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I have am working with a nonlinear pseudo-differential evolution equation of the form $$u_t + \mathcal{N}(u) + \mathcal{D}_{\epsilon} u = 0$$ where $\mathcal{N}$ is a nonlinear operator and $\mathcal{D}_{\epsilon}$ is a linear pseudo-differential operator depending on a parameter $\epsilon$. What I've done expanded the Fourier symbol of the linear part in small $\epsilon$ $$\widehat{\mathcal{D}_{\epsilon}} = \widehat{\mathcal{D}^{(0)}} + \epsilon \widehat{\mathcal{D}^{(1)}} + \ldots $$ and then truncate the symbol at $O(1)$ and get the equation $$u_t + \mathcal{N}(u) + \mathcal{D}^{(0)} u = 0$$ which is much nicer to work with in my case. I want to study the convergence of solutions of the first equation to this equation in the limit as $\epsilon \rightarrow 0$, but I can't find this kind of method used anywhere in the literature, even something without the nonlinearity ($\mathcal{N} = 0$) would be useful.

Thanks :)

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  • $\begingroup$ I'm sure this has been done before. Convergence of the perturbed solutions follows from a priori estimates for the solution to the perturbed equation plus a bound on the operator norm (with respect to the appropriate function spaces) of $D_{\epsilon}-D_0$. This in turn can be estimated in terms of the corresponding difference of the symbols. If in the end you want a $C^\infty$ result (and not, say, a $C^3$ result), then this is rather straightforward. I'm sure there are better references, but you could look at Appendix B in deaneyang.com/papers/goodman-yang.pdf $\endgroup$
    – Deane Yang
    Jan 12, 2017 at 16:15

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