Uniqueness of classical solutions of nonlinear first order pde.
$begingroup$
I want to prove that there is a unique classical solution (if it exists) to
$$
begin{cases}
partial_tu+A(u)partial_xu=0, quad tgeq 0, quad xinmathbb{R}\
u(x,0)=u_0(x),
end{cases}
$$
where $Ain C^1(mathbb{R})$. To do so, I take two solutions $u,v$ build their difference $d=u-v$, and note that it solves
$$
partial_td+A(u)partial_x d+(A(u)-A(v))partial_xv=0.
$$
and according to Lax' "Hyperbolic Systems of Conservation Laws", one can see from this uniqueness since $|A(u)-A(v)|leq C |d|$. What is the estimate that I need here, and can you sketch how to obtain it? I feel like we need a Gronwall like estimate. If I multiply with $d$ and integrate in space, I get an equation for the derivative of $|d|_{L^2}$, but I don't know what to do with the term $int A(u)(partial_x d )d$.
I guess I could also show uniqueness by the method of characteristics, but I want to understand this approach.
pde
edited Dec 20 '18 at 9:07
Brahadeesh
6,42442363
asked Nov 4 '15 at 15:39
BananachBananach
3,85111429
$endgroup$
add a comment |
$begingroup$
I want to prove that there is a unique classical solution (if it exists) to
$$
begin{cases}
partial_tu+A(u)partial_xu=0, quad tgeq 0, quad xinmathbb{R}\
u(x,0)=u_0(x),
end{cases}
$$
where $Ain C^1(mathbb{R})$. To do so, I take two solutions $u,v$ build their difference $d=u-v$, and note that it solves
$$
partial_td+A(u)partial_x d+(A(u)-A(v))partial_xv=0.
$$
and according to Lax' "Hyperbolic Systems of Conservation Laws", one can see from this uniqueness since $|A(u)-A(v)|leq C |d|$. What is the estimate that I need here, and can you sketch how to obtain it? I feel like we need a Gronwall like estimate. If I multiply with $d$ and integrate in space, I get an equation for the derivative of $|d|_{L^2}$, but I don't know what to do with the term $int A(u)(partial_x d )d$.
I guess I could also show uniqueness by the method of characteristics, but I want to understand this approach.
pde
edited Dec 20 '18 at 9:07
Brahadeesh
6,42442363
asked Nov 4 '15 at 15:39
BananachBananach
3,85111429
$endgroup$
1
$begingroup$
I think the author refers to Gronwall's lemma applied along a characteristic curve.
$endgroup$
– user147263
Nov 12 '15 at 1:00
add a comment |
$begingroup$
I want to prove that there is a unique classical solution (if it exists) to
$$
begin{cases}
partial_tu+A(u)partial_xu=0, quad tgeq 0, quad xinmathbb{R}\
u(x,0)=u_0(x),
end{cases}
$$
where $Ain C^1(mathbb{R})$. To do so, I take two solutions $u,v$ build their difference $d=u-v$, and note that it solves
$$
partial_td+A(u)partial_x d+(A(u)-A(v))partial_xv=0.
$$
and according to Lax' "Hyperbolic Systems of Conservation Laws", one can see from this uniqueness since $|A(u)-A(v)|leq C |d|$. What is the estimate that I need here, and can you sketch how to obtain it? I feel like we need a Gronwall like estimate. If I multiply with $d$ and integrate in space, I get an equation for the derivative of $|d|_{L^2}$, but I don't know what to do with the term $int A(u)(partial_x d )d$.
I guess I could also show uniqueness by the method of characteristics, but I want to understand this approach.
pde
edited Dec 20 '18 at 9:07
Brahadeesh
6,42442363
asked Nov 4 '15 at 15:39
BananachBananach
3,85111429
$endgroup$
I want to prove that there is a unique classical solution (if it exists) to
$$
begin{cases}
partial_tu+A(u)partial_xu=0, quad tgeq 0, quad xinmathbb{R}\
u(x,0)=u_0(x),
end{cases}
$$
where $Ain C^1(mathbb{R})$. To do so, I take two solutions $u,v$ build their difference $d=u-v$, and note that it solves
$$
partial_td+A(u)partial_x d+(A(u)-A(v))partial_xv=0.
$$
and according to Lax' "Hyperbolic Systems of Conservation Laws", one can see from this uniqueness since $|A(u)-A(v)|leq C |d|$. What is the estimate that I need here, and can you sketch how to obtain it? I feel like we need a Gronwall like estimate. If I multiply with $d$ and integrate in space, I get an equation for the derivative of $|d|_{L^2}$, but I don't know what to do with the term $int A(u)(partial_x d )d$.
I guess I could also show uniqueness by the method of characteristics, but I want to understand this approach.
pde
pde
edited Dec 20 '18 at 9:07
Brahadeesh
6,42442363
asked Nov 4 '15 at 15:39
BananachBananach
3,85111429
edited Dec 20 '18 at 9:07
Brahadeesh
6,42442363
asked Nov 4 '15 at 15:39
BananachBananach
3,85111429
edited Dec 20 '18 at 9:07
Brahadeesh
6,42442363
edited Dec 20 '18 at 9:07
Brahadeesh
6,42442363
edited Dec 20 '18 at 9:07
Brahadeesh
6,42442363
6,42442363
asked Nov 4 '15 at 15:39
BananachBananach
3,85111429
asked Nov 4 '15 at 15:39
BananachBananach
3,85111429
asked Nov 4 '15 at 15:39
BananachBananach
3,85111429
3,85111429
1
$begingroup$
I think the author refers to Gronwall's lemma applied along a characteristic curve.
$endgroup$
– user147263
Nov 12 '15 at 1:00
add a comment |
1
$begingroup$
I think the author refers to Gronwall's lemma applied along a characteristic curve.
$endgroup$
– user147263
Nov 12 '15 at 1:00
1
1
$begingroup$
I think the author refers to Gronwall's lemma applied along a characteristic curve.
$endgroup$
– user147263
Nov 12 '15 at 1:00
$begingroup$
I think the author refers to Gronwall's lemma applied along a characteristic curve.
$endgroup$
– user147263
Nov 12 '15 at 1:00
add a comment |
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$begingroup$
I think the author refers to Gronwall's lemma applied along a characteristic curve.
$endgroup$
– user147263
Nov 12 '15 at 1:00