Global classical solutions of 3D isentropic compressible MHD with general initial data

2015 ◽  
Vol 66 (4) ◽  
pp. 1777-1797 ◽  
Author(s):  
Yang Liu
Keyword(s):  
Initial Data ◽  
Classical Solutions ◽  
General Initial Data ◽  
Global Classical Solutions

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 On the Existence of Long-Time Classical Solutions for the 2D Inviscid Boussinesq Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Jiafa Xu ◽  
Lishan Liu
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Life Span ◽  
Boussinesq Equations ◽  
Classical Solutions ◽  
Buoyancy Frequency ◽  
General Initial Data ◽  
The Cauchy Problem ◽  
Long Time

In this paper, we consider the Cauchy problem for the 2D inviscid Boussinesq equations with N being the buoyancy frequency. It is proved that for general initial data u 0 ∈ H s with s > 3 , the life span of the classical solutions satisfies T > C ln     N 3 / 4 .

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 ¯ ) .

HN stability of the Vlasov–Poisson–Boltzmann system near Maxwellians

2007 ◽  
Vol 137 (2) ◽  
pp. 431-446 ◽  
Author(s):  
Hongjun Yu
Keyword(s):  
Initial Data ◽  
Stability Estimate ◽  
Classical Solutions ◽  
Energy Methods ◽  
Global Classical Solutions ◽  
Poisson Boltzmann

We study the HN stability of the Vlasov–Poisson Boltzmann system near Maxwellians. Under a suitable smallness assumption on initial data, we show that the global classical solutions constructed by Guo are HN stable. For a stability estimate, we employ the energy methods of Guo.

Sign in / Sign up

Export Citation Format

Share Document