The Benjamin-Ono-Burgers Equation: New Ideas and New Results (Journal of Nonlinear Modeling and Analysis, 2024)
Consider the Cauchy problem for the Benjamin-Ono-Burgers equation. There exists a unique global weak solution under appropriate conditionson the initial function and the external force. Here are many very importantand interesting questions.• Can we accomplish the exact limits for all order derivatives of the globalsmooth solution of the Benjamin-Ono-Burgers equation, in terms of some giveninformation, representing certain physical mechanisms?• What are the influences of various physical mechanisms (represented by theinitial function, the external force, the order of the derivatives and the diffusion coefficient) on the exact limits?• Can we establish improved decay estimates with sharp rates for all orderderivatives of the solution, so that the most important constants $\mathcal{A}$ and $\mathcal{C}$ areindependent of any norm of any order derivatives of the initial function, theexternal force and the solution, for all sufficiently large $t > 0?$ Other positiveconstants $\mathcal{B}$ and $\mathcal{D}$ in the estimates are much less important because $Bt^{−1}$ and $Dt^{−1}$ becomes arbitrarily small as $t → ∞.$ This kind of decay estimates mayplay a substantial role in long time, accurate numerical simulations.• Can we use the solution of the corresponding linear equation to approximatethe solution of the Benjamin-Ono-Burgers equation?• Can we couple together classical ideas (such as the Fourier transformation,the Parseval’s identity, Lebesgue’s dominated convergence theorem, squeezetheorem, etc) in an appropriate way to establish important and interestingresults for the Benjamin-Ono-Burgers equation?• For very similar dissipative dispersive wave equations, such as the Kortewegde Vries-Burgers equation and the Benjamin-Bona-Mahony-Burgers equation,can we apply the same ideas developed in this paper to establish the same orvery similar results?We will couple together a few novel ideas, several existing ideas and existingresults and use rigorous mathematical analysis to provide positive solutions tothese important and interesting questions.
Keywords: Benjamin-Ono-Burgers equation; global smooth solution; all order derivatives; exact limits; improved decay estimates with sharp rates.