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Smoothness of solutions
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applications apriori bounded apriori estimate argument assume Cauchy Chadam choose conclude conserved energy constant contraction mapping converges coupled Dirac define denote dense set Differential Equations Dirac equation dxdt Edited energy inequalities escalated energy spaces example existence theory finite interval follows free equation global solutions Hilbert space hold hypotheses H hypotheses of Theorem initial data integral equation interval of existence Iscat Klein-Gordon equation L2-norm lemma linear non-linear equations non-linear mapping non-linear term non-linear wave equations Notice problem prove global existence real-valued right hand side scat scattering operator scattering theory Section Segal self-adjoint operator sine-Gordon equation small data smoothness Sobolev estimate Sobolev inequality soliton solution exists special properties Strauss strongly continuously strongly differentiable Suppose Theorem 16 three dimensions topology uniformly bounded uniformly continuous uniformly equicontinuous unique global valued function Vlll wave equations wave operators zero