Mixing
Uniqueness of solutions to the wave equation
Clash Royale CLAN TAG#URR8PPP
up vote
2
down vote
favorite
we are given the problem $u_tt-c^2Delta u=0$ with conditions $u(x,0)=u_0(x)$ and $u_t(x,0)=u_1(x)$ where $xinmathbbR^n$ and $u_0,u_1inmathcalC^1$ having compact supports. If a solution exists, is it possible to conclude that this solution has a compact support, too?
Thanks.
pde wave-equation
asked Jan 21 '15 at 15:15
JohnSmith
529312
add a comment |Â
up vote
2
down vote
favorite
we are given the problem $u_tt-c^2Delta u=0$ with conditions $u(x,0)=u_0(x)$ and $u_t(x,0)=u_1(x)$ where $xinmathbbR^n$ and $u_0,u_1inmathcalC^1$ having compact supports. If a solution exists, is it possible to conclude that this solution has a compact support, too?
Thanks.
pde wave-equation
asked Jan 21 '15 at 15:15
JohnSmith
529312
The solution will not have global compact support for $t in [0,infty)$, but it will for every finite time because of finite wave propagation speed. Is that what you mean?
– Lukas Geyer
Jan 21 '15 at 15:48
Thanks for your answer! Basically, I want to prove uniqueness of the solution if the intial data have a compact support. Therefore, my idea was to use energy conservation which my professor proved in the case when the solution has a compact support. Unfortunately, this seems not to be true in general and I have no nice tool such as D'Alembert's formula as in the one-dimensional case available. Is there another way to prove uniqueness of the solution having compactly supported data?
– JohnSmith
Jan 21 '15 at 23:35
Hint: Consider two solutions say $u,v$ of your wave equation. The difference, $w$ will be a solution too, but with null data. Now write down the weak formulation (with null data) and use as test function $w_t$. Since you can write the term $(w_tt,w_t)$ as $fracd2dt(||w||^2_L^2)$, Gronwall's inequality should provide you uniqueness. This is a very standard routine, see e.g. Evans, Partial Differential Equations for further details.
– alemou
Jan 22 '15 at 13:39
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
we are given the problem $u_tt-c^2Delta u=0$ with conditions $u(x,0)=u_0(x)$ and $u_t(x,0)=u_1(x)$ where $xinmathbbR^n$ and $u_0,u_1inmathcalC^1$ having compact supports. If a solution exists, is it possible to conclude that this solution has a compact support, too?
Thanks.
pde wave-equation
asked Jan 21 '15 at 15:15
JohnSmith
529312
we are given the problem $u_tt-c^2Delta u=0$ with conditions $u(x,0)=u_0(x)$ and $u_t(x,0)=u_1(x)$ where $xinmathbbR^n$ and $u_0,u_1inmathcalC^1$ having compact supports. If a solution exists, is it possible to conclude that this solution has a compact support, too?
Thanks.
pde wave-equation
asked Jan 21 '15 at 15:15
JohnSmith
529312
asked Jan 21 '15 at 15:15
JohnSmith
529312
asked Jan 21 '15 at 15:15
JohnSmith
529312
asked Jan 21 '15 at 15:15
JohnSmith
529312
529312
The solution will not have global compact support for $t in [0,infty)$, but it will for every finite time because of finite wave propagation speed. Is that what you mean?
– Lukas Geyer
Jan 21 '15 at 15:48
Thanks for your answer! Basically, I want to prove uniqueness of the solution if the intial data have a compact support. Therefore, my idea was to use energy conservation which my professor proved in the case when the solution has a compact support. Unfortunately, this seems not to be true in general and I have no nice tool such as D'Alembert's formula as in the one-dimensional case available. Is there another way to prove uniqueness of the solution having compactly supported data?
– JohnSmith
Jan 21 '15 at 23:35
Hint: Consider two solutions say $u,v$ of your wave equation. The difference, $w$ will be a solution too, but with null data. Now write down the weak formulation (with null data) and use as test function $w_t$. Since you can write the term $(w_tt,w_t)$ as $fracd2dt(||w||^2_L^2)$, Gronwall's inequality should provide you uniqueness. This is a very standard routine, see e.g. Evans, Partial Differential Equations for further details.
