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Consider the following non linear PDE on $\mathbb{R}^n$ $$ \partial_t u_t(x) \,=\, F\big(x, u_t(x), D u_t(x)\big)$$ with given initial condition $u_0(x)$. Assume that:

  • $u_0$ is rotation invariant, namely $u_0(Ox)=u_0(x)$ for all rotation matrix $O$;
  • if $u$ is a rotation invariant function, then $F(x,u(x),Du(x))$ is rotation invariant

Can I conclude that the solution of the PDE $u_t$ is rotation invariant for any $t\geq0$? In discrete time this would be a very intuitive result...

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    $\begingroup$ We need to know more about the functions involved, I think. But if $F$ is real analytic, and we demand that $u_t$ be real analytic, then uniqueness of solution follows from the Cauchy--Kovalevskaya theorem, and then invariance is clear, since any noninvariant solution is taken by a rotation to another solution with the same initial conditions. $\endgroup$
    – Ben McKay
    Jun 5, 2020 at 9:32
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    $\begingroup$ If initial data give rise to unique solutions of your equation, then it is sufficient to prove that there exists a rotation invariant solution. Try to write $u_t(x) = v_t(r)$, where $r$ is the radius variable. Reduce the original PDE to a PDE for $v_t(r)$. If the reduced equation has a solution, it is guaranteed to be rotation invariant. $\endgroup$
    – Igor Khavkine
    Jun 5, 2020 at 9:46

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