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.

ON THE CAUCHY PROBLEM FOR SOME PSEUDO-DIFFERENTIAL EQUATIONS OF SCHRÖDINGER TYPE

1996 ◽  
Vol 06 (03) ◽  
pp. 295-314 ◽  
Author(s):  
R. AGLIARDI ◽  
D. MARI
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Fundamental Solution ◽  
Differential Equations ◽  
Lower Order ◽  
Order Term ◽  
Well Posedness ◽  
Gevrey Spaces ◽  
The Cauchy Problem ◽  
Pseudo Differential Equation

A fundamental solution of the Cauchy problem is constructed for a pseudo-differential equation generalizing some Schrödinger equations. Then well-posedness of the Cauchy problem is proved in some Gevrey spaces whose indices depend on the lower order term of the operator.

scholarly journals On the Stabilization Rate of Solutions of the Cauchy Problem for Nondivergent Parabolic Equations with Growing Lower-Order Term

2017 ◽  
Vol 63 (4) ◽  
pp. 586-598
Author(s):  
V N Denisov
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equation ◽  
Half Space ◽  
Lower Order ◽  
Parabolic Equations ◽  
Initial Function ◽  
Order Term ◽  
Sufficient Conditions ◽  
Lower Order Term ◽  
The Cauchy Problem

In the Cauchy problem L1u≡Lu+(b,∇u)+cu-ut=0,(x,t)∈D,u(x,0)=u0(x),x∈RN, for nondivergent parabolic equation with growing lower-order term in the half-space D=RN×[0,∞), N⩾3, we prove sufficient conditions for exponential stabilization rate of solution as t→+∞ uniformly with respect to x on any compact K in RN with any bounded and continuous in RN initial function u0(x).

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.

scholarly journals On the Cauchy problem for Dt2 − Dx(b(t)a(x))Dx

2020 ◽  
Vol 17 (01) ◽  
pp. 75-122
Author(s):  
Ferruccio Colombini ◽  
Tatsuo Nishitani
Keyword(s):  
Cauchy Problem ◽  
Differential Operators ◽  
Second Order ◽  
Gevrey Class ◽  
Independent Variables ◽  
The Cauchy Problem ◽  
Well Posed ◽  
Two Independent Variables

We consider the Cauchy problem for second-order differential operators with two independent variables [Formula: see text]. Assuming that [Formula: see text] is a nonnegative [Formula: see text] function and [Formula: see text] is a nonnegative Gevrey function of order [Formula: see text], we prove that the Cauchy problem for [Formula: see text] is well-posed in the Gevrey class of any order [Formula: see text] with [Formula: see text].

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.

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 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.

Global well-posedness and conservation laws for the water wave interaction equation

1997 ◽  
Vol 127 (2) ◽  
pp. 369-384 ◽  
Author(s):  
Takayoshi Ogawa
Keyword(s):  
Cauchy Problem ◽  
Conservation Laws ◽  
Water Wave ◽  
Wave Interaction ◽  
Energy Functional ◽  
Well Posedness ◽  
The Cauchy Problem ◽  
Interaction Equation ◽  
Image Position ◽  
Well Posed

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.

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