Tricki

## 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. ). To then establish control for much later (or much earlier) times , 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 , one often differentiates 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 be a smooth solution to the nonlinear wave equation which is compactly supported in space at each time. Show that the norm of grows at most linearly in .

Solution (Use conservation of the energy 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 is known to be monotone, but on the other hand for some times , then must be constant between and ; in particular, for all such . This fact can lead to some useful consequences (especially if one knows how to write in a "manifestly non-negative" way, e.g. as a sum or integral of squares).