Time translation symmetry: Difference between revisions
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A PDE is said to have '''time translation symmetry''' if for any solution <math>u(t,x)</math> the function <math>u(t + t_0 , x)</math> is also a solution for all (or some interval of) <math>t_0 \in \mathbb{R}</math>. | A PDE is said to have '''time translation symmetry''' if for any solution <math>u(t,x)</math> the function <math>u(t + t_0 , x)</math> is also a solution for all (or some interval of) <math>t_0 \in \mathbb{R}</math>. Generally speaking, a PDE has time translation symmetry when the coefficients and data do not explicitly involve the time variable t. | ||
A PDE which has time-translation symmetry is also called ''time-translation invariant'' or ''autonomous''. | |||
One can often use time-translation symmetry to bootstrap a local well-posedness result into a global well-posedness one, provided that one has some uniform control on the lifespan of the solution provided by the local well-posedness theory. | |||
Latest revision as of 07:18, 20 December 2007
A PDE is said to have time translation symmetry if for any solution the function is also a solution for all (or some interval of) . Generally speaking, a PDE has time translation symmetry when the coefficients and data do not explicitly involve the time variable t.
A PDE which has time-translation symmetry is also called time-translation invariant or autonomous.
One can often use time-translation symmetry to bootstrap a local well-posedness result into a global well-posedness one, provided that one has some uniform control on the lifespan of the solution provided by the local well-posedness theory.
