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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).