Weak analogues of gradient, divergence, and curl (collecting examples)
This question is mostly to help me understand the idea behind the "weak curl", but I also hope to accomplish other objectives with this question/post as well, partially inspired from some of the "proof collecting" posts I've seen.
$1)$ I want to better understand the notion of "weak curl" with some examples.
$2)$ To hopefully discuss theorems surrounding when the weak versions of gradient, divergence, and curl (if possible), are equal to their strong counterparts and what this implies for solutions for PDEs.
$3)$ Collect illustrative examples of weak gradient, weak curl, and weak divergence in any number of dimensions or subsets. Maybe we can consider compact vs. non-compact subsets, upper/lower bounds on these quantities, disconnected spaces, and any related topic of interest.
WEAK GRADIENT:
let $Omegasubset mathbb{R}^n$, and let $uin L^1_{loc}(Omega)$ and $phiin C^{infty}_c(Omega)$. The function $v$ is called the "weak gradient" of $u$ if $int_{Omega}uphi' dmu=-int_{Omega}vphi dmu$. The "a-th" weak gradient is just
$int_{Omega}uD^{a}phi dmu=-(1)^aint_{Omega}vphi dmu$ $forall phi in C^{infty}_c(Omega)$
WEAK DIVERGENCE:
v is called the "weak divergence" for $uin L^2(Omega)$ if we have
$int_{Omega}uphi dmu=-int_{Omega}langle v, nabla phi rangle$ $forall phi in C^{infty}_c(Omega)$
EDIT: I also realized I need clarification on the notation $int_{Omega}(v,nabla phi)$. I think this means we integrate w.r.t. each vector component of $nabla phi$, so for $mathbb{R}^2$ we have $int_{mathbb{R}}int_{mathbb{R}}|v_1 nabla phi_1 v_2 nabla phi_2|^2 dmu dlambda$.
WEAK CURL:
This is a bit more delicate, and I am only aware of the definition where $Omegasubset mathbb{R}^3$. First, we need to define $u:mathbb{R}^2rightarrow mathbb{R}^2$ with $u=(u_0, u_1)$ where $u_0in L^2(Omega)$, and $u_1in L^2(partial Omega)$. Let $n$ be a normal vector to the boundary $partial Omega$.
$v$ is called the "weak curl" if we have:
$v=curl(u)=langle u_0, (nabla times phi) rangle + langle u_1 times n, phi rangle $ $forall phi in C^{infty}_c(Omega)$ where the inner product here is the standard $L^2$ inner product.
Any interesting remarks/theorems are also welcome.
real-analysis vector-fields
asked Nov 28 '18 at 20:13
Kernel_Dirichlet
1,143416
add a comment |
This question is mostly to help me understand the idea behind the "weak curl", but I also hope to accomplish other objectives with this question/post as well, partially inspired from some of the "proof collecting" posts I've seen.
$1)$ I want to better understand the notion of "weak curl" with some examples.
$2)$ To hopefully discuss theorems surrounding when the weak versions of gradient, divergence, and curl (if possible), are equal to their strong counterparts and what this implies for solutions for PDEs.
$3)$ Collect illustrative examples of weak gradient, weak curl, and weak divergence in any number of dimensions or subsets. Maybe we can consider compact vs. non-compact subsets, upper/lower bounds on these quantities, disconnected spaces, and any related topic of interest.
WEAK GRADIENT:
let $Omegasubset mathbb{R}^n$, and let $uin L^1_{loc}(Omega)$ and $phiin C^{infty}_c(Omega)$. The function $v$ is called the "weak gradient" of $u$ if $int_{Omega}uphi' dmu=-int_{Omega}vphi dmu$. The "a-th" weak gradient is just
$int_{Omega}uD^{a}phi dmu=-(1)^aint_{Omega}vphi dmu$ $forall phi in C^{infty}_c(Omega)$
WEAK DIVERGENCE:
v is called the "weak divergence" for $uin L^2(Omega)$ if we have
$int_{Omega}uphi dmu=-int_{Omega}langle v, nabla phi rangle$ $forall phi in C^{infty}_c(Omega)$
EDIT: I also realized I need clarification on the notation $int_{Omega}(v,nabla phi)$. I think this means we integrate w.r.t. each vector component of $nabla phi$, so for $mathbb{R}^2$ we have $int_{mathbb{R}}int_{mathbb{R}}|v_1 nabla phi_1 v_2 nabla phi_2|^2 dmu dlambda$.
WEAK CURL:
This is a bit more delicate, and I am only aware of the definition where $Omegasubset mathbb{R}^3$. First, we need to define $u:mathbb{R}^2rightarrow mathbb{R}^2$ with $u=(u_0, u_1)$ where $u_0in L^2(Omega)$, and $u_1in L^2(partial Omega)$. Let $n$ be a normal vector to the boundary $partial Omega$.
$v$ is called the "weak curl" if we have:
$v=curl(u)=langle u_0, (nabla times phi) rangle + langle u_1 times n, phi rangle $ $forall phi in C^{infty}_c(Omega)$ where the inner product here is the standard $L^2$ inner product.
Any interesting remarks/theorems are also welcome.
real-analysis vector-fields
asked Nov 28 '18 at 20:13
Kernel_Dirichlet
1,143416
add a comment |
This question is mostly to help me understand the idea behind the "weak curl", but I also hope to accomplish other objectives with this question/post as well, partially inspired from some of the "proof collecting" posts I've seen.
