scholarly journals On Global Well-Posedness and Temporal Decay for 3D Magnetic Induction Equations with Hall Effect

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1847
Author(s):  
Xiaopeng Zhao
Keyword(s):  
Hall Effect ◽  
Magnetic Induction ◽  
Splitting Method ◽  
Three Dimensional ◽  
Well Posedness ◽  
Uniqueness Of Solutions ◽  
Algebraic Decay ◽  
Temporal Decay ◽  
Global Existence And Uniqueness ◽  
Higher Order Derivative

The main purpose of this paper is to study the global existence and uniqueness of solutions for three-dimensional incompressible magnetic induction equations with Hall effect provided that ∥u0∥H32+ε+∥b0∥H2(0<ε<1) is sufficiently small. Moreover, using the Fourier splitting method and the properties of decay character r*, one also shows the algebraic decay rate of a higher order derivative of solutions to magnetic induction equations with the Hall effect.

Global well-posedness and temporal decay estimates to the fractional Cahn–Hilliard equation in ℝ N \mathbb{R}^{N}

2019 ◽  
Vol 31 (3) ◽  
pp. 803-814
Author(s):  
Ning Duan ◽  
Xiaopeng Zhao
Keyword(s):  
Cauchy Problem ◽  
Decay Rate ◽  
Weak Solutions ◽  
Well Posedness ◽  
Decay Estimates ◽  
Hilliard Equation ◽  
Temporal Decay ◽  
The Cauchy Problem ◽  
Higher Order Derivative ◽  
Cahn Hilliard Equation

AbstractThis paper is devoted to study the global well-posedness of solutions for the Cauchy problem of the fractional Cahn–Hilliard equation in{\mathbb{R}^{N}}({N\in\mathbb{N}^{+}}), provided that the initial datum is sufficiently small. In addition, the{L^{p}}-norm ({1\leq p\leq\infty}) temporal decay rate for weak solutions and the higher-order derivative of solutions are also studied.

scholarly journals On the well-posedness of the anisotropically-reduced two-dimensional Kuramoto-Sivashinsky Equation

2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
David Massatt
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Existence And Uniqueness ◽  
Well Posedness ◽  
Two Dimensional ◽  
Periodic Domain ◽  
Uniqueness Of Solutions ◽  
Global Existence And Uniqueness ◽  
Sivashinsky Equation

<p style='text-indent:20px;'>We address the global existence and uniqueness of solutions for the anisotropically reduced 2D Kuramoto-Sivashinsky equations in a periodic domain with initial data <inline-formula><tex-math id="M1">\begin{document}$ u_{01} \in L^2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ u_{02} \in H^{-1 + \eta} $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M3">\begin{document}$ \eta &gt; 0 $\end{document}</tex-math></inline-formula>.</p>

scholarly journals Theory of the Hall effect in three-dimensional metamaterials

2018 ◽  
Vol 20 (8) ◽  
pp. 083034 ◽  
Author(s):  
Christian Kern ◽  
Graeme W Milton ◽  
Muamer Kadic ◽  
Martin Wegener
Keyword(s):  
Hall Effect ◽  
Three Dimensional

STEADY FLOWS OF VISCOUS COMPRESSIBLE FLUIDS IN EXTERIOR DOMAINS UNDER SMALL PERTURBATIONS OF GREAT POTENTIAL FORCES

1993 ◽  
Vol 03 (06) ◽  
pp. 725-757 ◽  
Author(s):  
ANTONÍN NOVOTNÝ
Keyword(s):  
Existence And Uniqueness ◽  
Compressible Flows ◽  
Three Dimensional ◽  
Compressible Fluids ◽  
Steady Flows ◽  
Exterior Domains ◽  
Uniqueness Of Solutions ◽  
Small Perturbations ◽  
Zero Velocity ◽  
Potential Forces

We investigate the steady compressible flows in three-dimensional exterior domains, in R3 and [Formula: see text], under the action of small perturbations of large potential forces and zero velocity at infinity. We prove existence and uniqueness of solutions in L2-spaces, and study their regularity as well as the decay at infinity.

