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jollage

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I'm considering the energy evolution of variables of different orders in a partial differential equation.

The PDE is nonlinear, which can be written as

## \frac{\partial u}{\partial t} = \mathcal{N}u ##

where ##\mathcal{N}## is a nonlinear operator in space and time. Now I want to check the linear stability analysis of this PDE and I assume

##u=u_0(x) + \epsilon u_1(x,t) + \epsilon^2 u_2(x,t) + H.O.T ##

where ##u_0(x)## is only a function of space, not of time (so, it's a time mean of the variable) and the other orders are modulated by a small quantity ##\epsilon##. Substituting this decomposition into the nonlinear PDE, neglecting the high order terms and collecting the terms of same order of ##\epsilon##, we will get the evolution equation for ##u_1## and ##u_2##, which are

##\frac{\partial u_1}{\partial t} = \mathcal{L} u_1##

##\frac{\partial u_2}{\partial t} = \mathcal{L} u_2 + \mathcal{N}_1##

where ##\mathcal{L}=\mathcal{L}(x)## is the linear operator and ##\mathcal{N}_1=\mathcal{N}_1(x,t)## is the nonlinear terms of the first order variables ##u_1## acting on the second order variable ##u_2##.

This derivation is trivial. Now I check the energy evolution of the variables ##\frac{1}{2}u_1^*u_1## and ##\frac{1}{2}u_2^*u_2##, and the two equations will become

##\frac{\partial}{\partial t} (\frac{1}{2}u_1^*u_1) = u_1^* \frac{1}{2}(\mathcal{L} + \mathcal{L}^*) u_1 ##

##\frac{\partial}{\partial t} (\frac{1}{2}u_2^*u_2) = u_2^* \frac{1}{2}(\mathcal{L} + \mathcal{L}^*) u_2 + \frac{1}{2}(u_2^*\mathcal{N}_1 + \mathcal{N}_1^* u_2)##

My question is regarding the above equations.

(1) Since ##\epsilon## is small, we expect that ##\epsilon u_1(x,t)## is much smaller than ##\epsilon^2 u_2(x,t)## provided that ##u_1(x,t)## is of the same order of ##u_2(x,t)##. In this case, is it meaningful to discuss the energy ##\frac{1}{2}u_1^*u_1## and ##\frac{1}{2}u_2^*u_2## in the same context? Though ##\frac{1}{2}u_1^*u_1## and ##\frac{1}{2}u_2^*u_2## is of the same order, but ##\epsilon^2\frac{1}{2}u_1^*u_1## and ##\epsilon^4\frac{1}{2}u_2^*u_2## is two orders of ##\epsilon## apart.

To be more specific, let me consider the following example. If the eigenvalue of ##\mathcal{L}## is negative, it means that the energy ##u_1## decays monotonically. If the nonlinear part ##\frac{1}{2}(u_2^*\mathcal{N}_1 + \mathcal{N}_1^* u_2)## causes a large energy growth of ##u_2##, will this lead to energy growth of ##u##?

(2) The two equations are only one-way coupled, so we can solve the first equation independently. I'm thinking a dynamical system. With different initial conditions for ##u_1##, the energy evolution of ##\frac{1}{2}u_1^*u_1## could be different. Assume that we manage to pick up the optimal initial condition of ##u_1##(to avoid ambiguity, its norm is 1) which gives rise to the optimal energy growth over all the initial conditions, then we use this path of energy evolution for ##\frac{1}{2}u_1^*u_1## in the second equation, my question is that is this the optimal energy growth path for ##\frac{1}{2}u_2^*u_2##? Or there exists some interaction of ##u_2## and ##\mathcal{N}_1## such that it can bypass this optimal path?

Thanks.