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I would really appreciate if someone could just write down for me one example of a second order, or higher, differential equation for which it is known that there are no smooth solutions; and it's fine if it's a partial differential equation.

At first I thought it would be easy to either come up with an example or else find one by searching google/wiki/arxiv; but now I am not so sure.

I have a thing for non-smooth functions, and it just bothers me that I don't even know a single example of this type of differential equation. Thanks!

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Couldn't you just take any old $C^1$ function that's not smooth and produce a differential equation from that? For example $y' = |x|$? Or if you want an even uglier answer, start with a continuous nowhere-differentiable function, $f: [a,b] \to \mathbb{R}$. By the fundamental theorem of calculus, the function $F: [a,b] \to \mathbb{R}$ defined by $F(x) = \int_a^x f(t)\;\mathrm{d}t$ is differentiable on $(a,b)$ with $F'(x) = f(x)$, or in other words: $F$ is a solution to the differential equation $y' = f$ on $(a,b)$. Or did you want your DE to satisfy some additional "niceness" property? – kahen Nov 2 '10 at 2:24
First of all, I got a bit stuck trying to integrate the Weierstrass function and then finding a second order differential equation; but I was even more confused about how I was going to go about showing the differential equation I got didn't have other solutions that were smooth. – Matt Calhoun Nov 2 '10 at 3:27
See the Picard-Lindelöf theorem. – kahen Nov 2 '10 at 8:09
up vote 11 down vote accepted

There are already first order linear partial differential equations with smooth coefficients which do not admit any smooth solutions.

Hans Lewy produced the first example of such a PDE. The equation reads $$\left[-i\partial_x+\partial_y-2(x+iy)\partial_z\right]u(x,y,z)=f(x,y,z),\qquad(x,y,z)\in\mathbb R^{3}.$$ The equation does not have distribution solutions in any neighbourhood of any point in $\mathbb R^3$ provided $f=f(x,y,z)$ is not a real analytic function (it can be smooth though).

The original paper by Lewy is nice, clear and less than 4 pages long (freely available here).

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An interesting detail about Lewy's paper is that it doesn't cite any other works. – Hans Lundmark Nov 2 '10 at 7:14

Consider the partial differential equations associated to the isometric embedding problem of the hyperbolic plane into Euclidean 3-space. In $C^1$ there exists a solution by Nash-Kuiper theorem, but it is known classically that there cannot be solutions that are twice or more continuously differentiable.

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Is the form of these partial differential equation known, or is it only known that they exist abstractly? – Matt Calhoun Nov 2 '10 at 14:28
They are known. See, e.g. – Willie Wong Nov 2 '10 at 16:40

How about you take the differential equation

$ \frac{dy}{dx} = |x| $

This is a linear non-homogeneous differential equation, whose solution is $C^1$ but not smooth at $x=0$.

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The solution is $y = \frac12 sgn(x) x^2 + c$. – Willie Wong Nov 2 '10 at 16:36
The solution is not $y=\frac{1}{2}|x|^2$, since we have $\frac{d\frac{1}{2}∣x∣}{dx}=sgn(x)|x|=x$. So I'm wondering if you know the solution about x = 0, and also how can I derive this solution? – Matt Calhoun Nov 2 '10 at 16:42
@Matt: The solution is a piecewise quadratic function. It is $-x^2/2 + C$ for $x \leq 0$ and $+x^2/2 + C$ for $x \geq 0$. I got this by integrating $|t|$ from $0$ to $x$. – Eric Haengel Nov 2 '10 at 20:05

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