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Nonlinear instability for a modified form of Burgers’ equation. (English) Zbl 0704.76023

In the nonlinear stability analysis of the problem of Couette flow between concentric cylinders very sharp estimates on the Reynolds number were derived elsewhere. To obtain such sharp results, however, they found it necessary to restrict the size of the initial kinetic energy, thereby introducing the concept of conditional stability into the theory of nonlinear energy stability. Since then, it has been found that very sharp nonlinear stability results may also be obtained in various thermal or salt convection problems of practical interest. However, these papers all obtain sharp conditions on the Rayleigh number at the expense of restricting the size of the amplitude of the initial perturbation. One of the motivations for this paper is to cast some light on the situation where the same restrictions apply to the Rayleigh or Reynolds number, but the initial energy restriction is exceeded. It is by no means obvious what will happen in this situation since the equations contain nonlinearities like \((temperature)^ 2\) or \((concentration)^ 2\) and in parabolic partial differential equations without convection it is well known that such terms can lead to blow-up of the solution in finite time.

MSC:

76E30 Nonlinear effects in hydrodynamic stability
35Q53 KdV equations (Korteweg-de Vries equations)
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