Basic Theory of ODE and Vector Fields

Abstract

This chapter examines basic topics in the field of ordinary differential equations (ODE), as it has developed from the era of Newton into modern times. This is closely tied to the development of a number of concepts in advanced calculus. We begin with a brief discussion of the derivative of a vector-valued function of several variables as a linear map.

A Nonsmooth vector fields

Here we establish properties of solutions to the ODE

$$\frac{dy} {dt} = F(t,y),\quad y({t}_{0}) = {x}_{0}$$

(A.1)

of a sort done in §§2–6, under weaker hypotheses than those used there; in particular, we do not require F to be Lipschitz in y. For existence, we can assume considerably less:

Proposition A.1.

Let ${x}_{0}\in\mathcal{O}$, an open subset of ℝⁿ, I ⊂ ℝ an interval containing t₀. Assume F is continuous on $I\times\mathcal{O}$. Then the (A.1) has a solution on some t-interval containing t₀.

Proof.

Without loss of generality, we can assume F is bounded and continuous on ℝ × ℝⁿ. Take F_j ∈ C^∞(ℝ × ℝⁿ) such that |F_j| ≤ K and F_j → F locally uniformly, and let y_j ∈ C^∞(ℝ) be the unique solution to

$$\frac{d{y}_{j}} {dt} = {F}_{j}(t,y),\quad {y}_{j}({t}_{0}) = {x}_{0}, $$

(A.2)

whose existence is guaranteed by the material of §2. Thus

$${y}_{j}(t) = {x}_{0} +{\int \nolimits\nolimits }_{{t}_{0}}^{t}{F}_{ j}{\bigl (s,{y}_{j}(s)\bigr)}\ ds.$$

(A.3)

Now

$$\vert {F}_{j}\vert\leq K\Longrightarrow\vert {y}_{j}(t') - {y}_{j}(t)\vert\leq K\vert t' - t\vert.$$

(A.4)

Hence, by Ascoli’s theorem (see Proposition 6.2 in Appendix A, Functional Analysis) the sequence (y_j) has a subsequence $({y}_{{j}_{\nu }})$ which converges locally uniformly: ${y}_{{j}_{\nu }}\rightarrow y$. It follows immediately that

$$y(t) = {x}_{0} +{\int \nolimits\nolimits }_{{t}_{0}}^{t}F{\bigl (s,y(s)\bigr)}\ ds,$$

(A.5)

so y solves (A.1).

Under the hypotheses of Proposition A.1, a solution to (A.1) may not be unique. The following family of examples illustrates the phenomenon. Take a ∈ (0,1) and consider

$$\frac{dy} {dt} =\vert y{\vert }^{a},\quad y(0) = 0.$$

(A.6)

Then one solution on [0, ∞) is given by

$${y}_{0}(t) = {(1 - a)}^{1/(1-a)}\ {t}^{1/(1-a)},$$

(A.7)

and another is given by

$${y}_{\ast}(t) = 0.$$

Note that, for any ε > 0, the problem $dy/dt =\vert y{\vert }^{a},\ y(0) =\epsilon$ has a unique solution on t ∈ [0, ∞), and lim_{ε → 0} y_ε(t) = y₀(t). Understanding this provides the key to the following uniqueness result, due to W. Osgood.

Let $\omega : {\mathbb{R}}^{+}\rightarrow{\mathbb{R}}^{+}$ be a modulus of continuity, i.e., ω(0) = 0, ω is continuous, and increasing. We may as well assume ω is bounded and C^∞ on (0, ∞).

Proposition A.2.

In the setting of Proposition A.1, assume F is continuous on $I\times\mathcal{O}$ and that

$$\vert F(t,{y}_{1}) - F(t,{y}_{2})\vert\leq \omega {\bigl (\vert {y}_{1} - {y}_{2}\vert\bigr)},$$

(A.8)

for all $t\in I,\ {y}_{j}\in\mathcal{O}$. Then solutions to (A.1) (with range in $\mathcal{O}$) are unique, provided

$${\int \nolimits\nolimits }_{0}^{1}\frac{ds} {\omega (s)} =\infty.$$

(A.9)

Proof.

