# Regularity of linear pde with smooth coefficients

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)

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 $$u\not\in C^\infty$$.