The Energy Method and Non- Linear Stability

Gavhale, Siddharth Balu and P A, Lakshmi Narayana (2016) The Energy Method and Non- Linear Stability. Masters thesis, Indian Institute of Technology Hyderabad.

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Abstract

This thesis is primarily a presentation of energy stability results obtained in some standard partial differential equations by means of an integral inequality technique. We are interested in the problem of stability or instability of different partial differential equations. Suppose for a given equation we have a solution. It is the stability of that solution, we wish to investigate. The idea is that, for solution to be stable that must be stable against any disturbance to which that may be subjected. Damping the disturbance rapidly is our ultimate goal, for this, energy method is very useful. To show that the solution is unstable, it is sufficient to find at least one disturbance that grows in amplitude or remains bounded away from the solution. Linear instability analysis and nonlinear stability analysis are the main parts. Nonlinear stability analysis are the main parts. Nonlinear stability we mean that following two condition are satisfied. Firstly, we can find an arbitrarily small bound on the size of any perturbation whose initial magnitude is small enough. Secondly, any perurbation whose initial magnitude is less than some critical value converges to 0 with time.

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