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I wish to show local well-posedness of the PDE $$u_t=\frac{u_{xx}}{1+u_x^2}$$ given initial conditions $u\vert_{t=0}=u_0\in C^2(\Bbb T^1)$. Now, I know how to prove local (and in fact global) well-posedness of the closely related heat-equation problem $$u_t=u_{xx}$$ by using Fourier space methods, and I suspect that this is the necessary ingredient to proving that $$u_t=\frac{u_{xx}}{1+u_x^2}$$ is locally, and in fact globally well-posed, however, I don't see what technique I should be using to do this. Intuitively, I see that when $u$ varies sharply in space, this equation should have a certain damping relative to the heat equation, but I don't see how to formalize this intuition into a proof of well-posedness. The only heat equation-like PDEs I'm familiar with are those of the form $$u_t=\Delta u+f(u)\ \ \hbox{ on }\Bbb R^d\hbox{ or }\Bbb T^d$$ where $f\in C^1(\Bbb R)$ with $f(0)=0$, but at least those are almost linear, while the equation I have isn't linear at all.

Dominic Wynter
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    It might be helpful to notice that the RHS is equal to $(\arctan u_x)_x$. Formally, it follows for example that $ \frac{d}{dt} \int u = \int (\arctan u_x)_x = 0$ or $\frac 12 \frac{d}{dt} \int u^2 = - \int u_x \cdot \arctan u_x \le 0$. – Michał Miśkiewicz Jun 07 '17 at 17:36
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    Following that remark, your equation becomes $u_t = (F'(u_x))_x$, where $F$ is the primitive function of $\arctan$. This is the gradient flow for the functional $\int F(u_x)$. The function $F$ is convex, but has linear growth, so maybe nonlinear semigroup theory might help. – Michał Miśkiewicz Jun 07 '17 at 18:43
  • Thanks, that's exactly the type of hint I was looking for. What's a good reference for nonlinear semigroup theory? – Dominic Wynter Jun 07 '17 at 19:15
  • Try Chapter 9.6 in Evans' PDE book and the references given in 9.8 (e.g. Operateurs Maximaux Monotones (...) by Brezis). – Michał Miśkiewicz Jun 08 '17 at 09:48

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