description of why local solution existence does not guarantee global existence for Euclidean-normed Euclidean vectors space ODE
Topics
About: normed vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of Euclidean-normed Euclidean vectors space.
- The reader admits the local unique solution existence for a closed interval domain for a Euclidean-normed Euclidean vectors space ordinary differential equation with an initial condition with a clarification on the solution domain area.
Target Context
- The reader will understand why the local solution existence for Euclidean-normed Euclidean vectors space ordinary differential equation does not guarantee the global solution existence for the entire domain interval.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(\mathbb{R}^d\): \(= \text{ the Euclidean-normed Euclidean vectors space }\)
\(\mathbb{R}\): \(= \text{ the Euclidean-normed Euclidean vectors space }\)
\([t_1, t_e]\): \(\subseteq \mathbb{R}\)
\(f\): \(: \mathbb{R}^d \times [t_1, t_e] \to \mathbb{R}^d\), \(\in \{\text{ the } C^0 \text{ maps }\}\)
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Statements:
\(\forall t_0 \in [t_1, t_e], \forall x_0 \in \mathbb{R}^d (\exists [t_0 - \epsilon_{t_0, x_0, 1}, t_0 + \epsilon_{t_0, x_0, 2}] \subseteq [t_1, t_e], \exists B_{x_0, K_{t_0, x_0}} \subseteq \mathbb{R}^d, \exists M_{t_0, x_0} \in \mathbb{R} (\forall x_1, x_2 \in B_{x_0, K_{t_0, x_0}}, \forall r \in [t_0 - \epsilon_{t_0, x_0, 1}, t_0 + \epsilon_{t_0, x_0, 2}] (\Vert f (x_1, r) - f (x_2, r) \Vert \le L_{t_0, x_0} \Vert x_1 - x_2 \Vert) \land Sup (\{\Vert f (x, r) \Vert \vert x \in B_{x_0, K_{t_0, x_0}}, r \in [t_0 - \epsilon_{t_0, x_0, 1}, t_0 + \epsilon_{t_0, x_0, 2}]\}) \le M_{t_0, x_0}))\)
\(\lnot \implies\)
\(\exists x: [t_1, t_e] \to \mathbb{R}^d (d x / d t = f (x, t))\)
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2: Proof
Let us get the local solution, \(x: [t_1, t_1 + \epsilon_{t_1, x_1, 2}] \to \mathbb{R}^d\), by the initial condition, \(x (t_1) = x_1\), which is possible, by the local unique solution existence for a closed interval domain for a Euclidean-normed Euclidean vectors space ordinary differential equation with an initial condition with a clarification on the solution domain area.
Then, let us get the connected next local solution by the initial condition at \(t_2 = t_1 + \epsilon_{t_1, x_1, 2}\) with the value of \(x (t_2)\), getting the extended solution, \(x: [t_1, t_2 + \epsilon_{t_2, x (t_2), 2}]\), and so on.
But the issue is that there is no guarantee that \(t_j + \epsilon_{t_j, x (t_j), 2}\) eventually reaches \(t_e\), because it may converge to a value, \(t_l \lt t_e\), failing to extend the solution to the entire \([t_1, t_e]\).
But, why do we not get a local solution by an initial condition at the limit point \(t_l\) having an interval \([t_l - \epsilon_{t_l, x_{t_l}, 1}, t_l + \epsilon_{t_l, x_{t_l}, 2}]\)?
But what is really \(x_{t_l}\)?
As \(x\) has not been extended to \(t_l\), it cannot be chosen to be \(x (t_l)\), and there is no guarantee that there is an \(x_{t_l}\) that makes the local solution coincide with the so-far-extended solution on the so-far-extended domain (note that the local existence guarantees that any initial value at \(t_l\) can be realized, not that any value at another point can be realized).
3: Note
A sufficient condition to guarantee the global solution existence is the bijective-ness of the map, \(x (t_j) \mapsto x (t_k)\), between the set of possible values at \(t_j\) and the set of possible values at \(t_k\) where \(t_j\) and \(t_k\) are any points in \([t_1, t_e]\), which guarantees the \([t_l - \epsilon_{t_l, x_l, 1}, t_l + \epsilon_{t_l, x_l, 2}]\) local solution to be able to be connected to the not-fully-extended solution.
