# Research topics in Microlocal Analysis

Before asking this question here I did some research on web but I would like to get the opinion of those directly interested if there are any , (as I did in this thread Research topics in distribution theory since a good connection exists). I have seen that the applications and/or research topics are varied. I also tried to see more about mathematical analysis, but every time I am more passionate about these topics.

I also mention this thread for applications: Applications of Microlocal Analysis?

For example, even if the question I'm doing is quite generic, these theories that are mostly applied to the study of linear PDEs, as can be applied in non-linear PDE contexts (I read that they are very important for this case too very difficult). The non-linear case techniques on what are based? Clearly I refer for example to the existence and uniqueness of the solution or regularity.

Another thing to add, I have read of applications in various classification of PDEs, ellipticals obviously (fairly well known), but also hyperbolic or parabolic. I would be grateful for every answer.

Clearly any reference is welcome

• It is not clear what exactly are you looking for. Did you talk to your advisor for a topic? The applications can vary from earthquake modelling to construct new invariants in low dimensional topology. – Bombyx mori Apr 22 '19 at 1:48
• I don't have an advisor. I could be interested in anything,and I know the question is a bit generic. Mainly I would be interested to know some technique for the study of non-linear PDE, and of any type. Most of the literature I find is more related to linear PDEs – Andrew Apr 22 '19 at 11:05
• I see. Good luck! – Bombyx mori Apr 23 '19 at 3:35

I can give a brief description of a (perhaps, currently the main) topic pertaining the application of microlocal analysis to the study of nonlinear PDEs, which seem the argument you are most interested to, since it is also the one to which I am more accustomed, having been interested in it many years ago: to my knowledge, the main (if not the only) method used to extend this analysis to nonlinear PDEs, is the use of paradifferential operators and paraproducts (introduced by Jean-Michel Bony), so let's sketch how this approach works.

Notation
I am taking the monograph [2] as a reference, and assume as known, at least formally, the basic linear theory of pseudodifferential operators calculus. As it is customary when dealing with $$\psi$$DOs, we have $$\begin{split} D_k&\overset{\text{Def}}{=}\frac{1}{i}\frac{\partial}{\partial x_k}=\frac{1}{i}\partial_{x_k}\quad k=1,\ldots,n\\ D&\overset{\text{Def}}{=}\frac{1}{i}\left(\frac{\partial}{\partial x_1},\cdots,\frac{\partial}{\partial x_n}\right)\\ D^\beta&\overset{\text{Def}}{=}\left(\frac{1}{i}\frac{\partial}{\partial x_1}\right)^{\beta_1}\!\!\!\!\cdot\ldots\cdot\left(\frac{1}{i}\frac{\partial}{\partial x_n}\right)^{\beta_n}\quad\beta=(\beta_1,\ldots,\beta_n)\in\Bbb N^n\\ \end{split}$$

Introduction
Microlocal analysis is, roughly speaking, the analysis of the wavefront set $${W\!F}(u)$$ of a distribution $$u\in \mathscr{D}^\prime(\Omega)$$, $$\Omega\subseteq\Bbb R^n$$ which solves, in the standard case, a given linear PDE (or $$\psi$$DE). This is generically achieved with the use of the instruments of pseudodifferental calculus (parametrices, composition operators and commutator formulas) and a priori estimates. Applying this machinery to nonlinear problems implies being able to represent nonlinear operators in terms of standard pseudodifferential operators: this is the approach taken by Bony, Meyer and their pupils.

Consider a $$C^\infty$$ nonlinear function $$F:\Bbb R\to\Bbb R$$ (just to focus on the basic ideas: the method is well suited also for vector functions), a function $$u\in C^r(\Bbb R^n)$$ and a dyadic partition of unity of $$\Bbb R^n$$, $$\{\psi_j(\xi)\}_{j\in \Bbb N}$$ (see Taylor [2], §1.3 pp. 41-42, for the details): then we can define a countable family constant coefficients pseudodifferential operators $$\{\psi_j(D)\}_{j\in \Bbb N}$$ and the associated family $$\{\Psi_j(D)\}_{j\in \Bbb N}$$ $$\Psi_j(D)=\sum_{i\le j}\psi_i(D)\quad j\in \Bbb N$$ Applying it to $$u$$ we obtain $$u_j=\Psi_j(D)u\implies u(x)=\sum_{j=0}^\infty \psi_j(D)u(x)=\lim_{j\to\infty}u_j(x)$$ and $$F(u)=F(u_0)+\big[F(u_1)-F(u_0)\big]+\big[F(u_2)-F(u_1)\big]+\cdots+\big[F(u_{k+1})-F(u_k)\big]+\cdots$$ Now we have that $$\begin{split} F(u_{k+1})-F(u_k)&= F\big(u_k+\psi_{k+1}(D)u\big)-F(u_k)\\ &=m_k(x)\psi_k(D)u \end{split}\quad\forall k\in\Bbb N$$ where $$m_k(x)=\int_0^1\!\!\! F^\prime\big(u_k+t\psi_{k+1}(D)u\big)\mathrm{d}t = \int_0^1\!\!\! F^\prime\big(\Psi_j(D)u+t\psi_{k+1}(D)u\big)\mathrm{d}t\quad\forall k\in\Bbb N$$ Thus we have obtained the following decomposition $$F(u) = F(u_0)+\sum_{k=1}^\infty m_k(x)\psi_k(D)u = M_F(x,D)u + R(u)\label{1}\tag{1}$$ where

