下载免费PDF全文

Linghai Zhang 《Journal of Nonlinear Modeling and Analysis》2024,6(3):841-872

Consider the Cauchy problem for the Benjamin-Ono-Burgers equation. There exists a unique global weak solution under appropriate conditions on the initial function and the external force. Here are many very important and interesting questions.$bullet$ Can we accomplish the exact limits for all order derivatives of the global smooth solution of the Benjamin-Ono-Burgers equation, in terms of some given information, representing certain physical mechanisms?$bullet$ What are the influences of various physical mechanisms (represented by the initial function, the external force, the order of the derivatives and the diffusion coefficient) on the exact limits?$bullet$ Can we establish improved decay estimates with sharp rates for all order derivatives of the solution, so that the most important constants ${mathcal A}$ and ${mathcal C}$ are independent of any norm of any order derivatives of the initial function, the external force and the solution, for all sufficiently large $t>0$? Other positive constants ${mathcal B}$ and ${mathcal D}$ in the estimates are much less important because ${mathcal B}t^{-1}$ and ${mathcal D}t^{-1}$ becomes arbitrarily small as $ttoinfty$. This kind of decay estimates may play a substantial role in long time, accurate numerical simulations.$bullet$ Can we use the solution of the corresponding linear equation to approximate the solution of the Benjamin-Ono-Burgers equation?$bullet$ Can we couple together classical ideas (such as the Fourier transformation, the Parseval''s identity, Lebesgue''s dominated convergence theorem, squeeze theorem, etc) in an appropriate way to establish important and interesting results for the Benjamin-Ono-Burgers equation?$bullet$ For very similar dissipative dispersive wave equations, such as the Korteweg-de Vries-Burgers equation and the Benjamin-Bona-Mahony-Burgers equation, can we apply the same ideas developed in this paper to establish the same or very similar results?We will couple together a few novel ideas, several existing ideas and existing results and use rigorous mathematical analysis to provide positive solutions to these important and interesting questions.  相似文献   

    

Linghai Zhang 《Journal of Nonlinear Modeling and Analysis》2023,5(1):146-202

We will accomplish the exact limits for all order derivatives of the global weak solutions to a two-dimensional incompressible dissipative quasi-geostrophic equation. We will also establish the improved decay estimates with sharp rates for all order derivatives. We will consider two cases for the initial function and the external force and prove the optimal results for both cases. We will couple together existing ideas (including the Fourier transformation and its properties, Parseval''s identity, iteration technique, Lebesgue''s dominated convergence theorem, Gagliardo-Nirenberg-Sobolev interpolation inequality, squeeze theorem, Cauchy-Schwartz''s inequality, etc) existing results (the existence of global weak solutions, the existence of local smooth solution on $(T,infty)$ and the elementary decay estimate with a sharp rate) and a few novel ideas to obtain the main results.  相似文献   

    

Linghai Zhang 《Journal of Nonlinear Modeling and Analysis》2025,7(1):334-382

We consider the Cauchy problem for a general Korteweg-de Vries-Burgers equation and the Cauchy problem for the corresponding linear equation. We will couple together a few novel ideas, several existing ideas and existing results and use rigorous mathematical analysis to accomplish several very important and very interesting results for these Cauchy problems.  相似文献   

    

Linghai Zhang 《Journal of Nonlinear Modeling and Analysis》2023,5(4):803-903

First of all, the author accomplishes the exact limits for all order derivatives of the global weak solutions of the $n$-dimensional incompressible magnetohydrodynamics equations, the $n$-dimensional incompressible Navier-Stokes equations and the two-dimensional incompressible dissipative quasi-geostrophic equation. Secondly, by making use of the exact limits, he establishes the improved decay estimates with sharp rates for all order derivatives of the global weak solutions, for all sufficiently large $t$. The author proves these results by making use of existing ideas, existing results and several new, novel ideas.  相似文献