How to prove that the L2 norm is a non-increasing function of time?

[Su3liminal]

Homework Statement

Homework Equations

How can I start the proof? Shall I use the Poincare inequality?

The Attempt at a Solution

Well, I know that this norm is defined by

, but still I don't know how to start constructing the proof?

 

[Ray Vickson]

Homework Statement

Homework Equations

How can I start the proof? Shall I use the Poincare inequality?

The Attempt at a Solution

Well, I know that this norm is defined by

, but still I don't know how to start constructing the proof?

Start by omitting the square root. Take $\partial / \partial t$ inside the integral sign (justify!), then use the DE to eliminate, or at least, modify $\partial{u^2}/\partial t$.

 

[Su3liminal]

Start by omitting the square root. Take $\partial / \partial t$ inside the integral sign (justify!), then use the DE to eliminate, or at least, modify $\partial{u^2}/\partial t$.

Thanks! I have done what you said (note that I just made a change in variables so I stick to the symbol convention of integration by parts.):

\\2\int_{0}^{L}\frac{\partial^2 s}{\partial x^2}s dx<br /> \\<br /> \\<br /> \\u=s, dv=\frac{\partial^2 s}{\partial x^2}dx<br /> \\du=\frac{\partial s}{\partial x}dx,v=\frac{\partial s}{\partial x}<br /> <br /> \\<br /> \\<br /> \\\therefore 2\int_{0}^{L}\frac{\partial^2 s}{\partial x^2}s dx=2s\frac{\partial s}{\partial x}\mid-2\int_{0}^{L}\frac{\partial s}{\partial x}\frac{\partial s}{\partial x}dx=-2\int_{0}^{L}(\frac{\partial s}{\partial x})^{2}dx

Is that sufficient?

 

[Quesadilla]

Standard for me would be to start with $\frac{\partial u}{\partial t}= \frac{\partial^2 u}{\partial x^2}$ and multiply by $u$ then integrate over the spatial domain $[0,L]$.