WELL-POSEDNESS AND CONVERGENCE FOR TIME-SPACE FRACTIONAL STOCHASTIC SCHRÖGER-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.

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Sharp well-posedness results for the BBM equation

Discrete and Continuous Dynamical Systems ◽

10.3934/dcds.2009.23.1241 ◽

2009 ◽

Vol 23 (4) ◽

pp. 1241-1252 ◽

Cited By ~ 50

Author(s):

Jerry Bona ◽

◽

Nikolay Tzvetkov ◽

Keyword(s):

Well Posedness ◽

Bbm Equation

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Well-posedness of the time-space fractional stochastic Navier-Stokes equations driven by fractional Brownian motion

Mathematical Modelling of Natural Phenomena ◽

10.1051/mmnp/2018003 ◽

2018 ◽

Vol 13 (1) ◽

pp. 11 ◽

Cited By ~ 1

Author(s):

Pengfei Xu ◽

Caibin Zeng ◽

Jianhua Huang

Keyword(s):

Brownian Motion ◽

Fractional Brownian Motion ◽

Existence And Uniqueness ◽

Stokes Equations ◽

Navier Stokes ◽

Banach Fixed Point Theorem ◽

Well Posedness ◽

Stochastic Convolution ◽

Navier Stokes Equations ◽

Time Space

The current paper is devoted to the time-space fractional Navier-Stokes equations driven by fractional Brownian motion. The spatial-temporal regularity of the nonlocal stochastic convolution is firstly established, and then the existence and uniqueness of mild solution are obtained by Banach Fixed Point theorem and Mittag-Leffler families operators.

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

Computational Methods in Applied Mathematics ◽

10.1515/cmam-2019-0031 ◽

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.

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