Quasigeostrophic equation with random initial data in negative-order Sobolev space

2019 ◽  
Vol 70 (4) ◽  
Author(s):  
Lihuai Du ◽  
Huaqiao Wang
Keyword(s):  
Sobolev Space ◽  
Initial Data ◽  
Random Initial Data ◽  
Negative Order

Local well-posedness for the quantum Zakharov system in one spatial dimension

2017 ◽  
Vol 14 (01) ◽  
pp. 157-192 ◽  
Author(s):  
Yung-Fu Fang ◽  
Hsi-Wei Shih ◽  
Kuan-Hsiang Wang
Keyword(s):  
Sobolev Space ◽  
Initial Data ◽  
Electric Field ◽  
Classical Result ◽  
Spatial Dimension ◽  
Well Posedness ◽  
Ion Density ◽  
Zakharov System ◽  
Quantum Parameter ◽  
Quantum Zakharov System

We consider the quantum Zakharov system in one spatial dimension and establish a local well-posedness theory when the initial data of the electric field and the deviation of the ion density lie in a Sobolev space with suitable regularity. As the quantum parameter approaches zero, we formally recover a classical result by Ginibre, Tsutsumi, and Velo. We also improve their result concerning the Zakharov system and a result by Jiang, Lin, and Shao concerning the quantum Zakharov system.

scholarly journals SCALING LIMITS FOR TIME-FRACTIONAL DIFFUSION-WAVE SYSTEMS WITH RANDOM INITIAL DATA

2010 ◽  
Vol 10 (01) ◽  
pp. 1-35 ◽  
Author(s):  
GI-REN LIU ◽  
NARN-RUEIH SHIEH
Keyword(s):  
Initial Data ◽  
Initial Conditions ◽  
Scaling Limit ◽  
Reaction Diffusion ◽  
Equation System ◽  
Random Initial Data ◽  
Diffusion Type ◽  
Diffusion Wave ◽  
Decoupling Method ◽  
Time Space

Let w (x, t) := (u, v)(x, t), x ∈ ℝ3, t > 0, be the ℝ2-valued spatial-temporal random field w = (u, v) arising from a certain two-equation system of time-fractional linear partial differential equations of reaction-diffusion-wave type, with given random initial data u(x,0), ut(x,0), and v(x,0), vt(x,0). We discuss the scaling limit, under proper homogenization and renormalization, of w(x,t), subject to suitable assumptions on the random initial conditions. Since the component fields u,v depend on the interactions present within the system, we employ a certain stochastic decoupling method to tackle this component dependence. The work shows, in particular, the various non-Gaussian scenarios proposed in [4, 13, 17] and the references therein, for the single diffusion type equations, in classical or in fractional time/space derivatives, can be studied for the two-equation system, in a significant way.

scholarly journals The Cauchy Problem for a Fifth-Order Dispersive Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Hongjun Wang ◽  
Yongqi Liu ◽  
Yongqiang Chen
Keyword(s):  
Cauchy Problem ◽  
Sobolev Space ◽  
Initial Data ◽  
Order Equation ◽  
Dispersive Equations ◽  
Dispersive Equation ◽  
The Cauchy Problem ◽  
Fifth Order ◽  
Well Posed

This paper is devoted to studying the Cauchy problem for a fifth-order equation. We prove that it is locally well-posed for the initial data in the Sobolev spaceHs(R)withs≥1/4. We also establish the ill-posedness for the initial data inHs(R)withs<1/4. Thus, the regularity requirement for the fifth-order dispersive equationss≥1/4is sharp.

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