– alemou
Jan 22 '15 at 13:39
add a comment |Â
The solution will not have global compact support for $t in [0,infty)$, but it will for every finite time because of finite wave propagation speed. Is that what you mean?
– Lukas Geyer
Jan 21 '15 at 15:48
Thanks for your answer! Basically, I want to prove uniqueness of the solution if the intial data have a compact support. Therefore, my idea was to use energy conservation which my professor proved in the case when the solution has a compact support. Unfortunately, this seems not to be true in general and I have no nice tool such as D'Alembert's formula as in the one-dimensional case available. Is there another way to prove uniqueness of the solution having compactly supported data?
– JohnSmith
Jan 21 '15 at 23:35
Hint: Consider two solutions say $u,v$ of your wave equation. The difference, $w$ will be a solution too, but with null data. Now write down the weak formulation (with null data) and use as test function $w_t$. Since you can write the term $(w_tt,w_t)$ as $fracd2dt(||w||^2_L^2)$, Gronwall's inequality should provide you uniqueness. This is a very standard routine, see e.g. Evans, Partial Differential Equations for further details.
– alemou
Jan 22 '15 at 13:39
The solution will not have global compact support for $t in [0,infty)$, but it will for every finite time because of finite wave propagation speed. Is that what you mean?
– Lukas Geyer
Jan 21 '15 at 15:48
The solution will not have global compact support for $t in [0,infty)$, but it will for every finite time because of finite wave propagation speed. Is that what you mean?
– Lukas Geyer
Jan 21 '15 at 15:48
Thanks for your answer! Basically, I want to prove uniqueness of the solution if the intial data have a compact support. Therefore, my idea was to use energy conservation which my professor proved in the case when the solution has a compact support. Unfortunately, this seems not to be true in general and I have no nice tool such as D'Alembert's formula as in the one-dimensional case available. Is there another way to prove uniqueness of the solution having compactly supported data?
– JohnSmith
Jan 21 '15 at 23:35
Thanks for your answer! Basically, I want to prove uniqueness of the solution if the intial data have a compact support. Therefore, my idea was to use energy conservation which my professor proved in the case when the solution has a compact support. Unfortunately, this seems not to be true in general and I have no nice tool such as D'Alembert's formula as in the one-dimensional case available. Is there another way to prove uniqueness of the solution having compactly supported data?
– JohnSmith
Jan 21 '15 at 23:35
Hint: Consider two solutions say $u,v$ of your wave equation. The difference, $w$ will be a solution too, but with null data. Now write down the weak formulation (with null data) and use as test function $w_t$. Since you can write the term $(w_tt,w_t)$ as $fracd2dt(||w||^2_L^2)$, Gronwall's inequality should provide you uniqueness. This is a very standard routine, see e.g. Evans, Partial Differential Equations for further details.
– alemou
Jan 22 '15 at 13:39
Hint: Consider two solutions say $u,v$ of your wave equation. The difference, $w$ will be a solution too, but with null data. Now write down the weak formulation (with null data) and use as test function $w_t$. Since you can write the term $(w_tt,w_t)$ as $fracd2dt(||w||^2_L^2)$, Gronwall's inequality should provide you uniqueness. This is a very standard routine, see e.g. Evans, Partial Differential Equations for further details.
– alemou
Jan 22 '15 at 13:39
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
2
down vote
accepted
Both uniqueness and finite propagation speed follow by an energy method just like in the one-dimensional case. For the sake of simplicity the following is for $c=1$, the same argument works for arbitrary $c>0$.
Fix some point $x_0 in mathbbR^n$ and time $t_0 > 0$, and define
$$
E(t) = int_B(x_0, t_0-t) left( u_t(x,t)^2 + |nabla u (x,t)|^2 right) , dx
$$
for $0 le t le t_0$.
