scholarly journals Well-posedness for the heat flow of polyharmonic maps with rough initial data

2011 ◽  
Vol 4 (2) ◽  
Author(s):  
Tao Huang ◽  
Changyou Wang
Keyword(s):  
Heat Flow ◽  
Initial Data ◽  
Well Posedness ◽  
Rough Initial Data

Global well-posedness of the fractional Klein-Gordon-Schrödinger system with rough initial data

2016 ◽  
Vol 59 (7) ◽  
pp. 1345-1366 ◽  
Author(s):  
ChunYan Huang ◽  
BoLing Guo ◽  
DaiWen Huang ◽  
QiaoXin Li
Keyword(s):  
Initial Data ◽  
Well Posedness ◽  
Klein Gordon ◽  
Rough Initial Data ◽  
Schrödinger System

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.

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 Well posedness for the Kawahara equation on the half-line

2020 ◽  
Author(s):  
Boling Guo ◽  
fengxia liu
Keyword(s):  
Initial Data ◽  
Local Existence ◽  
Half Line ◽  
Well Posedness ◽  
Kawahara Equation ◽  
Low Regularity ◽  
Regularity Properties

We study the low-regularity properties of the Kawahara equation on the half line. We obtain the local existence, uniqueness, and continuity of the solution. Moreover, We obtain that the nonlinear terms of the solution are smoother than the initial data.

scholarly journals Almost Sure Global Well-posedness for the Fractional Cubic Schrödinger Equation on the Torus

2015 ◽  
Vol 58 (3) ◽  
pp. 471-485 ◽  
Author(s):  
Seckin Demirbas
Keyword(s):  
Probability Measure ◽  
Initial Data ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Invariant Probability Measure ◽  
Well Posedness ◽  
Cubic Nonlinearity ◽  
Cubic Schrödinger Equation ◽  
Well Posed ◽  
Fractional Schrodinger Equation

AbstractIn a previous paper, we proved that the 1-d periodic fractional Schrödinger equation with cubic nonlinearity is locally well-posed inHsfors> 1 −α/2 and globally well-posed fors> 10α− 1/12. In this paper we define an invariant probability measureμonHsfors<α− 1/2, so that for any ∊ > 0 there is a set Ω ⊂Hssuch thatμ(Ωc) <∊and the equation is globally well-posed for initial data in Ω. We see that this fills the gap between the local well-posedness and the global well-posedness range in an almost sure sense forin an almost sure sense.

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