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

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.

2013 ◽  
Vol 10 (04) ◽  
pp. 735-771 ◽  
Author(s):  
MAMORU OKAMOTO

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.


2009 ◽  
Vol 06 (03) ◽  
pp. 631-661 ◽  
Author(s):  
ACHENEF TESFAHUN

We prove that the Cauchy problem for the Dirac–Klein–Gordon system of equations in 1D is globally well-posed in a range of Sobolev spaces of negative index for the Dirac spinor and positive index for the scalar field. The main ingredient in the proof is the theory of "almost conservation law" and "I-method" introduced by Colliander, Keel, Staffilani, Takaoka, and Tao. Our proof also relies on the null structure in the system, and bilinear space–time estimates of Klainerman–Machedon type.


2020 ◽  
Vol 32 (6) ◽  
pp. 1575-1598
Author(s):  
Zhaohui Huo ◽  
Yueling Jia

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.


2020 ◽  
Vol 32 (1) ◽  
pp. 151-187
Author(s):  
Boling Guo ◽  
Zhaohui Huo

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}}.


2011 ◽  
Vol 08 (04) ◽  
pp. 615-650 ◽  
Author(s):  
ENRICO BERNARDI ◽  
TATSUO NISHITANI

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.


2019 ◽  
Vol 18 (03) ◽  
pp. 469-522
Author(s):  
Wei Yan ◽  
Yongsheng Li ◽  
Jianhua Huang ◽  
Jinqiao Duan

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].


2014 ◽  
Vol 11 (05) ◽  
pp. 1450042 ◽  
Author(s):  
Salvatore Capozziello ◽  
Tiberiu Harko ◽  
Francisco S. N. Lobo ◽  
Gonzalo J. Olmo ◽  
Stefano Vignolo

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.


Author(s):  
Takayoshi Ogawa

Interaction equations of long and short water wave are considered. It is shown that the Cauchy problem foris locally well posed in the largest space where the three conservationscan be justified. Here E(u,v) is the energy functional associated to the system. By these conservation laws, we establish the global well-posedness of the system in the largest class of initial data.


2003 ◽  
Vol 8 (1) ◽  
pp. 61-75
Author(s):  
V. Litovchenko

The well-posedness of the Cauchy problem, mentioned in title, is studied. The main result means that the solution of this problem is usual C∞ - function on the space argument, if the initial function is a real functional on the conjugate space to the space, containing the fundamental solution of the corresponding problem. The basic tool for the proof is the functional analysis technique.


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