Existance of $phi in L^2$ such as $L(vec v)=int_Omegaphi ;textdiv vec v ;dx$ The 2019 Stack Overflow Developer Survey Results Are Indual of $H^1_0$: $H^-1$ or $H_0^1$?Dual space of a closed subspace of a Hilbert spaceExample of an $H^-1$ function that isn't $L^2$Variational formulation curl-div equationHow can we extend $operatornamediv:C_c^infty(Omega)^dto L^p(Omega)$ to $W_0^1,:p(Omega)^d$?Definition of the Laplacian as an operator from $H_0^1(Omega)$ to $H_0^1(Omega)'$About the dual space of $V=uin H_0^1(Omega): textdivu=0 $ and its relations to $H^-1(Omega)$.Relation of $lVert dot u rVert_W^1(Omega)$ to $ lVert u rVert_W^1(Omega)$Continuous extension of an operator $textPI : H^s(partialOmega)to H^s+1/2(Omega)$Divergence operator: is it a linear continuous map on Sobolev Spaces?
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Existance of $phi in L^2$ such as $L(vec v)=int_Omegaphi ;textdiv vec v ;dx$
The 2019 Stack Overflow Developer Survey Results Are Indual of $H^1_0$: $H^-1$ or $H_0^1$?Dual space of a closed subspace of a Hilbert spaceExample of an $H^-1$ function that isn't $L^2$Variational formulation curl-div equationHow can we extend $operatornamediv:C_c^infty(Omega)^dto L^p(Omega)$ to $W_0^1,:p(Omega)^d$?Definition of the Laplacian as an operator from $H_0^1(Omega)$ to $H_0^1(Omega)'$About the dual space of $V=uin H_0^1(Omega): textdivu=0 $ and its relations to $H^-1(Omega)$.Relation of $lVert dot u rVert_W^1(Omega)$ to $ lVert u rVert_W^1(Omega)$Continuous extension of an operator $textPI : H^s(partialOmega)to H^s+1/2(Omega)$Divergence operator: is it a linear continuous map on Sobolev Spaces?
$begingroup$
[I was working on Stokes pde, and I'm stuck in proving this, couldn't find it anywhere]
Let $Omega in mathbbR^N$ an open bounded connected set such as $partial Omega$ is $mathcalC^1$.
And we define $$V=vec v in (H^1_0(Omega))^N; textdiv vec v=0$$
With $textdiv vec v=sum_n=1^Nfracpartial v_ipartial x_i$.
I'm stuck in proving : if $L$ is a linear continious form on $(H_0^1(Omega))^N$ such as $L_V=0$ then there exists a function $phi in L^2(Omega)$, such as $$forall vec vin (H_0^1(Omega))^N : L(vec v)=int_Omegaphi ;textdiv vec v ;dx$$
My idea : I proved that $V$ is a closed space of $(H_0^1(Omega))^N$ then its a Hilbert space, and I tried to apply Riesz 's theorem, but ut doesn't work.
pde sobolev-spaces stokes-theorem
asked Mar 23 at 17:22
AnasAnas
395
$endgroup$
add a comment |
$begingroup$
[I was working on Stokes pde, and I'm stuck in proving this, couldn't find it anywhere]
Let $Omega in mathbbR^N$ an open bounded connected set such as $partial Omega$ is $mathcalC^1$.
And we define $$V=vec v in (H^1_0(Omega))^N; textdiv vec v=0$$
With $textdiv vec v=sum_n=1^Nfracpartial v_ipartial x_i$.
I'm stuck in proving : if $L$ is a linear continious form on $(H_0^1(Omega))^N$ such as $L_V=0$ then there exists a function $phi in L^2(Omega)$, such as $$forall vec vin (H_0^1(Omega))^N : L(vec v)=int_Omegaphi ;textdiv vec v ;dx$$
My idea : I proved that $V$ is a closed space of $(H_0^1(Omega))^N$ then its a Hilbert space, and I tried to apply Riesz 's theorem, but ut doesn't work.
pde sobolev-spaces stokes-theorem
asked Mar 23 at 17:22
AnasAnas
395
$endgroup$
$begingroup$
$L$ is a function of $textdiv v$ since it maps those $v$ with the same div to the same value. Now the space of all $textdiv v$ is a subspace $Esubset L^2$, and $L$ is linear and bounded on $E$, so by Hahn-Banach then by Riesz there is $phiin L^2$ so that for all $uin E$ we have $L(u)=int phi u$, note $u=textdiv v$.
$endgroup$
– Yu Ding
Mar 23 at 22:34
$begingroup$
I get the idea, but I m not convinced, because saying that $L $ is a function of $div vec v $ (to me) does not make sense, because we suppose at beginning that $L$ is defined on $(H_0^1(Omega))^N$(note that this is a product space), and $div vec v$ is not a vector!
