scholarly journals Global well-posedness and inviscid limit for the Korteweg–de Vries–Burgers equation

2009 ◽  
Vol 246 (10) ◽  
pp. 3864-3901 ◽  
Author(s):  
Zihua Guo ◽  
Baoxiang Wang
Keyword(s):  
Burgers Equation ◽  
Inviscid Limit ◽  
Well Posedness ◽  
De Vries

scholarly journals Uniform well-posedness and inviscid limit for the Benjamin–Ono–Burgers equation

2011 ◽  
Vol 228 (2) ◽  
pp. 647-677 ◽  
Author(s):  
Zihua Guo ◽  
Lizhong Peng ◽  
Baoxiang Wang ◽  
Yuzhao Wang
Keyword(s):  
Burgers Equation ◽  
Inviscid Limit ◽  
Well Posedness

On the well-posedness and asymptotic behaviour of the generalized Korteweg–de Vries–Burgers equation

2018 ◽  
Vol 149 (1) ◽  
pp. 219-260 ◽  
Author(s):  
F. A. Gallego ◽  
A. F. Pazoto
Keyword(s):  
Burgers Equation ◽  
Unique Continuation ◽  
Global Solutions ◽  
Exponential Stabilization ◽  
Well Posedness ◽  
Interpolation Theory ◽  
Damping Term ◽  
Unique Continuation Property ◽  
De Vries ◽  
Compactness Arguments

In this paper we are concerned with the well-posedness and the exponential stabilization of the generalized Korteweg–de Vries–Burgers equation, posed on the whole real line, under the effect of a damping term. Both problems are investigated when the exponent p in the nonlinear term ranges over the interval [1, 5). We first prove the global well-posedness in Hs(ℝ) for 0 ≤ s ≤ 3 and 1 ≤ p < 2, and in H3(ℝ) when p ≥ 2. For 2 ≤ p < 5, we prove the existence of global solutions in the L2-setting. Then, by using multiplier techniques and interpolation theory, the exponential stabilization is obtained with an indefinite damping term and 1 ≤ p < 2. Under the effect of a localized damping term the result is obtained when 2 ≤ p < 5. Combining multiplier techniques and compactness arguments, we show that the problem of exponential decay is reduced to proving the unique continuation property of weak solutions. Here, the unique continuation is obtained via the usual Carleman estimate.

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.

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