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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
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See the Picard-Lindelöf theorem. –  kahen Nov 2 '10 at 8:09
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3 Answers

up vote 9 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
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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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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
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They are known. See, e.g. www.deaneyang.com/papers/gunther.pdf –  Willie Wong Nov 2 '10 at 16:40
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