Smoothness of Autonomous ODE Solutions

Lee Smooth Manifolds

Theorem) Let $U\subset {\mathbb{R}}^n$ be open and $V:U\to {\mathbb{R}}^n$ be locally Lipschitz continuous. Suppose $U_0\subset U$ is an open subset and $J_0\subset \mathbb{R}$ is an open interval containing $t_0$, and $\theta :J_0\times U_0\to U$ is a map such that for each $x\in U_0$, $y(t)=\theta (t,x)$ ($y:J_0\to U$) solves the initial value problem

$$\dot{y}(t)=V(y(t)),y(t_0)=x$$

If $V$ is $C^k$ for some $k\ge 0$, then so is $\theta$.

Proof) The proof is, as expected, inductive. It suffices to show that for any $(t_1,x_1)\in J_0\times U_0$, there exists some neighborhood of $(t_1,x_1)$ in which $\theta$ is $C^k$. This neighborhood will take the form of $\overline{J}\times \overline{U}$ where $J$ is an open interval containing $t_0$ and $t_1$ in the time dimension and $U$ is a neighborhood of $x_1$. Thus we are working with a “tube” in the time-state space.

We first show that $\theta$ is $C^0$. Let $J_1$ be a bounded open interval containing $t_0$ and $t_1$ such that $\overline{J_1}\subset J_0$. Clearly $K=\theta (\overline{J_1}\times \{x_1\})$ is a compact subset of $U$; there is a $c>0$ such that for any $k\in K$, $B_{2c}(k)\subset U$. Consider the precompact subset

$$W=\bigcup _{k\in K}{B}_c(k)\subset U$$

As $\overline{W}$ is compact we can take a Lipschitz constant $C>0$ of $V$ on $\overline{W}$ and $M={\sup }_{\overline{W}}|V|$. We additionally take the constant $T={\sup }_{t\in \overline{J_1}}|t-t_0|$. As $V$ is a Lipschitz field in $\overline{W}$, if we take $x,\tilde{x}\in W\cap U_0$ and the trajectories $\theta (J_1\times \{x\})$, $\theta (J_1\times \{\tilde{x}\})$ stay in $W$, we have that

$$|\theta (t,x)-\theta (t,\tilde{x})|\le e^{CT}|x-\tilde{x}|$$

for any $t\in J_1$ (by continuity, it actually holds for $t\in \overline{J_1}$). Take some $R>0$ such that $R{e}^{CT}<c$; we claim $\theta$ must then map $\overline{J_1}\times \overline{B_R(x_1)}$ into $W$. Assume there is some $(t_2,x_2)\in \overline{J_1}\times \overline{B_R(x_1)}$ such that $\theta (t_2,x_2)\notin W$. WLOG, assume $t_2>t_0$ ($t_2\ne t_0$ as $x_2\in \overline{B_R(x_1)}\subset W$) and let $\tau =\inf \{t\in [t_0,t_2]|\theta (t,x_2)\notin W\}$. Then, $\theta (\tau ,x_2)\in \partial W$ which is impossible as $\theta (t,x_2)\in W$ for $t\in [t_0,\tau )$, forcing

$$|\theta (\tau ,x_2)-\theta (\tau ,x_1)|\le e^{C(\tau -t_0)}|x_2-x_1|\le R{e}^{CT}<c$$

which is a contradiction as $B_c(\theta (\tau ,x_1))\subset W$. We now have the sufficient tools to show that $\theta$ is continuous at $(t_1,x_1)$. Clearly $\theta$ is continuous along the time and space coordinates separately in the neighborhood $J_1\times U_1$ of $(t_1,x_1)$; it is our task to adjoin them into one space. For any $(t,x)\in J_1\times B_R(x_1)$,

$$|\theta (t_1,x_1)-\theta (t,x)|\le |x_1-x|+|{\int }_{t_0}^{t_1}V(\theta (s,x_1))ds-{\int }_{t_0}^{t}V(\theta (s,x))ds|$$ $$\le |x_1-x|+|{\int }_{t_0}^t(V(\theta (s,x_1))-V(\theta (s,x)))ds|+|{\int }_t^{t_1}V(\theta (s,x_1))ds|$$ $$\le |x_1-x|+|{\int }_{t_0}^{t}C|\theta (s,x_1)-\theta (s,x)|ds|+|{\int }_t^{t_1}Mds|\le |x_1-x|+CR{e}^{CT}|x-x_1|+M|t-t_1|$$

Thus, $\theta$ is $C^0$.

We now take $r$ such that $R=2r$ and, under the hypothesis that $V$ is $C^1$, prove that $\theta$ is $C^1$ in $\overline{J_1}\times \overline{B_r(x_1)}$. Note that, as $\partial \theta /\partial t=V(\theta )$ and $\theta$ is $C^0$, $\partial \theta /\partial t$ exists and is guaranteed to be continuous. It does suffices to show the existence and continuity of $\partial \theta /\partial x$. We do so by defining, for real numbers $0<|h|<r$, the quotient $({\Delta }_h{)}_j^i:\overline{J_1}\times \overline{B_r(x_1)}\to \mathbb{R}$ such that

$$({\Delta }_h{)}_j^i(t,x)=\frac{{\theta }^i(t,x+h{e}_j)-{\theta }^i(t,x)}{h}$$

If we can show that the (clearly continuous) matrix function ${\Delta }_h$ uniformly converges under the Frobenius norm as $h\to 0$, this is sufficient to show $\theta$ is $C^1$ (note there is a mild but harmless abuse of the concept of uniform convergence; we actually mean taking a subsequence $h_k$ such that $h_k\to 0$). We have the initial observation that, as all relevant trajectories stay in $W$, $|{\Delta }_h|\le n{e}^{CT}$.

