Regularity of linear pde with smooth coefficients

Question

Consider $au_x+bu_y+cu_z=f$ on $\mathbb{T}^3$ where $a,b,c,f$ are in $C^\infty$ and $\forall (x,y,z)\in\mathbb{T}^3,|a|,|b|,|c|>1$.

If there exists $C^1$ solution to this pde, can we say that it is actually in $C^\infty$?

($u$ is real-valued, and $\mathbb{T}^3$ means three dimensional torus)

Answer 1

You can't. Take $a=b=c=1$, $f=0$, and a function $g(x,y)$ on $\mathbb{T}^2$ that is $C^1$ but not $C^\infty$. Then $$ u(x,y,z)=g(x-z,y-z) $$ satisfies $u_x+u_y+u_z=0$ on $\mathbb{T}^3$, and $u\in C^1$ but $u\not\in C^\infty$.