$endgroup$
– Anas
Mar 24 at 7:42
$begingroup$
Actually I mean there is $F$ so that $L(v)=F(textdiv v)$, and $F$ is linear, bounded on $L^2$, etc. as argued above......
$endgroup$
– Yu Ding
Mar 24 at 8:07
1
$begingroup$
Are you aware that you are asking about the de Rham lemma? You can find it in most literature on the mathematical theory of the (Navier-)Stokes equations, e.g. this book.
$endgroup$
– Fritz
Mar 24 at 8:19
$begingroup$
I Didn't know, I will check the book, thank you
$endgroup$
– Anas
Mar 24 at 10:54
add a comment |
$begingroup$
[I was working on Stokes pde, and I'm stuck in proving this, couldn't find it anywhere]
Let $Omega in mathbbR^N$ an open bounded connected set such as $partial Omega$ is $mathcalC^1$.
And we define $$V=vec v in (H^1_0(Omega))^N; textdiv vec v=0$$
With $textdiv vec v=sum_n=1^Nfracpartial v_ipartial x_i$.
I'm stuck in proving : if $L$ is a linear continious form on $(H_0^1(Omega))^N$ such as $L_V=0$ then there exists a function $phi in L^2(Omega)$, such as $$forall vec vin (H_0^1(Omega))^N : L(vec v)=int_Omegaphi ;textdiv vec v ;dx$$
My idea : I proved that $V$ is a closed space of $(H_0^1(Omega))^N$ then its a Hilbert space, and I tried to apply Riesz 's theorem, but ut doesn't work.
pde sobolev-spaces stokes-theorem
asked Mar 23 at 17:22
AnasAnas
395
$endgroup$
[I was working on Stokes pde, and I'm stuck in proving this, couldn't find it anywhere]
Let $Omega in mathbbR^N$ an open bounded connected set such as $partial Omega$ is $mathcalC^1$.
And we define $$V=vec v in (H^1_0(Omega))^N; textdiv vec v=0$$
With $textdiv vec v=sum_n=1^Nfracpartial v_ipartial x_i$.
I'm stuck in proving : if $L$ is a linear continious form on $(H_0^1(Omega))^N$ such as $L_V=0$ then there exists a function $phi in L^2(Omega)$, such as $$forall vec vin (H_0^1(Omega))^N : L(vec v)=int_Omegaphi ;textdiv vec v ;dx$$
My idea : I proved that $V$ is a closed space of $(H_0^1(Omega))^N$ then its a Hilbert space, and I tried to apply Riesz 's theorem, but ut doesn't work.
pde sobolev-spaces stokes-theorem
pde sobolev-spaces stokes-theorem
asked Mar 23 at 17:22
AnasAnas
395
asked Mar 23 at 17:22
AnasAnas
395
edited Mar 23 at 17:30
Anas
asked Mar 23 at 17:22
AnasAnas
395
asked Mar 23 at 17:22
AnasAnas
395
asked Mar 23 at 17:22
AnasAnas
395
395
$begingroup$
$L$ is a function of $textdiv v$ since it maps those $v$ with the same div to the same value. Now the space of all $textdiv v$ is a subspace $Esubset L^2$, and $L$ is linear and bounded on $E$, so by Hahn-Banach then by Riesz there is $phiin L^2$ so that for all $uin E$ we have $L(u)=int phi u$, note $u=textdiv v$.
$endgroup$
– Yu Ding
Mar 23 at 22:34
$begingroup$
I get the idea, but I m not convinced, because saying that $L $ is a function of $div vec v $ (to me) does not make sense, because we suppose at beginning that $L$ is defined on $(H_0^1(Omega))^N$(note that this is a product space), and $div vec v$ is not a vector!
$endgroup$
– Anas
Mar 24 at 7:42
$begingroup$
Actually I mean there is $F$ so that $L(v)=F(textdiv v)$, and $F$ is linear, bounded on $L^2$, etc. as argued above......
$endgroup$
– Yu Ding
Mar 24 at 8:07
1
$begingroup$
Are you aware that you are asking about the de Rham lemma? You can find it in most literature on the mathematical theory of the (Navier-)Stokes equations, e.g. this book.
$endgroup$
– Fritz
Mar 24 at 8:19
$begingroup$
I Didn't know, I will check the book, thank you
$endgroup$
– Anas
Mar 24 at 10:54
add a comment |
$begingroup$
$L$ is a function of $textdiv v$ since it maps those $v$ with the same div to the same value. Now the space of all $textdiv v$ is a subspace $Esubset L^2$, and $L$ is linear and bounded on $E$, so by Hahn-Banach then by Riesz there is $phiin L^2$ so that for all $uin E$ we have $L(u)=int phi u$, note $u=textdiv v$.
$endgroup$
– Yu Ding
Mar 23 at 22:34
$begingroup$
I get the idea, but I m not convinced, because saying that $L $ is a function of $div vec v $ (to me) does not make sense, because we suppose at beginning that $L$ is defined on $(H_0^1(Omega))^N$(note that this is a product space), and $div vec v$ is not a vector!
