WELL-POSEDNESS OF THE CAUCHY PROBLEM FOR THE CHERN–SIMONS–DIRAC SYSTEM IN TWO DIMENSIONS

2013 ◽  
Vol 10 (04) ◽  
pp. 735-771 ◽  
Author(s):  
MAMORU OKAMOTO
Keyword(s):  
Cauchy Problem ◽  
Dirac Equation ◽  
Sobolev Space ◽  
Gauge Invariance ◽  
Two Dimensions ◽  
Well Posedness ◽  
Dirac System ◽  
The Cauchy Problem ◽  
Chern Simons ◽  
Well Posed

We consider the Cauchy problem associated with the Chern–Simons–Dirac system in ℝ1+2. Using gauge invariance, we reduce the Chern–Simons–Dirac system to a Dirac equation and we uncover the null structure of this Dirac equation. Next, relying on null structure estimates, we establish that the Cauchy problem associated with this Dirac equation is locally-in-time well-posed in the Sobolev space Hs for all s > 1/4. Our proof uses modified L4-type estimates.

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.

scholarly journals Global well-posedness in energy space for the Chern–Simons–Higgs system in temporal gauge

2016 ◽  
Vol 13 (02) ◽  
pp. 331-351 ◽  
Author(s):  
Hartmut Pecher
Keyword(s):  
Cauchy Problem ◽  
Energy Space ◽  
Well Posedness ◽  
Null Condition ◽  
Temporal Gauge ◽  
Time Estimates ◽  
Dimensional Minkowski Space ◽  
The Cauchy Problem ◽  
Chern Simons ◽  
Well Posed

The Cauchy problem for the Chern–Simons–Higgs system in the [Formula: see text]-dimensional Minkowski space in temporal gauge is globally well-posed in energy space improving a result of Huh. The proof uses the bilinear space-time estimates in wave-Sobolev spaces by d’Ancona, Foschi and Selberg, an [Formula: see text]-estimate for solutions of the wave equation, and also takes advantage of a null condition.

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.

scholarly journals Low regularity well-posedness of Hartree type Dirac equations in 2, 3-dimensions

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Kiyeon Lee
Keyword(s):  
Cauchy Problem ◽  
Dirac Equation ◽  
Critical Case ◽  
Nonlinear Term ◽  
Well Posedness ◽  
Low Regularity ◽  
Bilinear Estimates ◽  
Dirac Equations ◽  
The Cauchy Problem

<p style='text-indent:20px;'>In this paper, we consider the Cauchy problem of <inline-formula><tex-math id="M1">\begin{document}$ d $\end{document}</tex-math></inline-formula>-dimension Hartree type Dirac equation with nonlinearity <inline-formula><tex-math id="M2">\begin{document}$ c|x|^{-\gamma} * \langle \psi, \beta \psi\rangle $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ c\in \mathbb R\setminus\{0\} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ 0 &lt; \gamma &lt; d $\end{document}</tex-math></inline-formula>(<inline-formula><tex-math id="M5">\begin{document}$ d = 2,3 $\end{document}</tex-math></inline-formula>). Our aim is to show the local well-posedness in <inline-formula><tex-math id="M6">\begin{document}$ H^s $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M7">\begin{document}$ s &gt; \frac{\gamma-1}2 $\end{document}</tex-math></inline-formula> with mass-supercritical cases(<inline-formula><tex-math id="M8">\begin{document}$ 1 &lt; \gamma&lt;d $\end{document}</tex-math></inline-formula>) and mass-critical case(<inline-formula><tex-math id="M9">\begin{document}$ {\gamma} = 1 $\end{document}</tex-math></inline-formula>) via bilinear estimates and angular decomposition for which we use the null structure of nonlinear term effectively. We also provide the flow of Dirac equations cannot be <inline-formula><tex-math id="M10">\begin{document}$ C^3 $\end{document}</tex-math></inline-formula> at the origin for <inline-formula><tex-math id="M11">\begin{document}$ H^s $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M12">\begin{document}$ s &lt; \frac{\gamma-1}2 $\end{document}</tex-math></inline-formula>.</p>

Modified energy method and applications for the well-posedness for the higher-order Benjamin–Ono equation and the higher-order intermediate long wave equation

2020 ◽  
Vol 32 (1) ◽  
pp. 151-187
Author(s):  
Boling Guo ◽  
Zhaohui Huo
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Gauge Transformation ◽  
Energy Method ◽  
Higher Order ◽  
Well Posedness ◽  
Long Wave ◽  
The Cauchy Problem ◽  
Well Posed

AbstractIn this paper, the well-posedness of the higher-order Benjamin–Ono equationu_{t}+\mathcal{H}(u_{xx})+u_{xxx}=uu_{x}-\partial_{x}(u\mathcal{H}\partial_{x}% u+\mathcal{H}(u\partial_{x}u))is considered. The modified energy method is introduced to consider the equation. It is shown that the Cauchy problem of the higher-order Benjamin–Ono equation is locally well-posed in {H^{3/4}} without using the gauge transformation. Moreover, the well-posedness of the higher-order intermediate long wave equationu_{t}+\mathcal{G}_{\delta}(u_{xx})+u_{xxx}=uu_{x}-\partial_{x}(u\mathcal{G}_{% \delta}\partial_{x}u+\mathcal{G}_{\delta}(u\partial_{x}u)),\quad\mathcal{G}_{% \delta}=\mathcal{F}_{x}^{-1}i(\coth(\delta\xi))\mathcal{F}_{x},is considered. It is shown that the Cauchy problem of the higher-order intermediate long wave equation is locally well-posed in {H^{3/4}}.

