scholarly journals Construction of 2-Peakon Solutions and Nonuniqueness for a Generalized mCH Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Hao Yu ◽  
Aiyong Chen ◽  
Kelei Zhang
Keyword(s):  
Initial Data ◽  
Sobolev Spaces ◽  
Collision Time ◽  
Initial Value ◽  
Two Peaks

For the generalized mCH equation, we construct a 2-peakon solution on both the line and the circle, and we can control the size of the initial data. The two peaks at different speeds move in the same direction and eventually collide. This phenomenon is that the solution at the collision time is consistent with another solitary peakon solution. By reversing the time, we get two new solutions with the same initial value and different values at the rest of the time, which means the nonuniqueness for the equation in Sobolev spaces H s is proved for s < 3 / 2 .

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.

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&gt; ‒3/4.

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].

DECAY ESTIMATES OF SOLUTIONS TO A SEMI-LINEAR DISSIPATIVE PLATE EQUATION

2010 ◽  
Vol 07 (03) ◽  
pp. 471-501 ◽  
Author(s):  
YOUSUKE SUGITANI ◽  
SHUICHI KAWASHIMA
Keyword(s):  
Initial Data ◽  
Sobolev Spaces ◽  
Linear Problem ◽  
Dimensional Space ◽  
Weighted Sobolev Spaces ◽  
Decay Estimates ◽  
Plate Equation ◽  
Estimates Of Solutions ◽  
Optimal Decay ◽  
Additional Regularity

We study the initial value problem for a semi-linear dissipative plate equation in n-dimensional space. We observe that the dissipative structure of the linearized equation is of the regularity-loss type. This means that we have the optimal decay estimates of solutions under the additional regularity assumption on the initial data. This regularity-loss property causes the difficulty in solving the nonlinear problem. For our semi-linear problem, this difficulty can be overcome by introducing a set of time-weighted Sobolev spaces, where the time-weights and the regularity of the Sobolev spaces are determined by our regularity-loss property. Consequently, under smallness condition on the initial data, we prove the global existence and optimal decay of the solution in the corresponding Sobolev spaces.

Non-uniform dependence on initial data for the generalized Degasperis–Procesi equation on the line

2016 ◽  
Vol 27 (03) ◽  
pp. 1650026
Author(s):  
Yanggeng Fu ◽  
Zanping Yu ◽  
Jianhe Shen
Keyword(s):  
Initial Data ◽  
Sobolev Spaces ◽  
High Frequency ◽  
Low Frequency ◽  
Approximate Solutions ◽  
Solution Map ◽  
High Frequency Part ◽  
Uniformly Continuous

In this paper, we show that the solution map of the generalized Degasperis–Procesi (gDP) equation is not uniformly continuous in Sobolev spaces [Formula: see text] for [Formula: see text]. Our proof is based on the estimates for the actual solutions and the approximate solutions, which consist of a low frequency and a high frequency part. It also exploits the fact that the gDP equation conserves a quantity which is equivalent to the [Formula: see text] norm.

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