scholarly journals Global smooth solutions to the multidimensional hydrodynamic model for plasmas with insulating boundary conditions

2007 ◽  
Vol 336 (2) ◽  
pp. 888-904 ◽  
Author(s):  
Qiangchang Ju
Keyword(s):  
Boundary Conditions ◽  
Hydrodynamic Model ◽  
Smooth Solutions ◽  
Global Smooth Solutions

scholarly journals On the Existence of Global Smooth Solutions to the Parabolic–Elliptic Keller–Segel System with Irregular Initial Data

2021 ◽  
Author(s):  
Frederic Heihoff
Keyword(s):  
Initial Data ◽  
Boundary Conditions ◽  
Neumann Boundary ◽  
Three Dimensions ◽  
Classical Solutions ◽  
List Type ◽  
Smooth Solutions ◽  
Smooth Bounded Domain ◽  
Global Classical Solutions ◽  
Global Smooth Solutions

AbstractWe consider the parabolic–elliptic Keller–Segel system $$\begin{aligned} \left\{ \begin{aligned} u_t&= \Delta u - \chi \nabla \cdot (u \nabla v), \\ 0&= \Delta v - v + u \end{aligned} \right. \end{aligned}$$ u t = Δ u - χ ∇ · ( u ∇ v ) , 0 = Δ v - v + u in a smooth bounded domain $$\Omega \subseteq {\mathbb {R}}^n$$ Ω ⊆ R n , $$n\in {\mathbb {N}}$$ n ∈ N , with Neumann boundary conditions. We look at both chemotactic attraction ($$\chi > 0$$ χ > 0 ) and repulsion ($$\chi < 0$$ χ < 0 ) scenarios in two and three dimensions. The key feature of interest for the purposes of this paper is under which conditions said system still admits global classical solutions due to the smoothing properties of the Laplacian even if the initial data is very irregular. Regarding this, we show for initial data $$\mu \in {\mathcal {M}}_+({\overline{\Omega }})$$ μ ∈ M + ( Ω ¯ ) that, if either $$n = 2$$ n = 2 , $$\chi < 0$$ χ < 0 or $$n = 2$$ n = 2 , $$\chi > 0$$ χ > 0 and the initial mass is small or $$n = 3$$ n = 3 , $$\chi < 0$$ χ < 0 and $$\mu = f \in L^p(\Omega )$$ μ = f ∈ L p ( Ω ) , $$p > 1$$ p > 1 holds, it is still possible to construct global classical solutions to ($$\star $$ ⋆ ), which are continuous in $$t = 0$$ t = 0 in the vague topology on $${\mathcal {M}}_+({\overline{\Omega }})$$ M + ( Ω ¯ ) .

ASYMPTOTIC BEHAVIOR OF GLOBAL SMOOTH SOLUTIONS TO THE FULL 1D HYDRODYNAMIC MODEL FOR SEMICONDUCTORS

2002 ◽  
Vol 12 (06) ◽  
pp. 777-796 ◽  
Author(s):  
LING HSIAO ◽  
SHU WANG
Keyword(s):  
Asymptotic Behavior ◽  
Boundary Value Problem ◽  
Stationary Solution ◽  
Hydrodynamic Model ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Smooth Solutions ◽  
One Dimensional ◽  
Global Smooth Solutions ◽  
Initial Boundary Value

In this paper, we study the asymptotic behavior of smooth solutions to the initial boundary value problem for the full one-dimensional hydrodynamic model for semiconductors. We prove that the solution to the problem converges to the unique stationary solution time asymptotically exponentially fast.

Singularities of solutions for first order quasilinear hyperbolic systems

1983 ◽  
Vol 94 (1-2) ◽  
pp. 137-147 ◽  
Author(s):  
Lee Da-tsin(Li Ta-tsien) ◽  
Shi Jia-hong
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Hyperbolic Systems ◽  
Smooth Solutions ◽  
Small Initial Data ◽  
Quasilinear Hyperbolic Systems ◽  
First Order ◽  
Global Smooth Solutions ◽  
The Cauchy Problem

SynopsisIn this paper, the existence of global smooth solutions and the formation of singularities of solutions for strictly hyperbolic systems with general eigenvalues are discussed for the Cauchy problem with essentially periodic small initial data or nonperiodic initial data. A result of Klainerman and Majda is thus extended to the general case.

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