Leseprobe
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Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time
Philip Isett
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Naturwissenschaften, Medizin, Informatik, Technik / Naturwissenschaften allgemein
Beschreibung
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.
Kundenbewertungen
Support (mathematics), Exterior algebra, Requirement, Boundedness, Relative velocity, Theorem, Energy level, Energy density, Summation, Amplitude, Commutator, Bilinear form, Lipschitz continuity, Linear equation, Baire category theorem, Einstein notation, Third derivative, Weak topology, Weierstrass function, Galilean transformation, Partial differential equation, Invertible matrix, Quantity, Derivative, Equation, Material derivative, Dissipation, Integration by parts, Three-dimensional space (mathematics), Calculation, High frequency, Galilean invariance, Parametrix, Continuous function (set theory), Lebesgue measure, C0, Orthogonality, Square root, Degrees of freedom (statistics), Symmetric tensor, Weak convergence (Hilbert space), Initial value problem, Error term, Convection–diffusion equation, Pointwise, Spatial frequency, Dimension, Sign (mathematics), Simultaneous equations, Vector field, Oscillation, Weak solution, Time derivative, Estimation, Mollifier, Eigenfunction, Euler equations (fluid dynamics), Dimensional analysis, Parameter, Absolute value, Hilbert space, Navier–Stokes equations, Computation, Conjecture, Momentum, Probability measure, Approximation, Iteration, Continuous function, Linear function