scholarly journals Global well-posedness for a system of KdV-type equations with coupled quadratic nonlinearities

2014 ◽  
Vol 215 ◽  
pp. 67-149 ◽  
Author(s):  
Jerry L. bona ◽  
Jonathan Cohen ◽  
Gang Wang
Keyword(s):  
Initial Data ◽  
Sobolev Spaces ◽  
Initial Value Problem ◽  
Coupled Systems ◽  
Well Posedness ◽  
Initial Value ◽  
Quadratic Polynomials ◽  
Partial Differentiation ◽  
Quadratic Nonlinearities ◽  
Kdv Type Equations

AbstractIn this paper, coupled systemsof Korteweg-de Vries type are considered, whereu=u(x, t),v=v(x, t) are real-valued functions and wherex, t∈R. Here, subscripts connote partial differentiation andare quadratic polynomials in the variablesuandv. Attention is given to the pure initial-value problem in whichu(x, t) andv(x, t) are both specified att= 0, namely,forx∈ ℝ. Under suitable conditions onPandQ, global well-posedness of this problem is established for initial data in theL2-based Sobolev spacesHs(ℝ) ×Hs(ℝ) for anys> ‒3/4.

scholarly journals Global well-posedness for a system of KdV-type equations with coupled quadratic nonlinearities

2014 ◽  
Vol 215 ◽  
pp. 67-149 ◽  
Author(s):  
Jerry L. bona ◽  
Jonathan Cohen ◽  
Gang Wang
Keyword(s):  
Initial Data ◽  
Sobolev Spaces ◽  
Initial Value Problem ◽  
Coupled Systems ◽  
Well Posedness ◽  
Initial Value ◽  
Quadratic Polynomials ◽  
Partial Differentiation ◽  
Quadratic Nonlinearities ◽  
Kdv Type Equations

AbstractIn this paper, coupled systemsof Korteweg-de Vries type are considered, where u = u(x, t), v = v(x, t) are real-valued functions and where x, t∈R. Here, subscripts connote partial differentiation andare quadratic polynomials in the variables u and v. Attention is given to the pure initial-value problem in which u(x, t) and v(x, t) are both specified at t = 0, namely,for x ∈ ℝ. Under suitable conditions on P and Q, global well-posedness of this problem is established for initial data in the L2-based Sobolev spaces Hs(ℝ) × Hs(ℝ) for any s > ‒3/4.

On the Benney equation

2009 ◽  
Vol 139 (6) ◽  
pp. 1121-1144 ◽  
Author(s):  
Amin Esfahani
Keyword(s):  
Sobolev Spaces ◽  
Initial Value Problem ◽  
A Priori ◽  
Travelling Wave ◽  
Well Posedness ◽  
Travelling Wave Solutions ◽  
Initial Value ◽  
Wave Solutions ◽  
Benney Equation ◽  
Well Posed

We study the Benney equation and show that the associated initial-value problem is locally well-posed in Sobolev spaces Hs(ℝ2) for s > −2. Furthermore, we use a priori estimates to establish the global well-posedness for s ≥ 0. We also prove that these results are in some sense sharp. In addition, we obtain some exact travelling-wave solutions of the equation.

scholarly journals Well-posedness, wave breaking, Holder continuity and periodic peakons for a nonlocal sine-µ-Camassa-Holm equation

2021 ◽  
Author(s):  
Guoquan Qin ◽  
Zhenya Yan ◽  
Boling Guo
Keyword(s):  
Initial Data ◽  
Explicit Formula ◽  
Initial Value Problem ◽  
Besov Spaces ◽  
Wave Breaking ◽  
Well Posedness ◽  
Sufficient Condition ◽  
Initial Value ◽  
Solution Map ◽  
Holm Equation

In this paper, we investigate the initial value problem of a nonlocal sine-type µ-Camassa-Holm (µCH) equation, which is the µ-version of the sine-type CH equation. We first discuss its local well-posedness in the framework of Besov spaces. Then a sufficient condition on the initial data is provided to ensure the occurance of the wave-breaking phenomenon. We finally prove the H¨older continuity of the data-to-solution map, and find the explicit formula of the global weak periodic peakon solution.

scholarly journals Existence and Continuous Dependence of the Local Solution of Non-Homogeneous KdV-K-S Equation in Periodic Sobolev Spaces

