$u$ is a spherical wave $iff$ $f$ and $g$ are radial
Consider the Cauchy problem for the wave
equation :
begin{cases} u_{tt}= Delta u \u(x,0)=f(x) \ u_t(x,0)= g(x)end{cases} with $t>0$ and $xinmathbb{R^d}$
This is an exercise from Stein's Fourier Analysis an Introduction chapter 6 :
A spherical wave is a solution $u(x; t)$ of the Cauchy problem for the wave
equation in $mathbb{R^d}$, which as a function of $x$ is radial. Prove that $u$ is a spherical
wave if and only if the initial data $f; g in mathcal S$ are both radial, where $mathcal S$ denotes the Schwartz space.
I have already done the $Leftarrow$ direction , my doubt is on the $Rightarrow$ :
My thought: if $u$ is a solution then $u(x,0) = f(x)$ and because $u(x,0)$ depends only on $x$ and $u$ is radial as a function of $x$ (by the definition of spherical wave above) then $f$ is radial by definition since $h(x)$ is radial if $h(x)= h_0(|x|)$ for some $h_0$ . In an analogous way we obtain the result for $g$.
Is this ok ? Or is there a better way to prove it ?
pde fourier-analysis fourier-transform wave-equation cauchy-problem
edited Nov 26 at 22:24
Davide Giraudo
125k16150259
asked Nov 26 at 21:06
math.pr
284
add a comment |
Consider the Cauchy problem for the wave
equation :
begin{cases} u_{tt}= Delta u \u(x,0)=f(x) \ u_t(x,0)= g(x)end{cases} with $t>0$ and $xinmathbb{R^d}$
This is an exercise from Stein's Fourier Analysis an Introduction chapter 6 :
A spherical wave is a solution $u(x; t)$ of the Cauchy problem for the wave
equation in $mathbb{R^d}$, which as a function of $x$ is radial. Prove that $u$ is a spherical
wave if and only if the initial data $f; g in mathcal S$ are both radial, where $mathcal S$ denotes the Schwartz space.
I have already done the $Leftarrow$ direction , my doubt is on the $Rightarrow$ :
My thought: if $u$ is a solution then $u(x,0) = f(x)$ and because $u(x,0)$ depends only on $x$ and $u$ is radial as a function of $x$ (by the definition of spherical wave above) then $f$ is radial by definition since $h(x)$ is radial if $h(x)= h_0(|x|)$ for some $h_0$ . In an analogous way we obtain the result for $g$.
Is this ok ? Or is there a better way to prove it ?
pde fourier-analysis fourier-transform wave-equation cauchy-problem
edited Nov 26 at 22:24
Davide Giraudo
125k16150259
asked Nov 26 at 21:06
math.pr
284
1
I'd look at the contrapositive (if $f$ is not radial....) and use the continuity at $t$ goes to $0$.
– Paul
Nov 26 at 21:11
The hard part of the "complete solution" is to find the Green function is radial so radial initial conditions is equivalent to radial solution.
– reuns
Nov 27 at 4:12
add a comment |
Consider the Cauchy problem for the wave
equation :
begin{cases} u_{tt}= Delta u \u(x,0)=f(x) \ u_t(x,0)= g(x)end{cases} with $t>0$ and $xinmathbb{R^d}$
This is an exercise from Stein's Fourier Analysis an Introduction chapter 6 :
A spherical wave is a solution $u(x; t)$ of the Cauchy problem for the wave
equation in $mathbb{R^d}$, which as a function of $x$ is radial. Prove that $u$ is a spherical
wave if and only if the initial data $f; g in mathcal S$ are both radial, where $mathcal S$ denotes the Schwartz space.
I have already done the $Leftarrow$ direction , my doubt is on the $Rightarrow$ :
My thought: if $u$ is a solution then $u(x,0) = f(x)$ and because $u(x,0)$ depends only on $x$ and $u$ is radial as a function of $x$ (by the definition of spherical wave above) then $f$ is radial by definition since $h(x)$ is radial if $h(x)= h_0(|x|)$ for some $h_0$ . In an analogous way we obtain the result for $g$.
