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Motivated by the theory of turbulence in fluids, the physicist and chemist Lars Onsager conjectured in 1949 that weak solutions to the incompressible Euler equations might fail to conserve energy if their spatial regularity was below 1/3-Hölder. In this book, Philip Isett uses the method of convex integration to achieve the best-known results regarding nonuniqueness of solutions and Onsager's conjecture. Focusing on the intuition behind the method, the ideas introduced now play a pivotal role in the ongoing study of weak solutions to fluid dynamics equations.The construction itself-an intricate algorithm with hidden symmetries-mixes together transport equations, algebra, the method of nonstationary phase, underdetermined partial differential equations (PDEs), and specially designed high-frequency waves built using nonlinear phase functions. The powerful "Main Lemma"-used here to construct nonzero solutions with compact support in time and to prove nonuniqueness of solutions to the initial value problem-has been extended to a broad range of applications that are surveyed in the appendix. Appropriate for students and researchers studying nonlinear PDEs, this book aims to be as robust as possible and pinpoints the main difficulties that presently stand in the way of a full solution to Onsager's conjecture.

Fluid dynamics --- Mathematics. --- Beltrami flows. --- Einstein summation convention. --- Euler equations. --- Euler flow. --- Euler-Reynolds equations. --- Euler-Reynolds system. --- Galilean invariance. --- Galilean transformation. --- HighЈigh Interference term. --- HighЈigh term. --- HighЌow Interaction term. --- Hlder norm. --- Hlder regularity. --- Lars Onsager. --- Main Lemma. --- Main Theorem. --- Mollification term. --- Newton's law. --- Noether's theorem. --- Onsager's conjecture. --- Reynolds stres. --- Reynolds stress. --- Stress equation. --- Stress term. --- Transport equation. --- Transport term. --- Transport-Elliptic equation. --- abstract index notation. --- algebra. --- amplitude. --- coarse scale flow. --- coarse scale velocity. --- coefficient. --- commutator estimate. --- commutator term. --- commutator. --- conservation of momentum. --- continuous solution. --- contravariant tensor. --- convergence. --- convex integration. --- correction term. --- correction. --- covariant tensor. --- dimensional analysis. --- divergence equation. --- divergence free vector field. --- divergence operator. --- energy approximation. --- energy function. --- energy increment. --- energy regularity. --- energy variation. --- energy. --- error term. --- error. --- finite time interval. --- first material derivative. --- fluid dynamics. --- frequencies. --- frequency energy levels. --- h-principle. --- integral. --- lifespan parameter. --- lower indices. --- material derivative. --- mollification. --- mollifier. --- moment vanishing condition. --- momentum. --- multi-index. --- non-negative function. --- nonzero solution. --- optimal regularity. --- oscillatory factor. --- oscillatory term. --- parameters. --- parametrix expansion. --- parametrix. --- phase direction. --- phase function. --- phase gradient. --- pressure correction. --- pressure. --- regularity. --- relative acceleration. --- relative velocity. --- scaling symmetry. --- second material derivative. --- smooth function. --- smooth stress tensor. --- smooth vector field. --- spatial derivative. --- stress. --- tensor. --- theorem. --- time cutoff function. --- time derivative. --- transport derivative. --- transport equations. --- transport estimate. --- transport. --- upper indices. --- vector amplitude. --- velocity correction. --- velocity field. --- velocity. --- weak limit. --- weak solution.