On global classical solutions of the three dimensional relativistic Vlasov-Darwin system

2016 ◽  
Vol 57 (8) ◽  
pp. 081508
Author(s):  
Xiuting Li ◽  
Xianwen Zhang
Keyword(s):  
Three Dimensional ◽  
Classical Solutions ◽  
Global Classical Solutions

scholarly journals Optimal control of the two-dimensional Vlasov-Maxwell system

2020 ◽  
Author(s):  
Jörg Weber
Keyword(s):  
Adjoint Equation ◽  
Collisionless Plasma ◽  
Three Dimensional ◽  
Classical Solutions ◽  
Two Dimensional ◽  
Maxwell System ◽  
Global Classical Solutions ◽  
Global Existence Of Solutions ◽  
Dimensional Version ◽  
The One

The time evolution of a collisionless plasma is modeled by the Vlasov-Maxwell system which couples the Vlasov equation (the transport equation) with the Maxwell equations of electrodynamics. We only consider a two-dimensional version of the problem since existence of global, classical solutions of the full three-dimensional problem is not known. We add external currents to the system, in applications generated by coils, to control the plasma properly. After considering global existence of solutions to this system, differentiability of the control-to-state operator is proved. In applications, on the one hand, we want the shape of the plasma to be close to some desired shape. On the other hand, a cost term penalizing the external currents shall be as small as possible. These two aims lead to minimizing some objective function. We restrict ourselves to only such control currents that are realizable in applications. After that, we prove existence of a minimizer and deduce first order optimality conditions and the adjoint equation.

Global classical solutions to the 3D Navier–Stokes–Korteweg equations with small initial energy

2017 ◽  
Vol 16 (01) ◽  
pp. 55-84 ◽  
Author(s):  
Xiaofeng Hou ◽  
Hongyun Peng ◽  
Changjiang Zhu
Keyword(s):  
Three Dimensional ◽  
Type System ◽  
Global Solutions ◽  
Initial Energy ◽  
Navier Stokes ◽  
Classical Solutions ◽  
Dimensional System ◽  
Well Posedness ◽  
Global Classical Solutions ◽  
Two Dimensional System

In this paper, we investigate the global well-posedness of classical solutions to three-dimensional Cauchy problem of the compressible Navier–Stokes type system with a Korteweg stress tensor under the condition that the initial energy is small. This result improves previous results obtained by Hattori–Li in [H. Hattori and D. Li, Solutions for two dimensional system for materials of Korteweg type, SIAM J. Math. Anal. 25 (1994) 85–98; H. Hattori and D. Li. Global solutions of a high-dimensional system for Korteweg materials. J. Math. Anal. Appl. 198 (1996) 84–97.], where the existence of the classical solution is established for initial data close to an equilibrium in some Sobolev space [Formula: see text].

scholarly journals Blow-up prevention by quadratic degradation in a higher-dimensional chemotaxis-growth model with indirect attractant production

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jianing Xie
Keyword(s):  
Mountain Pine Beetle ◽  
Blow Up ◽  
Three Dimensional ◽  
Regularity Theory ◽  
Classical Solutions ◽  
Sobolev Regularity ◽  
Global Classical Solutions ◽  
Flux Boundary Conditions ◽  
Growth System ◽  
Energy Type

<p style='text-indent:20px;'>This paper deals with a boundary-value problem in three-dimensional smoothly bounded domains for a coupled chemotaxis-growth system generalizing the prototype</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$\begin{align} \left\{\begin{array}{ll} u_t = \Delta u-\nabla\cdot(u\nabla v)+\mu u(1-u),\quad x\in \Omega, t&gt;0,\\ { }{ v_t = \Delta v- v +w},\quad x\in \Omega, t&gt;0,\\ { }{\tau w_t+\delta w = u},\quad x\in \Omega, t&gt;0\\ \end{array}\right. \end{align} (*)$ \end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>in a smoothly bounded domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset\mathbb{R}^N(N\geq1) $\end{document}</tex-math></inline-formula> under zero-flux boundary conditions, which describe the spread and aggregative behavior of the Mountain Pine Beetle in forest habitat, where the parameters <inline-formula><tex-math id="M2">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> as well as <inline-formula><tex-math id="M3">\begin{document}$ \delta $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \tau $\end{document}</tex-math></inline-formula> are positive. Based on an <b>new</b> energy-type argument combined with maximal Sobolev regularity theory, it is proved that global classical solutions exist whenever</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \mu&gt;\left\{ \begin{array}{ll} {0, \; \; \; {\rm{if}}\; \; N\leq4},\\ {\frac{(N-4)_{+}}{N-2}\max\{1,\lambda_{0}\},\; \; \; {\rm{if}}\; \; N\geq5}\\ \end{array} \right. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>and the initial data <inline-formula><tex-math id="M5">\begin{document}$ (u_0,v_0,w_0) $\end{document}</tex-math></inline-formula> are sufficiently regular. Here <inline-formula><tex-math id="M6">\begin{document}$ \lambda_0 $\end{document}</tex-math></inline-formula> is a positive constant which is corresponding to the maximal Sobolev regularity. This extends some recent results by several authors.</p>

GLOBAL CLASSICAL SOLUTIONS FOR A COMPRESSIBLE FLUID-PARTICLE INTERACTION MODEL

2013 ◽  
Vol 10 (03) ◽  
pp. 537-562 ◽  
Author(s):  
MYEONGJU CHAE ◽  
KYUNGKEUN KANG ◽  
JIHOON LEE
Keyword(s):  
Stokes Equations ◽  
Particle Interaction ◽  
Three Dimensional ◽  
Interaction Model ◽  
Planck Equation ◽  
Navier Stokes ◽  
Classical Solutions ◽  
Navier Stokes Equations ◽  
Global Classical Solutions ◽  
System Coupling

We consider a system coupling the compressible Navier–Stokes equations to the Vlasov–Fokker–Planck equation on three-dimensional torus. The coupling arises from a drag force exerted by each other. We establish the existence of the global classical solutions close to an equilibrium, and further prove that the solutions converge to the equilibrium exponentially fast.

scholarly journals Global classical solutions to the elastodynamic equations with damping

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mengmeng Liu ◽  
Xueyun Lin
Keyword(s):  
Initial Data ◽  
Periodic Boundary ◽  
Energy Estimate ◽  
Periodic Boundary Conditions ◽  
Deformation Tensor ◽  
Classical Solutions ◽  
Small Initial Data ◽  
Weighted Energy ◽  
Global Classical Solutions ◽  
Weighted Energy Estimate

AbstractIn this paper, we show the global existence of classical solutions to the incompressible elastodynamics equations with a damping mechanism on the stress tensor in dimension three for sufficiently small initial data on periodic boxes, that is, with periodic boundary conditions. The approach is based on a time-weighted energy estimate, under the assumptions that the initial deformation tensor is a small perturbation around an equilibrium state and the initial data have some symmetry.

scholarly journals Global Existence and Blow-Up for the Classical Solutions of the Long-Short Wave Equations with Viscosity

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Jincheng Shi ◽  
Shengzhong Xiao
Keyword(s):  
Global Existence ◽  
Life Span ◽  
General Model ◽  
Wave Equations ◽  
Blow Up ◽  
Initial Conditions ◽  
Short Wave ◽  
Classical Solutions ◽  
Global Classical Solutions

We are concerned with the global existence of classical solutions for a general model of viscosity long-short wave equations. Under suitable initial conditions, the existence of the global classical solutions for the viscosity long-short wave equations is proved. If it does not exist globally, the life span which is the largest time where the solutions exist is also obtained.

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