A priori estimates, local and global existence

An a priori estimate is a bound on a solution, in terms of the data, proved on the assumption that a solution exists. It sounds circular but is the engine of existence theory: build approximate solutions (truncated series, discretisations, smoothed equations), show that the a priori estimate holds for all of them uniformly, and use compactness to pass to a limit that solves the real problem.

The simplest nonlinear example shows why existence can be only local. The ODE \( y' = y^2 \), \( y(0) = y_0 > 0 \), has solution \( y = y_0/(1 - y_0t) \), which blows up at \( T = 1/y_0 \). The larger the data, the shorter the time of existence. The time can be read off without solving: \( T = \int_{y_0}^\infty dy/y^2 \). In contrast \( y' = -y^3 \) has solutions that exist for all time, because the nonlinearity pushes towards zero.

For evolution PDEs the pattern is a continuation criterion. One proves that a solution exists on some interval \( [0, T) \) with \( T \) depending on the size of the data in a controlling norm, and that the solution can be continued as long as that norm stays finite. Global existence is then equivalent to an a priori bound on the controlling norm for all time. Gronwall's inequality is the standard tool: if \( Y' \le a(t)Y \) then \( Y(t) \le Y(0)\exp\int_0^ta \), so a linear differential inequality never blows up.

Picture it: a solution is a car whose fuel gauge is the controlling norm. The local theorem says it can always drive a little further as long as the gauge reads a finite number. Blow-up means the needle hits infinity at a finite time; global existence means proving it never does.

Think it: for Navier-Stokes in \( \mathbb R^3 \) the enstrophy \( Y = \|\nabla u\|_{L^2}^2 \) obeys \( Y' \le CY^3/\nu^3 \), a superlinear inequality like \( y' = y^2 \). It gives a smooth solution on an interval of length at least about \( \nu^3/Y(0)^2 \), and it shows that if the solution does blow up at \( T^* \) then \( \|\nabla u(t)\|_{L^2}^2 \ge c\,\nu^{3/2}(T^* - t)^{-1/2} \). The energy inequality alone, a linear bound, is not strong enough to rule this out. That gap is the Millennium problem.