Using integration by parts (Green's identities) you should get
$$
E'(t) = int_partial B(x_0, t_0-t) left( 2 u_t fracpartial upartial n -u_t^2 - |nabla u|^2right) , dx
$$
Then, using the inequality
$$
left| 2u_t fracpartial upartial n right| le 2|u_t| , |nabla u| le u_t^2 + |nabla u|^2
$$
you get that $E'(t) le 0$. So if you know $u(x,0)equiv u_t(x,0)equiv 0$ for $x in B(x_0, t_0)$, then $E(0)=0$, and so $E(t)=0$ for $0 le t le t_0$, implying that $u_t(x,t) = 0$ and $nabla u (x,t) = 0$ for $0le t le t_0$ and $|x-x_0| le t_0 - t$, which implies $u(x,t) = 0$ for the same parameters. This directly implies uniqueness and propagation speed $le 1$.
answered Jan 22 '15 at 22:16
Lukas Geyer
12.8k1254
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
Both uniqueness and finite propagation speed follow by an energy method just like in the one-dimensional case. For the sake of simplicity the following is for $c=1$, the same argument works for arbitrary $c>0$.
Fix some point $x_0 in mathbbR^n$ and time $t_0 > 0$, and define
$$
E(t) = int_B(x_0, t_0-t) left( u_t(x,t)^2 + |nabla u (x,t)|^2 right) , dx
$$
for $0 le t le t_0$.
Using integration by parts (Green's identities) you should get
$$
E'(t) = int_partial B(x_0, t_0-t) left( 2 u_t fracpartial upartial n -u_t^2 - |nabla u|^2right) , dx
$$
Then, using the inequality
$$
left| 2u_t fracpartial upartial n right| le 2|u_t| , |nabla u| le u_t^2 + |nabla u|^2
$$
you get that $E'(t) le 0$. So if you know $u(x,0)equiv u_t(x,0)equiv 0$ for $x in B(x_0, t_0)$, then $E(0)=0$, and so $E(t)=0$ for $0 le t le t_0$, implying that $u_t(x,t) = 0$ and $nabla u (x,t) = 0$ for $0le t le t_0$ and $|x-x_0| le t_0 - t$, which implies $u(x,t) = 0$ for the same parameters. This directly implies uniqueness and propagation speed $le 1$.
answered Jan 22 '15 at 22:16
Lukas Geyer
12.8k1254
add a comment |Â
up vote
2
down vote
accepted
Both uniqueness and finite propagation speed follow by an energy method just like in the one-dimensional case. For the sake of simplicity the following is for $c=1$, the same argument works for arbitrary $c>0$.
Fix some point $x_0 in mathbbR^n$ and time $t_0 > 0$, and define
$$
E(t) = int_B(x_0, t_0-t) left( u_t(x,t)^2 + |nabla u (x,t)|^2 right) , dx
$$
for $0 le t le t_0$.
Using integration by parts (Green's identities) you should get
$$
E'(t) = int_partial B(x_0, t_0-t) left( 2 u_t fracpartial upartial n -u_t^2 - |nabla u|^2right) , dx
$$
Then, using the inequality
$$
left| 2u_t fracpartial upartial n right| le 2|u_t| , |nabla u| le u_t^2 + |nabla u|^2
$$
you get that $E'(t) le 0$. So if you know $u(x,0)equiv u_t(x,0)equiv 0$ for $x in B(x_0, t_0)$, then $E(0)=0$, and so $E(t)=0$ for $0 le t le t_0$, implying that $u_t(x,t) = 0$ and $nabla u (x,t) = 0$ for $0le t le t_0$ and $|x-x_0| le t_0 - t$, which implies $u(x,t) = 0$ for the same parameters. This directly implies uniqueness and propagation speed $le 1$.
answered Jan 22 '15 at 22:16
Lukas Geyer
12.8k1254
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
Both uniqueness and finite propagation speed follow by an energy method just like in the one-dimensional case. For the sake of simplicity the following is for $c=1$, the same argument works for arbitrary $c>0$.
