certain minimal smoothness, (of order of 1/3 fractional derivatives) and that they dissipate energy if they are rougher. In this paper we prove that energy is conserved for velocities in the function space $B^{1/3}_{3,c(NN)}$. We show that this space is sharp in a natural sense. We phrase the energy spectrum in terms of the Littlewood-Paley decomposition and show that the energy flux is controlled by local interactions. This locality is shown to hold also for the helicity flux; moreover, every weak solution of the Euler equations that belongs to $B^{2/3}_{3,c(NN)}$ conserves helicity. In contrast, in two dimensions, the strong locality of the enstrophy holds only in the ultraviolet range.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
