# Motivation for the fractional Sobolev spaces

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.

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.
• Mmm, strange, well, you could always google lunardi Analytic Semigroups and Reaction-Diffusion Problems' Mar 30, 2020 at 8:27
• By singularity I simply mean divide by zero', just as in complex analysis. Hence, the operator norm above from $H^2$ to $L^2$, which goes as $t^{-1}$, has a singularity of order 1 at $t=0$. Apr 3, 2020 at 8:06