Regularity of a hyper-surface defined through a flow
$begingroup$
Let $X:mathbb{R}^n to mathbb{R}^n$ be a smooth and bounded vector field, such that
$$
X_n ge c|(X_1, dots, X_{n-1})| ge epsilon > 0,.
$$
Under these assumptions one can prove that integral curves $Phi(cdot, x)$ exist for all times (and are unique) and they intersect every hyperplane of fixed $n$-th coordinate exactly once.
Thus, fixed the hyperplane $H_{h} := {x,:,x_n = h}$, we have that the flow $Phi : mathbb{R} times H_h to mathbb{R}^n$ is a bijection (I "foliate" $mathbb{R}^n$ with Lipschitz curves).
Fix a bounded smooth set $A subset H_h$ (wlog a ball of $mathbb{R}^{n-1}$). I want to look at this set as it gets transformed by the flow $Phi$ generated by $X$. More precisely, fix another hyperplane $H_k$, with $k<h$, and define the set
$$
C:= {x,:, kle x_n le h}cap bigcup_{xin A} Phi([-T,T],x) ,,
$$
where $T:= inf_t{Phi([-t,t],x) cap H_k neq emptyset,, forall xin A }$. Under our working hypotheses this guy is well defined since all integral curves reach $H_k$ in finite time.
Is it true that the "lateral surface" of this set is regular (define $partial C$ in the same way as $C$ by taking the union over $xin partial A$ rather than in $A$)? Can I say it is diffeomorphic to a cylinder?
How do I go about proving it?
ordinary-differential-equations differential-geometry dynamical-systems vector-fields
asked Dec 3 '18 at 14:47
Paolo IntuitoPaolo Intuito
953318
$endgroup$
add a comment |
$begingroup$
Let $X:mathbb{R}^n to mathbb{R}^n$ be a smooth and bounded vector field, such that
$$
X_n ge c|(X_1, dots, X_{n-1})| ge epsilon > 0,.
$$
Under these assumptions one can prove that integral curves $Phi(cdot, x)$ exist for all times (and are unique) and they intersect every hyperplane of fixed $n$-th coordinate exactly once.
Thus, fixed the hyperplane $H_{h} := {x,:,x_n = h}$, we have that the flow $Phi : mathbb{R} times H_h to mathbb{R}^n$ is a bijection (I "foliate" $mathbb{R}^n$ with Lipschitz curves).
Fix a bounded smooth set $A subset H_h$ (wlog a ball of $mathbb{R}^{n-1}$). I want to look at this set as it gets transformed by the flow $Phi$ generated by $X$. More precisely, fix another hyperplane $H_k$, with $k<h$, and define the set
$$
C:= {x,:, kle x_n le h}cap bigcup_{xin A} Phi([-T,T],x) ,,
$$
where $T:= inf_t{Phi([-t,t],x) cap H_k neq emptyset,, forall xin A }$. Under our working hypotheses this guy is well defined since all integral curves reach $H_k$ in finite time.
Is it true that the "lateral surface" of this set is regular (define $partial C$ in the same way as $C$ by taking the union over $xin partial A$ rather than in $A$)? Can I say it is diffeomorphic to a cylinder?
How do I go about proving it?
ordinary-differential-equations differential-geometry dynamical-systems vector-fields
asked Dec 3 '18 at 14:47
Paolo IntuitoPaolo Intuito
953318
$endgroup$
add a comment |
$begingroup$
Let $X:mathbb{R}^n to mathbb{R}^n$ be a smooth and bounded vector field, such that
$$
X_n ge c|(X_1, dots, X_{n-1})| ge epsilon > 0,.
$$
Under these assumptions one can prove that integral curves $Phi(cdot, x)$ exist for all times (and are unique) and they intersect every hyperplane of fixed $n$-th coordinate exactly once.
Thus, fixed the hyperplane $H_{h} := {x,:,x_n = h}$, we have that the flow $Phi : mathbb{R} times H_h to mathbb{R}^n$ is a bijection (I "foliate" $mathbb{R}^n$ with Lipschitz curves).
Fix a bounded smooth set $A subset H_h$ (wlog a ball of $mathbb{R}^{n-1}$). I want to look at this set as it gets transformed by the flow $Phi$ generated by $X$. More precisely, fix another hyperplane $H_k$, with $k<h$, and define the set
$$
C:= {x,:, kle x_n le h}cap bigcup_{xin A} Phi([-T,T],x) ,,
$$
where $T:= inf_t{Phi([-t,t],x) cap H_k neq emptyset,, forall xin A }$. Under our working hypotheses this guy is well defined since all integral curves reach $H_k$ in finite time.
Is it true that the "lateral surface" of this set is regular (define $partial C$ in the same way as $C$ by taking the union over $xin partial A$ rather than in $A$)? Can I say it is diffeomorphic to a cylinder?
How do I go about proving it?
ordinary-differential-equations differential-geometry dynamical-systems vector-fields
asked Dec 3 '18 at 14:47
Paolo IntuitoPaolo Intuito
953318
$endgroup$
Let $X:mathbb{R}^n to mathbb{R}^n$ be a smooth and bounded vector field, such that
$$
X_n ge c|(X_1, dots, X_{n-1})| ge epsilon > 0,.
$$
Under these assumptions one can prove that integral curves $Phi(cdot, x)$ exist for all times (and are unique) and they intersect every hyperplane of fixed $n$-th coordinate exactly once.
Thus, fixed the hyperplane $H_{h} := {x,:,x_n = h}$, we have that the flow $Phi : mathbb{R} times H_h to mathbb{R}^n$ is a bijection (I "foliate" $mathbb{R}^n$ with Lipschitz curves).
Fix a bounded smooth set $A subset H_h$ (wlog a ball of $mathbb{R}^{n-1}$). I want to look at this set as it gets transformed by the flow $Phi$ generated by $X$. More precisely, fix another hyperplane $H_k$, with $k<h$, and define the set
$$
C:= {x,:, kle x_n le h}cap bigcup_{xin A} Phi([-T,T],x) ,,
$$
where $T:= inf_t{Phi([-t,t],x) cap H_k neq emptyset,, forall xin A }$. Under our working hypotheses this guy is well defined since all integral curves reach $H_k$ in finite time.
Is it true that the "lateral surface" of this set is regular (define $partial C$ in the same way as $C$ by taking the union over $xin partial A$ rather than in $A$)? Can I say it is diffeomorphic to a cylinder?
How do I go about proving it?
ordinary-differential-equations differential-geometry dynamical-systems vector-fields
ordinary-differential-equations differential-geometry dynamical-systems vector-fields
asked Dec 3 '18 at 14:47
Paolo IntuitoPaolo Intuito
953318
asked Dec 3 '18 at 14:47
Paolo IntuitoPaolo Intuito
953318
edited Jan 17 at 10:47
Paolo Intuito
asked Dec 3 '18 at 14:47
Paolo IntuitoPaolo Intuito
953318
asked Dec 3 '18 at 14:47
Paolo IntuitoPaolo Intuito
953318
asked Dec 3 '18 at 14:47
Paolo IntuitoPaolo Intuito
953318
953318
add a comment |
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
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
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3024153%2fregularity-of-a-hyper-surface-defined-through-a-flow%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
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
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3024153%2fregularity-of-a-hyper-surface-defined-through-a-flow%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
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
Required, but never shown
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
Required, but never shown
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
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
