scholarly journals Sharp well-posedness results for the BBM equation

2009 ◽  
Vol 23 (4) ◽  
pp. 1241-1252 ◽  
Author(s):  
Jerry Bona ◽  
◽  
Nikolay Tzvetkov ◽  
Keyword(s):  
Well Posedness ◽  
Bbm Equation

scholarly journals Norm Inflation for Benjamin–Bona–Mahony Equation in Fourier Amalgam and Wiener Amalgam Spaces with Negative Regularity

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3145
Author(s):  
Divyang G. Bhimani ◽  
Saikatul Haque
Keyword(s):  
Initial Data ◽  
Modulation Spaces ◽  
Well Posedness ◽  
Wiener Amalgam Spaces ◽  
General Initial Data ◽  
Amalgam Spaces ◽  
Loss Of Regularity ◽  
Bbm Equation

We consider the Benjamin–Bona–Mahony (BBM) equation of the form ut+ux+uux−uxxt=0,(x,t)∈M×R where M=T or R. We establish norm inflation (NI) with infinite loss of regularity at general initial data in Fourier amalgam and Wiener amalgam spaces with negative regularity. This strengthens several known NI results at zero initial data in Hs(T) established by Bona–Dai (2017) and the ill-posedness result established by Bona–Tzvetkov (2008) and Panthee (2011) in Hs(R). Our result is sharp with respect to the local well-posedness result of Banquet–Villamizar–Roa (2021) in modulation spaces Ms2,1(R) for s≥0.

scholarly journals WELL-POSEDNESS AND CONVERGENCE FOR TIME-SPACE FRACTIONAL STOCHASTIC SCHRÖGER-BBM EQUATION

2021 ◽  
Vol 11 (4) ◽  
pp. 1749-1767
Author(s):  
Shang Wu ◽  
◽  
Jianhua Huang ◽  
Yuhong Li ◽  
Keyword(s):  
Well Posedness ◽  
Time Space ◽  
Bbm Equation

Reconstruction of a Space-Dependent Coefficient in a Linear Benjamin–Bona–Mahony Type Equation

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Felipe Alexander Pipicano ◽  
Juan Carlos Muñoz Grajales ◽  
Anibal Sosa
Keyword(s):  
Constrained Optimization ◽  
Regularization Parameter ◽  
Type Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Well Posedness ◽  
Time Stepping ◽  
Uniqueness Of The Solution ◽  
Initial Boundary Value ◽  
Bbm Equation

Abstract In this paper, we consider the problem of reconstructing a space-dependent coefficient in a linear Benjamin–Bona–Mahony (BBM)-type equation from a single measurement of its solution at a given time. We analyze the well-posedness of the forward initial-boundary value problem and characterize the inverse problem as a constrained optimization one. Our objective consists on reconstructing the variable coefficient in the BBM equation by minimizing an appropriate regularized Tikhonov-type functional constrained by the BBM equation. The well-posedness of the forward problem is studied and approximated numerically by combining a finite-element strategy for spatial discretization using the Python-FEniCS package, together with a second-order implicit scheme for time stepping. The minimization process of the Tikhonov-regularization adopted is performed by using an iterative L-BFGS-B quasi-Newton algorithm as described for instance by Byrd et al. (1995) and Zhu et al. (1997). Numerical simulations are presented to demonstrate the robustness of the proposed method with noisy data. The local stability and uniqueness of the solution to the constrained optimization problem for a fixed value of the regularization parameter are also proved and illustrated numerically.

scholarly journals Well-posedness for the BBM-equation in a quarter plane

2014 ◽  
Vol 7 (6) ◽  
pp. 1149-1163
Author(s):  
Jerry L. Bona ◽  
◽  
Hongqiu Chen ◽  
Chun-Hsiung Hsia ◽  
◽  
...  
Keyword(s):  
Well Posedness ◽  
Quarter Plane ◽  
Bbm Equation

Sharp global well-posedness for the fractional BBM equation

2018 ◽  
Vol 41 (15) ◽  
pp. 5906-5918
Author(s):  
Ming Wang ◽  
Zaiyun Zhang
Keyword(s):  
Well Posedness ◽  
Bbm Equation

Cauchy Problem for Equations with Fractional Differentiation Bessel Operator in the Space of Temperate Distributions

2003 ◽  
Vol 8 (1) ◽  
pp. 61-75
Author(s):  
V. Litovchenko
Keyword(s):  
Functional Analysis ◽  
Cauchy Problem ◽  
Fundamental Solution ◽  
Initial Function ◽  
Well Posedness ◽  
Fractional Differentiation ◽  
Basic Tool ◽  
Conjugate Space ◽  
Analysis Technique ◽  
The Cauchy Problem

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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