Fix some point $x_0 in mathbbR^n$ and time $t_0 > 0$, and define
$$
E(t) = int_B(x_0, t_0-t) left( u_t(x,t)^2 + |nabla u (x,t)|^2 right) , dx
$$
for $0 le t le t_0$.
Using integration by parts (Green's identities) you should get
$$
E'(t) = int_partial B(x_0, t_0-t) left( 2 u_t fracpartial upartial n -u_t^2 - |nabla u|^2right) , dx
$$
Then, using the inequality
$$
left| 2u_t fracpartial upartial n right| le 2|u_t| , |nabla u| le u_t^2 + |nabla u|^2
$$
you get that $E'(t) le 0$. So if you know $u(x,0)equiv u_t(x,0)equiv 0$ for $x in B(x_0, t_0)$, then $E(0)=0$, and so $E(t)=0$ for $0 le t le t_0$, implying that $u_t(x,t) = 0$ and $nabla u (x,t) = 0$ for $0le t le t_0$ and $|x-x_0| le t_0 - t$, which implies $u(x,t) = 0$ for the same parameters. This directly implies uniqueness and propagation speed $le 1$.
answered Jan 22 '15 at 22:16
Lukas Geyer
12.8k1254
Both uniqueness and finite propagation speed follow by an energy method just like in the one-dimensional case. For the sake of simplicity the following is for $c=1$, the same argument works for arbitrary $c>0$.
Fix some point $x_0 in mathbbR^n$ and time $t_0 > 0$, and define
$$
E(t) = int_B(x_0, t_0-t) left( u_t(x,t)^2 + |nabla u (x,t)|^2 right) , dx
$$
for $0 le t le t_0$.
Using integration by parts (Green's identities) you should get
$$
E'(t) = int_partial B(x_0, t_0-t) left( 2 u_t fracpartial upartial n -u_t^2 - |nabla u|^2right) , dx
$$
Then, using the inequality
$$
left| 2u_t fracpartial upartial n right| le 2|u_t| , |nabla u| le u_t^2 + |nabla u|^2
$$
you get that $E'(t) le 0$. So if you know $u(x,0)equiv u_t(x,0)equiv 0$ for $x in B(x_0, t_0)$, then $E(0)=0$, and so $E(t)=0$ for $0 le t le t_0$, implying that $u_t(x,t) = 0$ and $nabla u (x,t) = 0$ for $0le t le t_0$ and $|x-x_0| le t_0 - t$, which implies $u(x,t) = 0$ for the same parameters. This directly implies uniqueness and propagation speed $le 1$.
answered Jan 22 '15 at 22:16
Lukas Geyer
12.8k1254
answered Jan 22 '15 at 22:16
Lukas Geyer
12.8k1254
answered Jan 22 '15 at 22:16
Lukas Geyer
12.8k1254
answered Jan 22 '15 at 22:16
Lukas Geyer
12.8k1254
12.8k1254
add a comment |Â
add a comment |Â
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f1113622%2funiqueness-of-solutions-to-the-wave-equation%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
The solution will not have global compact support for $t in [0,infty)$, but it will for every finite time because of finite wave propagation speed. Is that what you mean?
– Lukas Geyer
Jan 21 '15 at 15:48
Thanks for your answer! Basically, I want to prove uniqueness of the solution if the intial data have a compact support. Therefore, my idea was to use energy conservation which my professor proved in the case when the solution has a compact support. Unfortunately, this seems not to be true in general and I have no nice tool such as D'Alembert's formula as in the one-dimensional case available. Is there another way to prove uniqueness of the solution having compactly supported data?
– JohnSmith
Jan 21 '15 at 23:35
Hint: Consider two solutions say $u,v$ of your wave equation. The difference, $w$ will be a solution too, but with null data. Now write down the weak formulation (with null data) and use as test function $w_t$. Since you can write the term $(w_tt,w_t)$ as $fracd2dt(||w||^2_L^2)$, Gronwall's inequality should provide you uniqueness. This is a very standard routine, see e.g. Evans, Partial Differential Equations for further details.
– alemou
Jan 22 '15 at 13:39