Lectures on Nonlinear Hyperbolic Differential Equations
In this introductory textbook, a revised and extended version of well-known lectures by L. Hörmander from 1986, four chapters are devoted to weak solutions of systems of conservation laws. Apart from that the book only studies classical solutions. Two chapters concern the existence of global solutions or estimates of the lifespan for solutions of nonlinear perturbations of the wave or Klein-Gordon equation with small initial data. Four chapters are devoted to microanalysis of the singularities of the solutions. This part assumes some familiarity with pseudodifferential operators which are standard in the theory of linear differential operators, but the extension to the more exotic classes of opertors needed in the nonlinear theory is presented in complete detail.
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Scalar first order equations with one space variable
Scalar first order equations with several variables
First order systems of conservation laws with
Nonlinear perturbations of the wave equation
Nonlinear perturbations of the KleinGordon
Pseudodifferential operators of type 11
Propagation of singularities
Appendix on pseudoRiemannian geometry
apply arbitrary assume asymptotic Cauchy data Cauchy problem choose coefficients compact support completes the proof conclude condition conic neighborhood constant continuous function converges convex coordinates Corollary curve d'Alembertian defined denote derivatives differential equation discuss du/dt entropy entropy pairs equal existence theorems factor finite follows Fourier transform gives Hence homogeneous of degree Hormander hyperbolic hypothesis implies induction inequality initial data integral Klainerman Klein-Gordon equation L2 norm lifespan linear Lipschitz Lipschitz continuous microlocal Minkowski space noncharacteristic nonlinear notation Note obtain proof is complete proof of Lemma proof of Theorem Proposition pseudo-differential operators quadratic form R1+n reduced order replaced respect result right-hand side satisfies second order Section seminorm Sobolev's lemma suffices to prove supp Taylor's formula tensor uniform bound uniformly unique values vanishes of second vector fields wave equation wave front set weak solution write