Use conservation laws and monotonicity formulae to obtain long-time control on solutions

Quick description

Suppose one has a solution to an evolution equation (such as a PDE) which one controls well at some initial time (e.g. $t=0$ ). To then establish control for much later (or much earlier) times $t$ , one effective approach is to exploit a conservation law or monotonicity formula, which propagates control of some quantity at one time to control of a related quantity at other times. To establish a conservation law or monotonicity formula for a quantity $Q(t)$ , one often differentiates $Q$ in time, and rearranges it (using such tools as integration by parts) until it is manifestly zero, non-negative, or non-positive.

Prerequisites

Partial differential equations

Example 1

Problem: Let $\R \times \R \to \R$ be a smooth solution to the nonlinear wave equation $-u_{tt} + u_{xx} = u^3$ which is compactly supported in space at each time. Show that the $H^1(\R)$ norm of $u(t)$ grows at most linearly in $t$ .

Solution (Use conservation of the energy $\int_\R \frac{1}{2} u_t^2 + \frac{1}{2} u_x^2 + \frac{1}{4} u^4\ dx$ and Sobolev embedding...)

General discussion

A more advanced version of this technique is to exploit almost conserved quantities or almost monotone quantities - quantities whose derivative is not quite zero (or non-negative, or non-positive), but is instead small (or non-negative up to a small error, etc.).

Monotonicity properties can also be used in the contrapositive: if a quantity $Q(t)$ is known to be monotone, but on the other hand $Q(t_1)=Q(t_2)$ for some times $t_1, t_2$ , then $Q$ must be constant between $t_1$ and $t_2$ ; in particular, $Q'(t)=0$ for all such $t$ . This fact can lead to some useful consequences (especially if one knows how to write $Q'(t)$ in a "manifestly non-negative" way, e.g. as a sum or integral of squares).