Advances in Sobolev Spaces and Their Applications to Modern Analysis and PDEs

Abstract

This paper surveys recent advances in Sobolev spaces (also known as fractional-order Sobolev spaces, Besov spaces, Bessel or Riesz potential spaces, and Triebel–Lizorkin spaces) and their applications to modern analysis and partial differential equations (PDEs). The studied fractional Sobolev spaces unifying these classical spaces have proved to be useful tools in both linear and nonlinear PDE theory. Various classical tools of nonlinear analysis, including fixed point theorems, Leray–Schauder degree theory, variational methods, and blow-up arguments, have been extended to these spaces, significantly expanding their nonlinear analysis capacity. The report summarizes recent progress on fractional Sobolev spaces and related fields of nonlinear analysis, applications, nonlinear PDEs, and the Navier–Stokes equations, clarifying the advantages of employing fractional Sobolev spaces in these contexts.

Keywords: Sobolev spaces, partial differential equations, linear and nonlinear