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I encountered these two inequalities in reading Sogge's lectures on nonlinear wave equations (page 17). It seems natural and straightforward such that the author didn't give any hint but I cannot work out a prrof. Appreciate any hints.

(1) Let $$u(0,x)=\partial_t u(0,x)=0$$ Then $$\int_0^\phi |u(t,x)|^2 dt \leq t_0^2 \int_0^\phi |\partial_t u(t,x)|^2 dt$$ where $t_0$ is an upper bound and $\phi<t_0$

(2) Let the source of wave equation $\square u =F(u, u', u'')$ satisfy $$F(0,0,u'')=0$$ then $$ 2|\partial_t u F| \leq C (|u|^2 + |u'|^2)$$

Thanks in advance!

Bowen Zhao
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1 Answers1

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If amplitude at t=0 is zero, and velocity at t=0 is zero for all x, the time mean of the amplitude squared at any point $x$ is smaller than a constant times the mean of the velocity squared

$$\left(\frac{ u(x,t)}{t_0}\right)^2 \ dt < \left( \partial_t u(x,t)\right) ^2 \ dt $$

For a linear equation, with $u=0$ the equilibrium configuration, both sides remain at 0.

If u=0 is not the equilibrium configuration, the amplitude is growing with $t^2$ independent of $x$ as for a single mass like $\partial_t u^2$.

For a nonlinear equation dominated by the linear approximation, Lipshitz smoothness conditions yield the same boundedness between time means of kinetic energy and the quadratic approximation of the potential energy.

The topic in point mechanics and its application in statistical mechanics is called the virial theorem, that connects time and ensemble means of kinetic and potential energies by their power laws.

Roland F
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