The Cauchy problem: the Leray projection and local existence

To prove anything we must say exactly what is being solved. The Cauchy problem for the incompressible Navier-Stokes equations prescribes a divergence-free initial velocity \(\mathbf{u}_0\) and asks for \(\mathbf{u}(\mathbf{x},t)\) and \(p(\mathbf{x},t)\), \(t \ge 0\), with \[ \partial_t\mathbf{u} + (\mathbf{u}\cdot\nabla)\mathbf{u} = -\nabla p + \nu\Delta\mathbf{u}, \quad \nabla\cdot\mathbf{u} = 0, \quad \mathbf{u}(\cdot,0) = \mathbf{u}_0. \] Two settings are standard. On \(\mathbb{R}^3\), the data are smooth and decay rapidly with all their derivatives, and one asks for solutions of finite energy, \(\int|\mathbf{u}(\mathbf{x},t)|^2dx\) bounded. On the torus \(\mathbb{T}^3\) the data and solution are smooth and periodic in each coordinate. A smooth solution means \(\mathbf{u}\) and \(p\) are infinitely differentiable in space and time.

Some growth condition at infinity is essential. With no condition, \(\mathbf{u} = (a(t),0,0)\), \(p = -a'(t)\,x\) solves the equations for any function \(a\) with \(a(0) = 0\), so even zero data has infinitely many solutions: the "fluid" is being pushed by a pressure growing at infinity. Requiring finite energy, or periodicity, rules these out.

The pressure can be removed exactly. Every square-integrable vector field splits uniquely as a divergence-free field plus a gradient (the Helmholtz or Leray decomposition), and the Leray projection \(\mathbb{P}\) keeps the divergence-free part. In Fourier variables it is the matrix \(\widehat{\mathbb{P}\mathbf{f}}(\mathbf{k}) = \left(I - \frac{\mathbf{k}\mathbf{k}^T}{|\mathbf{k}|^2}\right)\hat{\mathbf{f}}(\mathbf{k})\), the orthogonal projection onto the plane perpendicular to \(\mathbf{k}\). Applying \(\mathbb{P}\) kills \(\nabla p\), leaves \(\partial_t\mathbf{u}\) and \(\Delta\mathbf{u}\) alone, and gives \[ \partial_t\mathbf{u} - \nu\Delta\mathbf{u} + \mathbb{P}\,\nabla\cdot(\mathbf{u}\otimes\mathbf{u}) = 0, \] an evolution equation for \(\mathbf{u}\) alone. The pressure is recovered as \(p = (-\Delta)^{-1}\partial_i\partial_j(u_iu_j)\).

Treating the nonlinearity as a forcing of the heat equation gives the mild formulation (Duhamel's principle): \[ \mathbf{u}(t) = e^{\nu t\Delta}\mathbf{u}_0 - \int_0^t e^{\nu(t-s)\Delta}\,\mathbb{P}\nabla\cdot(\mathbf{u}\otimes\mathbf{u})(s)\,ds. \] The heat semigroup multiplies the Fourier mode \(\mathbf{k}\) by \(e^{-\nu|\mathbf{k}|^2t}\), damping high frequencies and gaining derivatives, which pays for the derivative in the nonlinear term. A contraction mapping argument on a short time interval then gives local existence and uniqueness of smooth solutions: Leray (1934) for smooth data, Fujita and Kato (1964) for data in the Sobolev space \(\dot H^{1/2}\), Kato (1984) for data in \(L^3\). The critical spaces \(\dot H^{1/2}\) and \(L^3\) have a bonus: if the data are small enough in them, the solution exists and stays smooth for all time.

For large data, local theory gives a maximal time \(T^*\) and a blow-up alternative: either \(T^* = \infty\) or certain norms become infinite as \(t \to T^*\). Leray showed that a finite \(T^*\) forces \[ \|\nabla\mathbf{u}(t)\|_{L^2} \ge \frac{c\,\nu^{3/4}}{(T^* - t)^{1/4}}. \] Whether \(T^*\) can be finite for smooth finite-energy data in 3D, with no external force, is not known.

Picture it: in Fourier space, at each wavevector \(\mathbf{k}\), the projection throws away the component of the nonlinear forcing along \(\mathbf{k}\), and that discarded component is the pressure. Think it: the problem is a semilinear heat equation on the space of divergence-free fields; local well-posedness is a fixed point theorem, and every global question is about whether the fixed point can be continued forever.