If y₁(t) and y₂(t) are two solutions to (A.1), then

$${y}_{1}(t) - {y}_{2}(t) ={\int \nolimits\nolimits }_{{t}_{0}}^{t}{\bigl\{F{\bigl (s,{y}_{ 1}(s)\bigr)} - F{\bigl (s,{y}_{2}(s)\bigr)}\bigr\}}\ ds. $$

(A.10)

Let us set $\theta (t) =\vert {y}_{1}(t) - {y}_{2}(t)\vert$. Hence, by (A.8), for t ≥ t₀,

$$\theta (t)\leq {\int \nolimits\nolimits }_{{t}_{0}}^{t}\omega {\bigl (\theta (s)\bigr)}\ ds.$$

(A.11)

In particular, for each $\epsilon > 0,\theta (t)\leq {\int \nolimits\nolimits }_{{t}_{0}}^{t}\omega {\bigl (\theta (s) +\epsilon\bigr)}\ ds$. Since we are assuming ω is smooth on (0, ∞), we can apply the Gronwall inequality, derived in (5.19)–(5.21), to deduce that

$$\theta (t)\leq{\varphi }_{\epsilon }(t),\quad\forall\ t\geq{t}_{0},\epsilon > 0,$$

(A.12)

where φ_ε is uniquely defined on [t₀, ∞) by

$${\varphi '}_{\epsilon }(t) =\omega {\bigl ({\varphi }_{\epsilon }(t) +\epsilon\bigr)},\quad {\varphi }_{\epsilon }({t}_{0}) = 0.$$

(A.13)

Thus

$${\int \nolimits\nolimits }_{0}^{{\varphi }_{\epsilon }(t)}\frac{d\zeta } {\omega (\zeta +\epsilon)} = t - {t}_{0}.$$

(A.14)

Now the hypothesis (A.9) implies

$$\lim\limits_{\epsilon\searrow 0}\ {\varphi }_{\epsilon }(t) = 0,\quad\forall\ t\geq{t}_{0}, $$

(A.15)

so we have θ(t) = 0, for all t ≥ t₀. Similarly, one shows θ(t) = 0, for t ≤ t₀, and uniqueness is proved.

An important example to which Proposition A.2 applies is

$$\omega (s) = s\log\frac{1} {s},\quad s\leq \frac{1} {2}.$$

(A.16)

This arises in the study of ideal fluid flow, as will be seen in Chap. 17.

A similar argument establishes continuous dependence on initial data. If

$$\frac{d{y}_{j}} {dt} = F(t,{y}_{j}),\quad {y}_{j}({t}_{0}) = {x}_{j},$$

(A.17)

then

$${y}_{1}(t) - {y}_{2}(t) = {x}_{1} - {x}_{2} +{\int \nolimits\nolimits }_{{t}_{0}}^{t}{\bigl\{F{\bigl (s,{y}_{ 1}(s)\bigr)} - F{\bigl (s,{y}_{2}(s)\bigr)}\bigr\}}\ ds, $$

(A.18)

so ${\theta }_{12}(t) =\vert {y}_{1}(t) - {y}_{2}(t)\vert$ satisfies

$${\theta }_{12}(t)\leq\vert {x}_{1} - {x}_{2}\vert +{\int \nolimits\nolimits }_{{t}_{0}}^{t}\omega {\bigl ({\theta }_{ 12}(s)\bigr)}\ ds.$$

(A.19)

An argument similar to that used above gives (for t ≥ t₀)

$${\theta }_{12}(t)\leq \theta (\vert {x}_{1} - {x}_{2}\vert,t),$$

(A.20)

where, for a > 0, t ≥ t₀, θ(a, t) is the unique solution to

$${\partial }_{t}\theta =\omega (\theta),\quad\theta (a,{t}_{0}) = a,$$

(A.21)

that is,

$${\int \nolimits\nolimits }_{a}^{\theta (a,t)}\frac{d\zeta } {\omega (\zeta)} = t - {t}_{0}.$$

(A.22)

Again, the hypothesis (A.9) implies

$$\lim\limits_{a\searrow 0}\theta (a,t) = 0,\quad\forall\ t\geq{t}_{0}.$$

(A.23)

By (A.20), we have

$$\vert {y}_{1}(t) - {y}_{2}(t)\vert\leq \theta {\bigl (\vert {x}_{1} - {x}_{2}\vert,t\bigr)},$$

(A.24)

for all t ≥ t₀, and a similar argument works for t ≤ t₀