Global Classical Solutions and Stabilization in a Two-Dimensional Parabolic-Elliptic Keller–Segel–Stokes System

2021 ◽  
Vol 23 (3) ◽  
Author(s):  
Jiashan Zheng
Keyword(s):  
Stokes System ◽  
Classical Solutions ◽  
Two Dimensional ◽  
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.

scholarly journals Existence and uniqueness of global classical solutions to a two dimensional two species cancer invasion haptotaxis model

2018 ◽  
Vol 23 (10) ◽  
pp. 4397-4431
Author(s):  
Jan Giesselmann ◽  
◽  
Niklas Kolbe ◽  
Lukacova-MedvidovaMaria ◽  
Nikolaos Sfakianakis ◽  
...  
Keyword(s):  
Existence And Uniqueness ◽  
Cancer Invasion ◽  
Classical Solutions ◽  
Two Dimensional ◽  
Global Classical Solutions
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