Note that for any $h$ and $\tilde{h}$, for any $x\in \overline{B_r(x_1)}=U_1$,

$$({\Delta }_h{)}_j^i(t_0,x)={\delta }_j^i=({\Delta }_{\tilde{h}}{)}_j^i(t_0,x)$$

Where $\delta$ is the Kronecker delta. To obtain a bound for $|{\Delta }_h-{\Delta }_{\tilde{h}}|$, therefore, a good starting point is time-integrating from the trivial $(t_0,x)$ case. Computing $\partial ({\Delta }_h{)}_j^i/\partial t$ gives

$$\frac{\partial ({\Delta }_h{)}_j^i}{\partial t}(t,x)=\frac{1}{h}(\frac{\partial }{\partial t}{\theta }^i(t,x+h{e}_j)-\frac{\partial }{\partial t}{\theta }^i(t,x))=\frac{1}{h}(V^i(\theta (t,x+h{e}_j))-V^i(\theta (t,x)))$$ $$=\frac{1}{h}(\theta (t,x+h{e}_j)-\theta (t,x))\cdot \frac{d{V}^i}{dy}(y_0)=({\Delta }_h{)}_j\cdot \frac{d{V}^i}{dy}(y_0)$$

where the last line follows from an application of MVT, where $y_0$ is a point on the segment connecting $\theta (t,x+h{e}_j)$ and $\theta (t,x)$ in ${\mathbb{R}}^n$. Thus, for any sufficiently small $h,\tilde{h}$,

$$\frac{\partial }{\partial t}(({\Delta }_h{)}_j^i(t,x)-({\Delta }_{\tilde{h}}{)}_j^i(t,x))=({\Delta }_h{)}_j\cdot \frac{d{V}^i}{dy}(y_0)-({\Delta }_{\tilde{h}}{)}_j\cdot \frac{d{V}^i}{dy}({\tilde{y}}_0)$$ $$=({\Delta }_h{)}_j\cdot (\frac{d{V}^i}{dy}(y_0)-\frac{d{V}^i}{dy}({\tilde{y}}_0))+({\Delta }_{\tilde{h}}-{\Delta }_h{)}_j\cdot \frac{d{V}^i}{dy}({\tilde{y}}_0))$$

From here is a standard bounding argument. We assume $h$ and $\tilde{h}$ are sufficiently small such that $y_0$ and $\widetilde{y_0}$ stay inside $W$. Note that, by the same argument as given before, $\theta$ maps $\overline{J_1}\times \overline{U_1}$ into $W$. Let $E$ be the supremum of $|DV|$ on $\overline{B_R(x_1)}$ and set any $ϵ>0$. As $V$ is $C^1$, there is a $\delta$ such that $|(d{V}^i/dy)(y_1)-(d{V}^i/dy)(y_2)|<ϵ$ for any $y_1,y_2\in \overline{W}$ such that $d(y_1,y_2)<\delta$. If $|h|$ and $|\tilde{h}|$ are both less than $\delta e^{-CT}$, $|y_0-\theta (t,x)|\le |\theta (t,x+h{e}_j)-\theta (t,x)|\le |h||{\Delta }_j(t,x)|<\delta$, and analogously $|y_1-\theta (t,x)|<\delta$; this implies that

$$|\frac{d{V}^i}{dy}(y_0)-\frac{d{V}^i}{dy}(\widetilde{y_0})|<2ϵ$$

This gives us the final bound

$$\frac{\partial }{\partial t}(({\Delta }_h{)}_j^i(t,x)-({\Delta }_{\tilde{h}}{)}_j^i(t,x))\le 2ϵn{e}^{CT}+E|{\Delta }_{\tilde{h}}-{\Delta }_h|$$

Using the comparison theorem gives

$$|{\Delta }_h(t,x)-{\Delta }_{\tilde{h}}(t,x)|\le \frac{2ϵn{e}^{CT}}{E}(e^{ET}-1)$$

as $ϵ$ was arbitrary, we are done.

Now assume $V$ is $C^{k+1}$ and that $\theta$ is $C^k$; it suffices to show $\theta$ is $C^{k+1}$. $\theta$ must satisfy the equation

$$\frac{\partial {\theta }^i}{\partial x^j}(t,x)={\delta }_j^i+\sum _{k=1}^n{\int }_{t_0}^t\frac{\partial V^i}{\partial y^k}(\theta (s,x))\frac{\partial {\theta }^k}{\partial x^j}(s,x)ds$$

which implies

$$\frac{\partial }{\partial t}\frac{\partial {\theta }^i}{\partial x^j}(t,x)=\sum _{k=1}^n\frac{\partial V^i}{\partial y^k}(\theta (t,x))\frac{\partial {\theta }^k}{\partial x^j}(t,x)ds$$

The $n^2+n$ functions ${\alpha }^i(t)={\theta }^i(t,x)$ and ${\beta }_i^j(t)=(\partial {\theta }^i/\partial x^j)(t,x)$ are thus solutions to the IVP

$${\dot{\alpha }}_i(t)=V^i(\alpha (t)),{\dot{\beta }}_i(t)=\sum _{k=1}^n\frac{\partial V^i}{\partial y^k}(\alpha (t)){\beta }_j^k(t),{\alpha }^i(t_0)=x^i,{\beta }_j^i(t_0)=(\partial {\theta }^i/\partial x^j)(t,x)$$

Existence and uniqueness, along with our inductive hypothesis, force $\alpha$ and $\beta$ to be $C^k$, completing our proof.