370 Index and cyclical monotonicity, 82 and gradient, 53, 54 and normal cone, 54 continuity, 53 T Talagrand inequality, 291, 342 dual formulation, 295 Tanaka’s theorem (Boltzmann equation), 229 tightness (for probability measures), 25 time-discretization (of gradient flow), 256 total variation (and transportation), 7, 36 transference plan, 2 approximation, 314 optimal, 3 transshipment, 37 U Ulam’s lemma, 25 Urysohn’s extension lemma, 39 V viscosity solution of Hamilton-Jacobi equation, 176 of Monge-Amp` ere equation, 130 Vlasov equation, 253, 336 Vlasov-Fokker-Planck equation, 256 volume distortion (and Hessian), 59 W Wasserstein distances, 207 quadratic, 12 weak KAM theory, 102 Wulff shape, 185 Y Youdovich’s theorem, 112 Young inequality, 193 optimal, 193 Young measures, 9
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