2021 ◽  
Vol 64 (1) ◽  
pp. 1-19
Author(s):  
Yolanda Silvia Santiago Ayala ◽  
◽  
Santiago Cesar Rojas Romero
Keyword(s):  
Initial Data ◽  
Sobolev Spaces ◽  
Initial Value Problem ◽  
Continuous Dependence ◽  
Local Solution ◽  
Dissipative Property ◽  
Homogeneous Part ◽  
Initial Value ◽  
Fourier Theory

In this article, we prove that initial value problem associated to the non-homogeneous KdV-Kuramoto-Sivashinsky (KdV-K-S) equation in periodic Sobolev spaces has a local solution in with and the solution has continuous dependence with respect to the initial data and the non-homogeneous part of the problem. We do this in an intuitive way using Fourier theory and introducing a inspired by the work of Iorio [2] and Ayala and Romero [8]. Also, we prove the uniqueness solution of the homogeneous and non-homogeneous KdV-K-S equation, using its dissipative property, inspired by the work of Iorio [2] and Ayala and Romero [9].

scholarly journals WELL-POSEDNESS FOR A FAMILY OF PERTURBATIONS OF THE KdV EQUATION IN PERIODIC SOBOLEV SPACES OF NEGATIVE ORDER

2013 ◽  
Vol 15 (06) ◽  
pp. 1350005
Author(s):  
XAVIER CARVAJAL PAREDES ◽  
RICARDO A. PASTRAN
Keyword(s):  
Sobolev Spaces ◽  
Initial Value Problem ◽  
Burgers Equation ◽  
Kdv Equation ◽  
Well Posedness ◽  
Initial Value ◽  
Negative Order ◽  
The Kdv Equation ◽  
De Vries ◽  
Sivashinsky Equation

We establish local well-posedness in Sobolev spaces Hs(𝕋), with s ≥ -1/2, for the initial value problem issues of the equation [Formula: see text] where η > 0, (Lu)∧(k) = -Φ(k)û(k), k ∈ ℤ and Φ ∈ ℝ is bounded above. Particular cases of this problem are the Korteweg–de Vries–Burgers equation for Φ(k) = -k2, the derivative Korteweg–de Vries–Kuramoto–Sivashinsky equation for Φ(k) = k2 - k4, and the Ostrovsky–Stepanyams–Tsimring equation for Φ(k) = |k| - |k|3.

scholarly journals ENERGY CRITICAL NLS IN TWO SPACE DIMENSIONS

2009 ◽  
Vol 06 (03) ◽  
pp. 549-575 ◽  
Author(s):  
J. COLLIANDER ◽  
S. IBRAHIM ◽  
M. MAJDOUB ◽  
N. MASMOUDI
Keyword(s):  
Schrödinger Equation ◽  
Initial Value Problem ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Energy Space ◽  
Well Posedness ◽  
Initial Value ◽  
Exponential Nonlinearity ◽  
Supercritical Case ◽  
Two Space Dimensions

We investigate the initial value problem for a defocusing nonlinear Schrödinger equation with exponential nonlinearity [Formula: see text] We identify subcritical, critical, and supercritical regimes in the energy space. We establish global well-posedness in the subcritical and critical regimes. Well-posedness fails to hold in the supercritical case.

scholarly journals Some Mathematical Aspects of f(R)-Gravity with Torsion: Cauchy Problem and Junction Conditions

Universe ◽  
2019 ◽  
Vol 5 (12) ◽  
pp. 224 ◽  
Author(s):  
Stefano Vignolo
Keyword(s):  
Cauchy Problem ◽  
Initial Value Problem ◽  
Sufficient Conditions ◽  
Well Posedness ◽  
Junction Conditions ◽  
Field Equations ◽  
Initial Value ◽  
The Cauchy Problem ◽  
General Conditions

We discuss the Cauchy problem and the junction conditions within the framework of f ( R ) -gravity with torsion. We derive sufficient conditions to ensure the well-posedness of the initial value problem, as well as general conditions to join together on a given hypersurface two different solutions of the field equations. The stated results can be useful to distinguish viable from nonviable f ( R ) -models with torsion.

Sign in / Sign up

Export Citation Format

Share Document