Can first order linear PDE be elliptic?

Question

In Corollary 3.1.15 of Hormander's Analysis of Linear Partial Differential Operators, he proves that if a distribution $u \in \mathcal D'(\Bbb R)$ satisfies $u' + a u = f \in C(\Bbb R)$, then $u \in C^1(\Bbb R)$ in the classical sense. Can this be generalized to several variables? Let's say that $$ P = \sum_{i = 1}^n a ^i(x)\partial_i + b(x) $$ is a first order partial differental operator with smooth coefficients. If in the weak sense we have $Pu = f \in X$ for some space $X$, when does it follow that $u \in X$? For instance, if $X = C_0^\infty(\Bbb R^n)$, I know I can solve $Pv = f$ to obtaine a smooth $v$, and $v$ would also be a weak solution, but I cannot see a way of showing that $u = v$ because I have no initial conditions for $u$.

Answer 1

No, first order equations you describe are known as transport equations. These are in general hyperbolic and do not preserve regularity of the right-hand side.

To see this, suppose $n=2$, and denote the variables by $(t,x)$. Then assuming the $a^i$ are constant with one non-zero and $b=0$, we can reduce to the form $$ \partial_tu + c \partial_xu = 0. $$ This admits solutions of the form $$ u(t,x) = \phi(x-ct) $$ for arbitrary $\phi$; in particular $u$ may be as irregular as you wish by choosing $\phi$ to be irregular.

The above construction can be generalised to higher dimensions. In general equations of this type, one can transport the initial data via the method of characteristics. In this case, the (ir)regularity of the initial data is preserved as it is transported along these characteristic curves. The validity of this method relies on the associated principal symbol $$ \sigma_P(x,\xi) = \sum_{i=1}^n a^i(x) \xi_i $$ vanishing along certain hypersurfaces. For general differential operators, ellipticity is often defined to require that $\sigma_P(x,\xi) \neq 0$ for all $x$ and all $\xi \neq 0$, which is evidently never satisfied for first order operators.

As an aside, first order systems of equations may be elliptic, such as with the Cauchy-Riemann equations.