Sobolev spaces and embeddings

The Sobolev space \( W^{k,p}(\Omega) \) contains the functions in \( L^p(\Omega) \) whose weak derivatives up to order \( k \) are also in \( L^p \), with norm \( \|u\|_{W^{k,p}} = \big(\sum_{|\alpha|\le k}\|\partial^\alpha u\|_{L^p}^p\big)^{1/p} \). The case \( p = 2 \) is written \( H^k \); it is a Hilbert space. These are the spaces in which energy methods naturally live: an energy estimate bounds \( \int|\nabla u|^2 \), which is an \( H^1 \) norm, and Sobolev spaces are complete, so limits of approximate solutions stay inside.

Sobolev functions can be unbounded. On the unit ball in \( \mathbb R^3 \), let \( u = |x|^{-a} \) with \( a > 0 \). Then \( |\nabla u| = a|x|^{-a-1} \), and in spherical coordinates \( \int_B|\nabla u|^2 = 4\pi a^2\int_0^1r^{-2a}\,dr \), finite exactly when \( a < 1/2 \). The example evaluates the case \( a = 1/4 \): the integral is \( \pi/2 \), so \( |x|^{-1/4} \in H^1(B) \) although it is infinite at the origin. In \( \mathbb R^3 \), \( H^1 \) functions need not be bounded.

Embeddings say how much integrability or continuity derivatives buy. For \( 1 \le p < n \), the Gagliardo-Nirenberg-Sobolev inequality \( \|u\|_{L^{p^*}(\mathbb R^n)} \le C\|\nabla u\|_{L^p(\mathbb R^n)} \) holds with \( \frac1{p^*} = \frac1p - \frac1n \); in three dimensions \( H^1 \subset L^6 \). For \( p > n \), Morrey's inequality makes \( W^{1,p} \) functions Hölder continuous. And \( H^s(\mathbb R^n) \subset C^0 \) when \( s > n/2 \).

Picture it: the exponent \( p^* \) comes from scaling. Set \( u_\lambda(x) = u(\lambda x) \). Changing variables, \( \|u_\lambda\|_{L^q} = \lambda^{-n/q}\|u\|_{L^q} \) and \( \|\nabla u_\lambda\|_{L^p} = \lambda^{1 - n/p}\|\nabla u\|_{L^p} \). If \( \|u\|_{L^q} \le C\|\nabla u\|_{L^p} \) held with \( -n/q \ne 1 - n/p \), letting \( \lambda \to 0 \) or \( \lambda \to \infty \) would make one side arbitrarily small compared with the other. The only possible exponent is \( -n/q = 1 - n/p \), which is \( q = p^* \).

Think it: every inequality between norms on \( \mathbb R^n \) must balance under scaling, so exponents can be predicted before any proof. For Navier-Stokes in three dimensions the energy inequality puts \( u \) in \( L^\infty_tL^2_x \) and \( \nabla u \) in \( L^2_tL^2_x \); Sobolev gives \( u \in L^2_tL^6_x \). Whether that is enough to control the solution is decided by the scaling count in the last lesson, and the answer is: not quite.