Weak solutions (Leray-Hopf) and Sobolev spaces for fluids

Without a bound that prevents blow-up, one can still ask for solutions in a weaker sense, defined by what the energy equality controls. The natural spaces are these. \(L^2_\sigma\) is the closure in \(L^2\) of smooth, compactly supported, divergence-free fields. The homogeneous Sobolev space \(\dot H^1\) measures \(\|\nabla\mathbf{u}\|_{L^2}\); in Fourier terms \(\|\mathbf{u}\|^2_{\dot H^s} = \int|\mathbf{k}|^{2s}|\hat{\mathbf{u}}(\mathbf{k})|^2\,d\mathbf{k}\), which defines \(\dot H^s\) for every real \(s\). Sobolev embedding in 3D says \(\dot H^1 \subset L^6\) and \(\dot H^{1/2} \subset L^3\); scaling explains the exponents, since both sides must change by the same power of \(\lambda\).

A Leray-Hopf weak solution with data \(\mathbf{u}_0 \in L^2_\sigma\) is a field \[ \mathbf{u} \in L^\infty(0,T;L^2_\sigma) \cap L^2(0,T;\dot H^1) \] that satisfies the equations in the sense of distributions against divergence-free test fields \(\boldsymbol{\varphi}\) (so the pressure never appears): \[ \int_0^T\!\!\int \left(-\mathbf{u}\cdot\partial_t\boldsymbol{\varphi} - (\mathbf{u}\otimes\mathbf{u}):\nabla\boldsymbol{\varphi} + \nu\nabla\mathbf{u}:\nabla\boldsymbol{\varphi}\right)dx\,dt = \int\mathbf{u}_0\cdot\boldsymbol{\varphi}(\cdot,0)\,dx, \] together with the energy inequality \(\tfrac12\|\mathbf{u}(t)\|^2_{L^2} + \nu\int_0^t\|\nabla\mathbf{u}\|^2_{L^2}\,ds \le \tfrac12\|\mathbf{u}_0\|^2_{L^2}\) for all \(t\). A smooth solution with enough decay is a weak solution: multiply by \(\boldsymbol{\varphi}\) and integrate by parts.

Leray (1934, on \(\mathbb{R}^3\)) and Hopf (1951, in bounded domains) proved that global weak solutions exist for every finite-energy initial datum. The method: solve an approximate problem (Hopf used the Galerkin method, truncating to finitely many Fourier modes), obtain the energy bound uniformly in the approximation, extract a weakly convergent subsequence, and use compactness in time (later formalised as the Aubin-Lions lemma) to get strong \(L^2\) convergence, which is what the quadratic term needs. Norms can drop under weak limits, which is why only an inequality for the energy survives.

What is known about these solutions. They are smooth and unique as long as a smooth solution exists (weak-strong uniqueness, Prodi 1959 and Serrin 1963). In two dimensions they are always unique and smooth (Lions and Prodi, 1959; Ladyzhenskaya). In three dimensions Leray showed they are smooth except on a closed set of times of measure zero, and smooth after some large time; Scheffer (1976) sharpened this to zero half-dimensional Hausdorff measure. Their uniqueness in 3D is open; Albritton, Brué and Colombo (2022) proved non-uniqueness of Leray-Hopf solutions with a specially constructed force, and Buckmaster and Vicol (2019) proved non-uniqueness for a weaker class of finite-energy weak solutions that need not satisfy the energy inequality.

The energy class controls \(\mathbf{u}\) in \(L^p_tL^q_x\) whenever \(\frac2p + \frac3q = \frac32\), \(2 \le q \le 6\), by interpolating between \(L^\infty L^2\) and \(L^2L^6\). Compare that with the scale-invariant relation \(\frac2p + \frac3q = 1\) in the next lesson: the energy falls short by exactly one half, the quantitative form of saying it is supercritical.

Picture it: a weak solution is a flow we can only see through frosted glass (averaged against smooth test fields), bright enough to know it exists and has bounded energy, too blurred to tell whether it has a singular point. Think it: existence comes from compactness plus the energy bound, and regularity needs more than the energy gives; that gap is the whole difficulty.