scholarly journals Local well-posedness for the Zakharov system in dimension $ d = 2, 3 $

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Zijun Chen ◽  
Shengkun Wu
Keyword(s):  
Normal Form ◽  
Unique Solution ◽  
Strichartz Estimates ◽  
Well Posedness ◽  
Zakharov System ◽  
Initial Values

<p style='text-indent:20px;'>The Zakharov system in dimension <inline-formula><tex-math id="M2">\begin{document}$ d = 2,3 $\end{document}</tex-math></inline-formula> is shown to have a local unique solution for any initial values in the space <inline-formula><tex-math id="M3">\begin{document}$ H^{s} \times H^{l} \times H^{l-1} $\end{document}</tex-math></inline-formula>, where a new range of regularity <inline-formula><tex-math id="M4">\begin{document}$ (s, l) $\end{document}</tex-math></inline-formula> is given, especially at the line <inline-formula><tex-math id="M5">\begin{document}$ s-l = -1 $\end{document}</tex-math></inline-formula>. The result is obtained mainly by the normal form reduction and the Strichartz estimates.</p>

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 Asymptotic behavior of supercritical wave equations driven by colored noise on unbounded domains

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Bixiang Wang
Keyword(s):  
Asymptotic Behavior ◽  
Existence And Uniqueness ◽  
Wave Equations ◽  
Colored Noise ◽  
Unbounded Domains ◽  
Strichartz Estimates ◽  
Well Posedness ◽  
Natural Energy ◽  
Random Attractors ◽  
Supercritical Nonlinearity

<p style='text-indent:20px;'>This paper deals with the asymptotic behavior of the non-autonomous random dynamical systems generated by the wave equations with supercritical nonlinearity driven by colored noise defined on <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M2">\begin{document}$ n\le 6 $\end{document}</tex-math></inline-formula>. Based on the uniform Strichartz estimates, we prove the well-posedness of the equation in the natural energy space and define a continuous cocycle associated with the solution operator. We also establish the existence and uniqueness of tempered random attractors of the equation by showing the uniform smallness of the tails of the solutions outside a bounded domain in order to overcome the non-compactness of Sobolev embeddings on unbounded domains.</p>

scholarly journals Well-posedness of the linear heat equation with a second order memory term

2020 ◽  
Vol 34 ◽  
pp. 03011
Author(s):  
Constantin Niţă ◽  
Laurenţiu Emanuel Temereancă
Keyword(s):  
Initial Data ◽  
Heat Equation ◽  
Unique Solution ◽  
Source Term ◽  
Memory Term ◽  
Well Posedness ◽  
Dimensional Torus ◽  
One Dimensional ◽  
Linear Heat ◽  
The One

In this article we prove that the heat equation with a memory term on the one-dimensional torus has a unique solution and we study the smoothness properties of this solution. These properties are related with some smoothness assumptions imposed to the initial data of the problem and to the source term.

scholarly journals IMPROVEMENT OF ACCURACY OF PARAMETRIC CLASSIFICATION IN THE SPACE OF N×2 FACTORS-ATTRIBUTES ON THE BASIS OF PRELIMINARY OBTAINED LINEAR DISCRIMINANT FUNCTION

2017 ◽  
Vol 3 ◽  
pp. 55-68 ◽  
Author(s):  
Mourad Aouati
Keyword(s):  
Normal Form ◽  
Discriminant Function ◽  
Classification Accuracy ◽  
Linear Discriminant Function ◽  
Linear Discriminant ◽  
Initial Values ◽  
Parametric Classification ◽  
Dividing Line

A procedure for classifying objects in the space of N×2 factors-attributes that are incorrectly classified as a result of constructing a linear discriminant function is proposed. The classification accuracy is defined as the proportion of correctly classified objects that are incorrectly classified at the first stage of constructing a linear discriminant function. It is shown that, for improperly classified objects, the transition from use as the factors-attributes of their initial values to the use of the centers of gravity (COGs) of local clusters provides the possibility of improving the classification accuracy by 14%. The procedure for constructing local clusters and the principle of forming a classifying rule are proposed, the latter being based on converting the equation of the dividing line to the normal form and determining the sign of the deviation magnitude of the COGs of local clusters from the dividing line

scholarly journals Local in Time Strichartz Estimates for the Dirac Equation on Spherically Symmetric Spaces

2020 ◽  
Author(s):  
Federico Cacciafesta ◽  
Anne-Sophie de Suzzoni
Keyword(s):  
Dirac Equation ◽  
Symmetric Spaces ◽  
Nonlinear Models ◽  
Strichartz Estimates ◽  
Spherically Symmetric ◽  
Well Posedness

Abstract We prove local in time Strichartz estimates for the Dirac equation on spherically symmetric manifolds. As an application, we give a result of local well-posedness for some nonlinear models.

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