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Extended my answer and altered the grammar in the title
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user71046
user71046

Reference Requestrequest for the Focussing Examplefocussing example

DefineI was reading "From Rotating Needles to Stability of Waves: Emerging Connections between Combinatorics, Analysis, and PDE" by Terence Tao, which is a Notice of the wave equation asAmerican Mathematical Society Vol. 48, No 3.

There is a mention on energy estimates being fixed time estimates at a specified time, but requiring a lot of regularity in $u_{tt}-\Delta u=0$$L^p.$ Then there is a mention of the focussing example.

The article talks of how it is an example with initial position some given $f$data dispersed near the unit sphere and initially zero velocitythe solution $u$ focuses at the origin at time $t=1$ with high $L^{\infty}$ norm. 

I was wondering if anyone can provide any references for the Focussing example in dispersive PDE theory, which (apparently) isor if they know anywhere I can find out more about the dispersion of the initial data around the unit sphere and the focusing of $u$ at the originit?

Thank you in advance for you comments.

Reference Request for the Focussing Example

Define the wave equation as $u_{tt}-\Delta u=0$ with initial position some given $f$ and initially zero velocity. I was wondering if anyone can provide any references for the Focussing example in dispersive PDE theory, which (apparently) is about the dispersion of the initial data around the unit sphere and the focusing of $u$ at the origin?

Reference request for the focussing example

I was reading "From Rotating Needles to Stability of Waves: Emerging Connections between Combinatorics, Analysis, and PDE" by Terence Tao, which is a Notice of the American Mathematical Society Vol. 48, No 3.

There is a mention on energy estimates being fixed time estimates at a specified time, but requiring a lot of regularity in $L^p.$ Then there is a mention of the focussing example.

The article talks of how it is an example with initial data dispersed near the unit sphere and the solution $u$ focuses at the origin at time $t=1$ with high $L^{\infty}$ norm. 

I was wondering if anyone can provide any references for the Focussing example, or if they know anywhere I can find out more about it?

Thank you in advance for you comments.

Source Link
user71046
user71046

Reference Request for the Focussing Example

Define the wave equation as $u_{tt}-\Delta u=0$ with initial position some given $f$ and initially zero velocity. I was wondering if anyone can provide any references for the Focussing example in dispersive PDE theory, which (apparently) is about the dispersion of the initial data around the unit sphere and the focusing of $u$ at the origin?