• $$M_F(x,D)u\overset{\text{Def}}{=} \sum_{k=1}^\infty m_k(x)\psi_k(D)u$$ is a paradifferential operator, which is a particular kind of $$\psi$$DO by construction.
• $$R(u)=F(u_0)$$ is a remainder term, which is $$C^\infty$$-smooth by construction, since each $$\psi_j(\xi)$$, $$j\in \Bbb N$$, is a compactly supported $$C^\infty$$-function.

We have thus obtained a general nonlinear function composition as the sum of a (infinitely) smooth remainder term $$R$$ and of a paradifferential operator: now, considering a nonlinear partial differential equation $$F(x,u,\ldots,\partial^\beta u,\ldots)_{|\beta|\le m}=f\quad m\in\Bbb N\label{nlpde}\tag{NLPDE}$$ we can proceed in a similar way as for \eqref{1} and obtain the following formula $$\begin{split} F(x,u,\ldots,\partial^\beta u,\ldots)_{|\beta|\le m}&=\sum_{|\beta|\le m} M_{\partial_{\partial^\beta u}F}(x,D)\partial_\beta u + F(x,u_0,\ldots,\partial^\beta u_0,\ldots)\\ &=\sum_{|\beta|\le m} M_{\partial_{\partial^\beta u}F}(x,D)\partial_\beta u + R(x,u_0,\ldots,\partial^\beta u_0,\ldots) \end{split}\label{2}\tag{2}$$ Thus again we have obtained an expansion expressing a nonlinear PDE as the sum of a finite number of paradifferential operators and a $$C^\infty$$-smooth remainder term: then you can apply to formula \eqref{2} the machinery of microlocal analysis developed for the linear theory and get a priori estimates and information on the structure of the wavefront set $$W\!F(u)$$.
The paraproduct is instead a linear operator related to the operator $$M_F(x,D)$$ in formula \eqref{1}. There are several versions of it: one is the following $$\pi(a,u)=\sum_{j\ge 1}^\infty \big(\Psi_{j-1}(D)a\big)\big(\psi_{j+1}(D)u\big) \quad a, u\in C^r\label{3}\tag{3}$$ Apart from its elementary properties, \eqref{3} can be used in approximating $$M_F(x,D)u$$ and obtain a formula analogous to \eqref{1} (see [2], p. 76, proposition 3.2.C) $$F(u)=\pi\big(F^\prime(u),u\big)+R(u)\label{4}\tag{4}$$ where $$R$$ belongs now to an appropriate Sobolev space. The analogies between this formula and \eqref{1} can be used in the study of \eqref{nlpde} to obtain from \eqref{4} a representation similar to \eqref{2} and study it by the techniques of microlocal analysis. Not being an expert in microlocal analysis nor having been involved in studying the subject recently, I address to the monographs [1] and [2] for the details of this sketchy answer.

Notes

• The book [1] is a nice and well organized illustration of known results in the microlocal analysis of nonlinear PDEs, and exposes many results on the Navier-Stokes system including the ones obtained by Marco Cannone, who proved an important global existence theorem for the solution of this system of PDEs, assuming the Cauchy data belongs to an appropriate Besov space.

• See also the MathOverflow question:-"Why is paraproduct or paradifferential calculus important in PDE theory? In particular, the original paper of J-M. Bony linked in the answer to that question is very clear and informative.

Bibliography

[1] Hajer Bahouri, Jean-Yves Chemin, Raphaël Danchin (2011), Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, vol. 343, Berlin-Heidelberg-Dordrecht-London-New York: Springer-Verlag, pp. XVI+524, DOI 10.1007/978-3-642-16830-7, ISBN 978-3-642-16829-1, e-ISBN 978-3-642-16830-7, MR2768550, Zbl 1227.35004.

[2] Michael Eugene Taylor (1991), Pseudodifferential operators and nonlinear PDE, Progress in Mathematics, vol. 100. Boston, MA-Basel-Berlin: Birkhäuser Verlag, pp. 213, ISBN: 0-8176-3595-5, MR1121019, Zbl 0746.35062.

• Thank you very much, I was hoping for such an answer – Andrew May 24 '19 at 12:39
• @Andrew You are welcome. – Daniele Tampieri May 24 '19 at 12:58