Differential Equation with forcing/fluids question

[nickthequick]

Hi,
An issue has come up in my research: I think the problem can be phrased as such:

Given a differential equation of the form
$$Lu(t) = f(t)$$
Where the forcing function is of the form $$f(t)=\gamma e^{-t/\epsilon},\gamma$$ is a constant, and L is some linear second order operator.

We want to see what happens to the particular solution in the limit as $$\epsilon \to 0$$ while keeping
$$\int_{0}^{\infty}f(t')dt' = const$$

Physically, it seems as if the forcing function will turn into a boundary condition, although I am not sure how to make this rigorous. Is this always the case? When I go through and evaluate the limits for the particular solution I am finding results that do not make sense.

In case a particular example helps, this is where this problem comes from.

Consider the time independent linearized Navier Stokes equation for a viscous rotating fluid with forcing. The equation governing the dynamics is:
$$u_{zz}-\frac{if}{\nu} u =-\frac{1}{\nu} A(z)$$
where $$A(z)=\gamma e^{-z/\epsilon}$$
I am getting results that do make sense physically for particular types of forcing functions A, so I would like to see what happens when A turns into a surface condition as opposed to a forcing function, since I know what to expect in that case.

 

[Valerie Park]

I am not sure how to approach this problem and would really appreciate any guidance or advice. Thank you in advance.