scholarly journals ON THE CAUCHY PROBLEM FOR NON-EFFECTIVELY HYPERBOLIC OPERATORS: THE GEVREY 3 WELL-POSEDNESS

2011 ◽  
Vol 08 (04) ◽  
pp. 615-650 ◽  
Author(s):  
ENRICO BERNARDI ◽  
TATSUO NISHITANI
Keyword(s):  
Cauchy Problem ◽  
Lower Order ◽  
Order Term ◽  
Jordan Block ◽  
Gevrey Class ◽  
Lower Order Term ◽  
Well Posedness ◽  
Class 1 ◽  
The Cauchy Problem ◽  
Well Posed

For hyperbolic differential operators P with double characteristics we study the relations between the maximal Gevrey index for the strong Gevrey well-posedness and the Hamilton map and flow of the associated principal symbol p. If the Hamilton map admits a Jordan block of size 4 on the double characteristic manifold denoted by Σ and by assuming that the Hamilton flow does not approach Σ tangentially, we proved earlier that the Cauchy problem for P is well-posed in the Gevrey class 1 ≤ s < 4 for any lower order term. In the present paper, we remove this restriction on the Hamilton flow and establish that the Cauchy problem for P is well-posed in the Gevrey class 1 ≤ s < 3 for any lower order term and we check that the Gevrey index 3 is optimal. Combining this with results already proved for the other cases, we conclude that the Hamilton map and flow completely characterizes the threshold for the strong Gevrey well-posedness and vice versa.

scholarly journals The Cauchy problem for a two-dimensional generalized Kadomtsev–Petviashvili-I equation in anisotropic Sobolev spaces

2019 ◽  
Vol 18 (03) ◽  
pp. 469-522
Author(s):  
Wei Yan ◽  
Yongsheng Li ◽  
Jianhua Huang ◽  
Jinqiao Duan
Keyword(s):  
Cauchy Problem ◽  
Sobolev Spaces ◽  
Well Posedness ◽  
Two Dimensional ◽  
Anisotropic Sobolev Spaces ◽  
The Cauchy Problem ◽  
Kp Equations ◽  
Fifth Order ◽  
Well Posed

The goal of this paper is three-fold. First, we prove that the Cauchy problem for a generalized KP-I equation [Formula: see text] is locally well-posed in the anisotropic Sobolev spaces [Formula: see text] with [Formula: see text] and [Formula: see text]. Second, we prove that the Cauchy problem is globally well-posed in [Formula: see text] with [Formula: see text] if [Formula: see text]. Finally, we show that the Cauchy problem is globally well-posed in [Formula: see text] with [Formula: see text] if [Formula: see text] Our result improves the result of Saut and Tzvetkov [The Cauchy problem for the fifth order KP equations, J. Math. Pures Appl. 79 (2000) 307–338] and Li and Xiao [Well-posedness of the fifth order Kadomtsev–Petviashvili-I equation in anisotropic Sobolev spaces with nonnegative indices, J. Math. Pures Appl. 90 (2008) 338–352].

scholarly journals The Cauchy problem in hybrid metric-Palatini f(X)-gravity

2014 ◽  
Vol 11 (05) ◽  
pp. 1450042 ◽  
Author(s):  
Salvatore Capozziello ◽  
Tiberiu Harko ◽  
Francisco S. N. Lobo ◽  
Gonzalo J. Olmo ◽  
Stefano Vignolo
Keyword(s):  
Cauchy Problem ◽  
Scalar Field ◽  
Solar System ◽  
Gravitational Theory ◽  
Cosmic Acceleration ◽  
Late Time ◽  
Well Posedness ◽  
Initial Value ◽  
The Cauchy Problem ◽  
Well Posed

The well-formulation and the well-posedness of the Cauchy problem are discussed for hybrid metric-Palatini gravity, a recently proposed modified gravitational theory consisting of adding to the Einstein–Hilbert Lagrangian an f(R)-term constructed à la Palatini. The theory can be recast as a scalar-tensor one predicting the existence of a light long-range scalar field that evades the local Solar System tests and is able to modify galactic and cosmological dynamics, leading to the late-time cosmic acceleration. In this work, adopting generalized harmonic coordinates, we show that the initial value problem can always be well-formulated and, furthermore, can be well-posed depending on the adopted matter sources.

scholarly journals Global well-posedness for the 3D magneto-micropolar equations with fractional dissipation

10.53733/161 ◽  
2021 ◽  
Vol 51 ◽  
pp. 119-130
Author(s):  
Baoquan Yuan ◽  
Panpan Zhang
Keyword(s):  
Cauchy Problem ◽  
Sobolev Space ◽  
Global Solutions ◽  
Well Posedness ◽  
The Cauchy Problem ◽  
Alpha Beta

This paper focus on the Cauchy problem of the 3D incompressible magneto-micropolar equations with fractional dissipation in the Sobolev space. Liu, Sun and Xin obtained the global solutions to the 3D magneto-micropolar equations with $\alpha=\beta=\gamma=\frac{5}{4}$. Deng and Shang established the global well-posedness of the 3D magneto-micropolar equations in the case of $\alpha\geq\frac{5}{4}$, $\alpha+\beta\geq\frac{5}{2}$ and $\gamma\geq2-\alpha\geq\frac{3}{4}$. In this paper, we establish the global well-posedness of the 3D magneto-micropolar equations with $\alpha=\beta=\frac{5}{4}$ and $\gamma=\frac{1}{2}$, which improves the results of Liu-Sun-Xin and Deng-Shang by reducing the value of $\gamma$ to $\frac{1}{2}$.

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