# Flow of ODE with monotone source

Let $$\Phi$$ be the flow (defined as in page 6 of this paper) of the ODE $$\begin{cases} \frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0 \\ \Phi(x,0) = x \quad x \in \mathbb{R}. \end{cases}$$

Is it true that if $$f$$ is monotone in the first variable then $$\Phi$$ is Lipschitz?

Suppose that $$f$$ is decreasing in $$x$$. Let $$x(t)$$, $$y(t)$$ be two solutions of the ode. Then $$\dot{x}-\dot{y}= f(x,t)-f(y,t).$$

Multiplying both sides by $$x-y$$ we deduce

$$(\dot{x}-\dot{y})(x-y) =\big(f(x,t)-f(y,t)\big)(x-y)\leq 0,$$ where the last equality holds because $$f$$ is decreasing.

Hence $$\frac{1}{2}\frac{d}{dt}\big(x-y)^2\leq 0.$$ Thus the function $$t\mapsto \big( x(t)-y(t)\big)^2$$ is decreasing so $$\big(x(t)-y(t)\big)^2\leq \big( x(0)-y(0)\big)^2,\;\;\forall t\geq 0,$$ i.e., $$\Big(\Phi(x_0,t)-\Phi(y_0,t)\Big)^2\leq \Big(x_0-y_0\Big)^2,\;\;\forall t\geq 0.$$ In other words, for $$t\geq 0$$, $$\Phi(x,t)$$ is Lipschitz in $$x$$ with Lipschitz constant $$1$$ if $$f$$ is decreasing.

• Thank you. How can the argument be made rigorous even when $f$ is not smooth and $\Phi$ is not a classical solution but a regular Lagrangian flow?
– user124345
Apr 16, 2019 at 11:48
• The function $f$ coud even be multivalued, and you can work in an infinite dimensional Hilbert space as well This is a special case of the general theory of maximal monotone operators and the associated differential equations. Perhaps the friendliest introduction is Brezis' book Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. The ultimate reference is however V. Barbu's book Nonlinear semigroups and differen tial equations in Banach spaces Apr 16, 2019 at 12:05
• The finite dimensional case is discussed in V. Barbu's recent book Differential Equations Springer 2016, Example 2.4 and Sec. 2.7. Apr 16, 2019 at 12:08
• In the scalar case all you need for existence and uniqueness is that $f$ is decreasing and the function $\mathbb{R}\ni x\mapsto f(x)-x\in\mathbb{R}$ is onto. Apr 16, 2019 at 14:30
• Thank you. What if $f$ is increasing?
– user124345
Apr 16, 2019 at 23:48