Regularity of a hyper-surface defined through a flow
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?
differential-equations differential-geometry dynamical-systems vector-fields
asked Dec 3 '18 at 14:47
Paolo IntuitoPaolo Intuito
988318
add a comment |
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?
differential-equations differential-geometry dynamical-systems vector-fields
asked Dec 3 '18 at 14:47
Paolo IntuitoPaolo Intuito
988318
add a comment |
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?
differential-equations differential-geometry dynamical-systems vector-fields
asked Dec 3 '18 at 14:47
Paolo IntuitoPaolo Intuito
988318
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?
differential-equations differential-geometry dynamical-systems vector-fields
differential-equations differential-geometry dynamical-systems vector-fields
asked Dec 3 '18 at 14:47
Paolo IntuitoPaolo Intuito
988318
asked Dec 3 '18 at 14:47
Paolo IntuitoPaolo Intuito
988318
edited Dec 3 '18 at 15:13
Paolo Intuito
asked Dec 3 '18 at 14:47
Paolo IntuitoPaolo Intuito
988318
asked Dec 3 '18 at 14:47
Paolo IntuitoPaolo Intuito
988318
asked Dec 3 '18 at 14:47
Paolo IntuitoPaolo Intuito
988318
988318
add a comment |
add a comment |
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