$endgroup$
– Anas
Mar 24 at 7:42
$begingroup$
Actually I mean there is $F$ so that $L(v)=F(textdiv v)$, and $F$ is linear, bounded on $L^2$, etc. as argued above......
$endgroup$
– Yu Ding
Mar 24 at 8:07
1
$begingroup$
Are you aware that you are asking about the de Rham lemma? You can find it in most literature on the mathematical theory of the (Navier-)Stokes equations, e.g. this book.
$endgroup$
– Fritz
Mar 24 at 8:19
$begingroup$
I Didn't know, I will check the book, thank you
$endgroup$
– Anas
Mar 24 at 10:54
$begingroup$
$L$ is a function of $textdiv v$ since it maps those $v$ with the same div to the same value. Now the space of all $textdiv v$ is a subspace $Esubset L^2$, and $L$ is linear and bounded on $E$, so by Hahn-Banach then by Riesz there is $phiin L^2$ so that for all $uin E$ we have $L(u)=int phi u$, note $u=textdiv v$.
$endgroup$
– Yu Ding
Mar 23 at 22:34
$begingroup$
$L$ is a function of $textdiv v$ since it maps those $v$ with the same div to the same value. Now the space of all $textdiv v$ is a subspace $Esubset L^2$, and $L$ is linear and bounded on $E$, so by Hahn-Banach then by Riesz there is $phiin L^2$ so that for all $uin E$ we have $L(u)=int phi u$, note $u=textdiv v$.
$endgroup$
– Yu Ding
Mar 23 at 22:34
$begingroup$
I get the idea, but I m not convinced, because saying that $L $ is a function of $div vec v $ (to me) does not make sense, because we suppose at beginning that $L$ is defined on $(H_0^1(Omega))^N$(note that this is a product space), and $div vec v$ is not a vector!
$endgroup$
– Anas
Mar 24 at 7:42
$begingroup$
I get the idea, but I m not convinced, because saying that $L $ is a function of $div vec v $ (to me) does not make sense, because we suppose at beginning that $L$ is defined on $(H_0^1(Omega))^N$(note that this is a product space), and $div vec v$ is not a vector!
$endgroup$
– Anas
Mar 24 at 7:42
$begingroup$
Actually I mean there is $F$ so that $L(v)=F(textdiv v)$, and $F$ is linear, bounded on $L^2$, etc. as argued above......
$endgroup$
– Yu Ding
Mar 24 at 8:07
$begingroup$
Actually I mean there is $F$ so that $L(v)=F(textdiv v)$, and $F$ is linear, bounded on $L^2$, etc. as argued above......
$endgroup$
– Yu Ding
Mar 24 at 8:07
1
1
$begingroup$
Are you aware that you are asking about the de Rham lemma? You can find it in most literature on the mathematical theory of the (Navier-)Stokes equations, e.g. this book.
$endgroup$
– Fritz
Mar 24 at 8:19
$begingroup$
Are you aware that you are asking about the de Rham lemma? You can find it in most literature on the mathematical theory of the (Navier-)Stokes equations, e.g. this book.
$endgroup$
– Fritz
Mar 24 at 8:19
$begingroup$
I Didn't know, I will check the book, thank you
$endgroup$
– Anas
Mar 24 at 10:54
$begingroup$
I Didn't know, I will check the book, thank you
$endgroup$
– Anas
Mar 24 at 10:54
add a comment |
0
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oldest
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$begingroup$
$L$ is a function of $textdiv v$ since it maps those $v$ with the same div to the same value. Now the space of all $textdiv v$ is a subspace $Esubset L^2$, and $L$ is linear and bounded on $E$, so by Hahn-Banach then by Riesz there is $phiin L^2$ so that for all $uin E$ we have $L(u)=int phi u$, note $u=textdiv v$.
$endgroup$
– Yu Ding
Mar 23 at 22:34
$begingroup$
I get the idea, but I m not convinced, because saying that $L $ is a function of $div vec v $ (to me) does not make sense, because we suppose at beginning that $L$ is defined on $(H_0^1(Omega))^N$(note that this is a product space), and $div vec v$ is not a vector!
$endgroup$
– Anas
Mar 24 at 7:42
$begingroup$
Actually I mean there is $F$ so that $L(v)=F(textdiv v)$, and $F$ is linear, bounded on $L^2$, etc. as argued above......
$endgroup$
– Yu Ding
Mar 24 at 8:07
1
$begingroup$
Are you aware that you are asking about the de Rham lemma? You can find it in most literature on the mathematical theory of the (Navier-)Stokes equations, e.g. this book.
$endgroup$
– Fritz
Mar 24 at 8:19
$begingroup$
I Didn't know, I will check the book, thank you
$endgroup$
– Anas
Mar 24 at 10:54