Why don't we study hyperbolic equations as elliptic and parabolic equations?

Question

In the research of elliptic and parabolic equations, the Schauder estimate is one of the most important issues for them. In this topic, we always bound the norm of higher regularity in the small ball by a bigger one. That is, for the elliptic equation $ \operatorname{div}(A(x)\nabla u)=0 $, we have estimates like $ \left\|u\right\|_{C^{0,\alpha}(B_1)}\leq C\left\|u\right\|_{L^2(B_2)} $, where $ B_r=B(0,r) $ is the ball with center $ 0 $ and radius $ r $. I want to ask why we do not study such estimates for hyperbolic equations.

Answer 1

Why we do not study such estimates for hyperbolic equations?

Because they are false.

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Now: you may ask "why are they false?" This is a fairly deep question, and answers often involve discussion of propagation of singularities and characteristics. Quite a few chapters in Hörmander's Analysis of Linear Partial Differential Operators are devoted to this and similar questions.

Answer 2

For hyperbolic PDE's, we have two aspects that distinguish them from elliptic and parabolic PDE's and thus forbid one from being able to obtain estimates in the form you want:

• Usually the loss of derivatives is worse than in the elliptic case — in fact, even worse than in the more general subelliptic case. Typically, you lose a full (weak = Sobolev) derivative in a priori regularity estimates for linear hyperbolic PDE's: the so-called energy estimates. This loss is essentially sharp and can be tracked back to the fundamental theorem of Calculus. As pointed by Willie Wong in his answer, this can be refined if you take into account the characteristic set of the principal symbol of the operator, by means of techniques collectively called microlocal analysis. More precisely, worse than elliptic-type regularity must propagate along the bicharacteristic curves of the principal symbol, which are nonexistent in the elliptic case.
• Even if you account for the above loss of derivatives, the "optimal" relative shape of the regions $B_1$, $B_2$ is different due to a hallmark property of hyperbolic PDE's — namely, the finite speed of propagation of (supports of) solutions. Typically, $B_2$ will be a "cone-shaped" shadow region cast by $B_1$, whose precise form is once again dictated by the principal symbol of the operator.