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

In the research of elliptic and parabolic equationequations, the Schauder estimate is the one one of the most important issues for them. In thesethis 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 the estimetesestimates 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.

In the research of elliptic and parabolic equation, the Schauder estimate is the one of the most important issues for them. In these topic, we always bound the norm of higher regularity in the small ball by a bigger one. That is, for elliptic equation $ \operatorname{div}(A(x)\nabla u)=0 $, we have the estimetes 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.

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.

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Why we do not study hyperbolic equations as elliptic and parabolic equations?

In the research of elliptic and parabolic equation, the Schauder estimate is the one of the most important issues for them. In these topic, we always bound the norm of higher regularity in the small ball by a bigger one. That is, for elliptic equation $ \operatorname{div}(A(x)\nabla u)=0 $, we have the estimetes 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.