Is this ok ? Or is there a better way to prove it ?
pde fourier-analysis fourier-transform wave-equation cauchy-problem
edited Nov 26 at 22:24
Davide Giraudo
125k16150259
asked Nov 26 at 21:06
math.pr
284
Consider the Cauchy problem for the wave
equation :
begin{cases} u_{tt}= Delta u \u(x,0)=f(x) \ u_t(x,0)= g(x)end{cases} with $t>0$ and $xinmathbb{R^d}$
This is an exercise from Stein's Fourier Analysis an Introduction chapter 6 :
A spherical wave is a solution $u(x; t)$ of the Cauchy problem for the wave
equation in $mathbb{R^d}$, which as a function of $x$ is radial. Prove that $u$ is a spherical
wave if and only if the initial data $f; g in mathcal S$ are both radial, where $mathcal S$ denotes the Schwartz space.
I have already done the $Leftarrow$ direction , my doubt is on the $Rightarrow$ :
My thought: if $u$ is a solution then $u(x,0) = f(x)$ and because $u(x,0)$ depends only on $x$ and $u$ is radial as a function of $x$ (by the definition of spherical wave above) then $f$ is radial by definition since $h(x)$ is radial if $h(x)= h_0(|x|)$ for some $h_0$ . In an analogous way we obtain the result for $g$.
Is this ok ? Or is there a better way to prove it ?
pde fourier-analysis fourier-transform wave-equation cauchy-problem
pde fourier-analysis fourier-transform wave-equation cauchy-problem
edited Nov 26 at 22:24
Davide Giraudo
125k16150259
asked Nov 26 at 21:06
math.pr
284
edited Nov 26 at 22:24
Davide Giraudo
125k16150259
asked Nov 26 at 21:06
math.pr
284
edited Nov 26 at 22:24
Davide Giraudo
125k16150259
edited Nov 26 at 22:24
Davide Giraudo
125k16150259
edited Nov 26 at 22:24
Davide Giraudo
125k16150259
125k16150259
asked Nov 26 at 21:06
math.pr
284
asked Nov 26 at 21:06
math.pr
284
asked Nov 26 at 21:06
math.pr
284
284
1
I'd look at the contrapositive (if $f$ is not radial....) and use the continuity at $t$ goes to $0$.
– Paul
Nov 26 at 21:11
The hard part of the "complete solution" is to find the Green function is radial so radial initial conditions is equivalent to radial solution.
– reuns
Nov 27 at 4:12
add a comment |
1
I'd look at the contrapositive (if $f$ is not radial....) and use the continuity at $t$ goes to $0$.
– Paul
Nov 26 at 21:11
The hard part of the "complete solution" is to find the Green function is radial so radial initial conditions is equivalent to radial solution.
– reuns
Nov 27 at 4:12
1
1
I'd look at the contrapositive (if $f$ is not radial....) and use the continuity at $t$ goes to $0$.
– Paul
Nov 26 at 21:11
I'd look at the contrapositive (if $f$ is not radial....) and use the continuity at $t$ goes to $0$.
– Paul
Nov 26 at 21:11
The hard part of the "complete solution" is to find the Green function is radial so radial initial conditions is equivalent to radial solution.
– reuns
Nov 27 at 4:12
The hard part of the "complete solution" is to find the Green function is radial so radial initial conditions is equivalent to radial solution.
– reuns
Nov 27 at 4:12
add a comment |
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I'd look at the contrapositive (if $f$ is not radial....) and use the continuity at $t$ goes to $0$.
– Paul
Nov 26 at 21:11
The hard part of the "complete solution" is to find the Green function is radial so radial initial conditions is equivalent to radial solution.
– reuns
Nov 27 at 4:12