Three-dimensional quantum Hall effect in Weyl and double-Weyl semimetals

2018 ◽  
Vol 382 (44) ◽  
pp. 3205-3210
Author(s):  
Zhi-Peng Gao ◽  
Zhi Li ◽  
Dan-Wei Zhang
Keyword(s):  
Hall Effect ◽  
Quantum Hall Effect ◽  
Quantum Hall ◽  
Three Dimensional ◽  
Weyl Semimetals

scholarly journals The Yang-Mills heat equation with finite action in three dimensions

2022 ◽  
Vol 275 (1349) ◽  
Author(s):  
Leonard Gross
Keyword(s):  
Heat Equation ◽  
Three Dimensional ◽  
Strong Solutions ◽  
Three Dimensions ◽  
Dimensional Manifold ◽  
Uniqueness Of Solutions ◽  
Reconstruction Procedure ◽  
Content Type ◽  
Yang Mills ◽  
Gauge Functions

The existence and uniqueness of solutions to the Yang-Mills heat equation is proven over R 3 \mathbb {R}^3 and over a bounded open convex set in R 3 \mathbb {R}^3 . The initial data is taken to lie in the Sobolev space of order one half, which is the critical Sobolev index for this equation over a three dimensional manifold. The existence is proven by solving first an augmented, strictly parabolic equation and then gauge transforming the solution to a solution of the Yang-Mills heat equation itself. The gauge functions needed to carry out this procedure lie in the critical gauge group of Sobolev regularity three halves, which is a complete topological group in a natural metric but is not a Hilbert Lie group. The nature of this group must be understood in order to carry out the reconstruction procedure. Solutions to the Yang-Mills heat equation are shown to be strong solutions modulo these gauge functions. Energy inequalities and Neumann domination inequalities are used to establish needed initial behavior properties of solutions to the augmented equation.

scholarly journals Well-posedness and long-time behavior for the modified phase-field crystal equation

2014 ◽  
Vol 24 (14) ◽  
pp. 2743-2783 ◽  
Author(s):  
Maurizio Grasselli ◽  
Hao Wu
Keyword(s):  
Phase Field ◽  
Time Derivative ◽  
Exponential Attractor ◽  
Well Posedness ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Long Time ◽  
Global Existence And Uniqueness ◽  
And Diffusion ◽  
Phase Field Crystal

We consider a modification of the so-called phase-field crystal (PFC) equation introduced by K. R. Elder et al. This variant has recently been proposed by P. Stefanovic et al. to distinguish between elastic relaxation and diffusion time scales. It consists of adding an inertial term (i.e. a second-order time derivative) into the PFC equation. The mathematical analysis of the resulting equation is more challenging with respect to the PFC equation, even at the well-posedness level. Moreover, its solutions do not regularize in finite time as in the case of PFC equation. Here we analyze the modified PFC (MPFC) equation endowed with periodic boundary conditions. We first prove the global existence and uniqueness of a solution with initial data in a bounded energy space. This solution satisfies some uniform dissipative estimates which allow us to study the long-time behavior of the corresponding dynamical system. In particular, we establish the existence of the global attractor as well as an exponential attractor. Then we demonstrate that any trajectory originating from the bounded energy phase space converges to a single equilibrium. This is done by means of a suitable version of the Łojasiewicz–Simon inequality. An estimate on the convergence rate is also given.

scholarly journals Some Remarks Concerning Jones Eigenfrequencies and Jones Modes

2005 ◽  
Vol 12 (2) ◽  
pp. 337-348
Author(s):  
David Natroshvili ◽  
Guram Sadunishvili ◽  
Irine Sigua
Keyword(s):  
Exterior Domain ◽  
Thermal Stresses ◽  
Acoustic Pressure ◽  
Pressure Field ◽  
Three Dimensional ◽  
Boundary Surface ◽  
Isotropic Body ◽  
Oscillation Parameter ◽  
Uniqueness Of Solutions ◽  
Fluid Solid Interaction

Abstract Three-dimensional fluid-solid interaction problems with regard for thermal stresses are considered. An elastic structure is assumed to be a bounded homogeneous isotropic body occupying a domain , where the thermoelastic four dimensional field is defined, while in the unbounded exterior domain there is defined the scalar (acoustic pressure) field. These two fields satisfy the differential equations of steady state oscillations in the corresponding domains along with the transmission conditions of special type on the interface ∂Ω±. We show that uniqueness of solutions strongly depends on the geometry of the boundary ∂Ω±. In particular, we prove that for the corresponding homogeneous transmission problem for a ball there exist infinitely many exceptional values of the oscillation parameter (Jones eigenfrequencies). The corresponding eigenvectors (Jones modes) are written explicitly. On the other hand, we show that if the boundary surface ∂Ω± contains two flat, non-parallel sub-manifolds then there are no Jones eigenfrequencies for such domains.

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