Motivation for the fractional Sobolev spaces

Question

I am strying studying on my own parabolic PDEs throughout the book "Linear and Quasi-linear Equations of Parabolic Type" by Vsevolod A. Solonnikov, Nina Uraltseva, Olga Ladyzhenskaya, but there is one thing that I do not understand: why we are looking for the solutions of linear and quasi-linear parabolic PDEs in fractional Sobolev spaces instead of the classical Sobolev spaces. I believe that there is some problem that we find when we try looking for these solutions in the classical Sobolev spaces, but I could not realize what is this problem and the only thing that I could find about the motivation for fractional Sobolev spaces is this. I will be grateful if someone can explain why we work in fractional Sobolev spaces instead of the classical Sobolev spaces.

Answer 1

The simple answer is that you can find better and sharper estimates using fractional spaces, or interpolation spaces. Let me give an example, our favorite parabolic pde: \begin{align} u_t=u_{xx}. \end{align} When we denote by $S(t)$ the semigroup generated by the laplacian, we can solve the equation as \begin{align} u(t)=S(t)u_0. \end{align} Suppose we want to measure $u(t)$ in some Hilbert space $X$, and the initial condition is from a space $Y$, then we find \begin{align} ||u(t)||_X\leq ||S(t)||_{L(Y,X)}||u_0||_Y. \end{align} The key question is now how the operator norm depends on time. For $X=H^2$ and $Y=L^2$, we know that the operator norm has a singularity of $t^{-1}$, but when we take $Y=H^2$ there is no singularity. Now what if we take an initial condition that is smoother then $L^2$, but not as smooth as $H^2$? How strong will the singularity be? In order to answer these questions, you need interpolation spaces between $H^2$ and $L^2$, in other words, you want to construct a family of spaces $H^\alpha$ in between $H^2$ and $L^2$, and fractional Sobolev spaces are a nice explicit way to construct these spaces. I do recommend to study these lecture notes, http://people.dmi.unipr.it/alessandra.lunardi/. Corollary 4.1.11 is the famous Ladyzhenskaja – Solonnikov – Ural’ceva-theorem, and the use of interpolation spaces becomes very clear here.