$1)$ I want to better understand the notion of "weak curl" with some examples.
$2)$ To hopefully discuss theorems surrounding when the weak versions of gradient, divergence, and curl (if possible), are equal to their strong counterparts and what this implies for solutions for PDEs.
$3)$ Collect illustrative examples of weak gradient, weak curl, and weak divergence in any number of dimensions or subsets. Maybe we can consider compact vs. non-compact subsets, upper/lower bounds on these quantities, disconnected spaces, and any related topic of interest.
WEAK GRADIENT:
let $Omegasubset mathbb{R}^n$, and let $uin L^1_{loc}(Omega)$ and $phiin C^{infty}_c(Omega)$. The function $v$ is called the "weak gradient" of $u$ if $int_{Omega}uphi' dmu=-int_{Omega}vphi dmu$. The "a-th" weak gradient is just
$int_{Omega}uD^{a}phi dmu=-(1)^aint_{Omega}vphi dmu$ $forall phi in C^{infty}_c(Omega)$
WEAK DIVERGENCE:
v is called the "weak divergence" for $uin L^2(Omega)$ if we have
$int_{Omega}uphi dmu=-int_{Omega}langle v, nabla phi rangle$ $forall phi in C^{infty}_c(Omega)$
EDIT: I also realized I need clarification on the notation $int_{Omega}(v,nabla phi)$. I think this means we integrate w.r.t. each vector component of $nabla phi$, so for $mathbb{R}^2$ we have $int_{mathbb{R}}int_{mathbb{R}}|v_1 nabla phi_1 v_2 nabla phi_2|^2 dmu dlambda$.
WEAK CURL:
This is a bit more delicate, and I am only aware of the definition where $Omegasubset mathbb{R}^3$. First, we need to define $u:mathbb{R}^2rightarrow mathbb{R}^2$ with $u=(u_0, u_1)$ where $u_0in L^2(Omega)$, and $u_1in L^2(partial Omega)$. Let $n$ be a normal vector to the boundary $partial Omega$.
$v$ is called the "weak curl" if we have:
$v=curl(u)=langle u_0, (nabla times phi) rangle + langle u_1 times n, phi rangle $ $forall phi in C^{infty}_c(Omega)$ where the inner product here is the standard $L^2$ inner product.
Any interesting remarks/theorems are also welcome.
real-analysis vector-fields
asked Nov 28 '18 at 20:13
Kernel_Dirichlet
1,143416
This question is mostly to help me understand the idea behind the "weak curl", but I also hope to accomplish other objectives with this question/post as well, partially inspired from some of the "proof collecting" posts I've seen.
$1)$ I want to better understand the notion of "weak curl" with some examples.
$2)$ To hopefully discuss theorems surrounding when the weak versions of gradient, divergence, and curl (if possible), are equal to their strong counterparts and what this implies for solutions for PDEs.
$3)$ Collect illustrative examples of weak gradient, weak curl, and weak divergence in any number of dimensions or subsets. Maybe we can consider compact vs. non-compact subsets, upper/lower bounds on these quantities, disconnected spaces, and any related topic of interest.
WEAK GRADIENT:
let $Omegasubset mathbb{R}^n$, and let $uin L^1_{loc}(Omega)$ and $phiin C^{infty}_c(Omega)$. The function $v$ is called the "weak gradient" of $u$ if $int_{Omega}uphi' dmu=-int_{Omega}vphi dmu$. The "a-th" weak gradient is just
$int_{Omega}uD^{a}phi dmu=-(1)^aint_{Omega}vphi dmu$ $forall phi in C^{infty}_c(Omega)$
WEAK DIVERGENCE:
v is called the "weak divergence" for $uin L^2(Omega)$ if we have
$int_{Omega}uphi dmu=-int_{Omega}langle v, nabla phi rangle$ $forall phi in C^{infty}_c(Omega)$
EDIT: I also realized I need clarification on the notation $int_{Omega}(v,nabla phi)$. I think this means we integrate w.r.t. each vector component of $nabla phi$, so for $mathbb{R}^2$ we have $int_{mathbb{R}}int_{mathbb{R}}|v_1 nabla phi_1 v_2 nabla phi_2|^2 dmu dlambda$.
WEAK CURL:
This is a bit more delicate, and I am only aware of the definition where $Omegasubset mathbb{R}^3$. First, we need to define $u:mathbb{R}^2rightarrow mathbb{R}^2$ with $u=(u_0, u_1)$ where $u_0in L^2(Omega)$, and $u_1in L^2(partial Omega)$. Let $n$ be a normal vector to the boundary $partial Omega$.
$v$ is called the "weak curl" if we have:
$v=curl(u)=langle u_0, (nabla times phi) rangle + langle u_1 times n, phi rangle $ $forall phi in C^{infty}_c(Omega)$ where the inner product here is the standard $L^2$ inner product.
Any interesting remarks/theorems are also welcome.
real-analysis vector-fields
real-analysis vector-fields
asked Nov 28 '18 at 20:13
Kernel_Dirichlet
1,143416
asked Nov 28 '18 at 20:13
Kernel_Dirichlet
1,143416
edited Dec 2 '18 at 1:46
asked Nov 28 '18 at 20:13
Kernel_Dirichlet
1,143416
asked Nov 28 '18 at 20:13
Kernel_Dirichlet
1,143416
asked Nov 28 '18 at 20:13
Kernel_Dirichlet
1,143416
1,143416
add a comment |
add a comment |
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