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.

Local well-posedness for the quantum Zakharov system in three and higher dimensions

2021 ◽  
Vol 18 (02) ◽  
pp. 257-270
Author(s):  
Isao Kato
Keyword(s):  
Cauchy Problem ◽  
Type System ◽  
Higher Dimensions ◽  
Well Posedness ◽  
Zakharov System ◽  
Quantum Parameter ◽  
Fourier Restriction ◽  
The Cauchy Problem ◽  
The One ◽  
Quantum Zakharov System

We study the Cauchy problem associated with a quantum Zakharov-type system in three and higher spatial dimensions.Taking the quantum parameter to unit and developing Fourier restriction norm arguments, we establish local well-posedness property for wider range than the one known for the Zakharov system.

Global well-posedness of the 2-D magnetic Prandtl model in the Prandtl–Hartmann regime

2020 ◽  
Vol 120 (3-4) ◽  
pp. 373-393
Author(s):  
Dongxiang Chen ◽  
Siqi Ren ◽  
Yuxi Wang ◽  
Zhifei Zhang
Keyword(s):  
Sobolev Space ◽  
Initial Data ◽  
Small Perturbation ◽  
Well Posedness ◽  
Prandtl Model

In this paper, we prove the global well-posedness of the 2-D magnetic Prandtl model in the mixed Prandtl/Hartmann regime when the initial data is a small perturbation of the Hartmann layer in Sobolev space.

Dyadic bilinear estimates and applications to the well-posedness for the 2D Zakharov–Kuznetsov equation in the endpoint space 𝐻−1/4

2020 ◽  
Vol 32 (6) ◽  
pp. 1575-1598
Author(s):  
Zhaohui Huo ◽  
Yueling Jia
Keyword(s):  
Cauchy Problem ◽  
Sobolev Space ◽  
Initial Data ◽  
Well Posedness ◽  
Small Initial Data ◽  
Bilinear Estimates ◽  
The Cauchy Problem ◽  
Univariate Polynomial ◽  
Well Posed ◽  
Resonance Function

AbstractThe Cauchy problem of the 2D Zakharov–Kuznetsov equation {\partial_{t}u+\partial_{x}(\partial_{xx}+\partial_{yy})u+uu_{x}=0} is considered. It is shown that the 2D Z-K equation is locally well-posed in the endpoint Sobolev space {H^{-1/4}}, and it is globally well-posed in {H^{-1/4}} with small initial data. In this paper, we mainly establish some new dyadic bilinear estimates to obtain the results, where the main novelty is to parametrize the singularity of the resonance function in terms of a univariate polynomial.

Local well-posedness for the quantum Zakharov system

2020 ◽  
Vol 18 (5) ◽  
pp. 1383-1411
Author(s):  
Yung-Fu Fang ◽  
Kuan-Hsiang Wang
Keyword(s):  
Well Posedness ◽  
Zakharov System ◽  
Quantum Zakharov System

scholarly journals Low Regularity Global Well-posedness for the Quantum Zakharov System in $1D$

2017 ◽  
Vol 21 (2) ◽  
pp. 341-361
Author(s):  
Tsai-Jung Chen ◽  
Yung-Fu Fang ◽  
Kuan-Hsiang Wang
Keyword(s):  
Well Posedness ◽  
Low Regularity ◽  
Zakharov System ◽  
Quantum Zakharov System

scholarly journals Well/ill-posedness for the dissipative Navier–Stokes system in generalized Carleson measure spaces

2017 ◽  
Vol 8 (1) ◽  
pp. 203-224 ◽  
Author(s):  
Yuzhao Wang ◽  
Jie Xiao
Keyword(s):  
Sobolev Space ◽  
Initial Data ◽  
Carleson Measure ◽  
Stokes System ◽  
Mild Solutions ◽  
Navier Stokes ◽  
Well Posedness ◽  
Solution Map ◽  
Measure Spaces ◽  
Dissipative Case

Abstract As an essential extension of the well known case {\beta\kern-1.0pt\in\kern-1.0pt({\frac{1}{2}},1]} to the hyper-dissipative case {\beta\kern-1.0pt\in\kern-1.0pt(1,\infty)} , this paper establishes both well-posedness and ill-posedness (not only norm inflation but also indifferentiability of the solution map) for the mild solutions of the incompressible Navier–Stokes system with dissipation {(-\Delta)^{{\frac{1}{2}}<\beta<\infty}} through the generalized Carleson measure spaces of initial data that unify many diverse spaces, including the Q space {(Q_{-s=-\alpha})^{n}} , the BMO-Sobolev space {((-\Delta)^{-{\frac{s}{2}}}\mathrm{BMO})^{n}} , the Lip-Sobolev space {((-\Delta)^{-{\frac{s}{2}}}\mathrm{Lip}\alpha)^{n}} , and the Besov space {(\dot{B}^{s}_{\infty,\infty})^{n}} .

scholarly journals GAUGE CHOICE FOR THE YANG–MILLS EQUATIONS USING THE YANG–MILLS HEAT FLOW AND LOCAL WELL-POSEDNESS IN H1

2014 ◽  
Vol 11 (01) ◽  
pp. 1-108 ◽  
Author(s):  
SUNG-JIN OH
Keyword(s):  
Heat Flow ◽  
Initial Data ◽  
Classical Result ◽  
Finite Energy ◽  
Previous Method ◽  
Alternative Proof ◽  
Well Posedness ◽  
Yang Mills ◽  
Gauge Choice ◽  
Novel Approach

We introduce a novel approach to the problem of gauge choice for the Yang–Mills equations on the Minkowski space ℝ1+3, which uses the Yang–Mills heat flow in a crucial way. As this approach does not possess the drawbacks of the previous approaches, it is expected to be more robust and easily adaptable to other settings. As a first application, we give an alternative proof of the local well-posedness of the Yang–Mills equations for initial data [Formula: see text], which is a classical result of Klainerman and Machedon (1995) that had been proved using a different method (local Coulomb gauges). The new proof does not involve localization in space–time, which had been the key drawback of the previous method. Based on the results proved in this paper, a new proof of finite energy global well-posedness of the Yang–Mills equations, also using the Yang–Mills heat flow, is established in a companion article.

Sign in / Sign up

Export Citation Format

Share Document