Holder Continuous Euler Flows in Three Dimensions with Compact Support in Time (eBook)

(AMS-196)

Philip Isett (Autor)

eBook Download: PDF

2017
216 Seiten
Princeton University Press (Verlag)
978-1-4008-8542-8 (ISBN)

Infos

Philip Isett is assistant professor of mathematics at the University of Texas, Austin.

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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-Holder. 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 "e;Main Lemma"e;-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.

Produkt-Details

• Erscheint lt. Verlag: 21.2.2017
• Reihe/Serie: Annals of Mathematics Studies
• Reihe/Serie: Annals of Mathematics Studies
• Verlagsort: Princeton
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Naturwissenschaften ► Chemie
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Schlagworte: absolute value • abstract index notation • Algebra • Amplitude • Ansatz • Approximation • arc length • asymptotic analysis • Baire category theorem • Beltrami flows • bilinear form • boundedness • C0 • Calculation • Cauchy stress tensor • Center of mass (relativistic) • classical mechanics • coarse scale flow • coarse scale velocity • coefficient • commutator • commutator estimate • commutator term • complex number • Computation • conjecture • Conservation of Momentum • continuous function • Continuous function (set theory) • continuous solution • contravariant tensor • convection–diffusion equation • convergence • convex integration • Correction • correction term • covariant tensor • Degrees of freedom (statistics) • Derivative • differential equation • Dimension • Dimensional Analysis • Dissipation • divergence equation • divergence free vector field • divergence operator • Eigenfunction • Einstein notation • einstein summation convention • Energy • energy approximation • Energy Density • energy function • energy increment • energy level • energy regularity • energy variation • Equation • error • Error Term • estimation • Euler equations • Euler equations (fluid dynamics) • Euler flow • Euler-Reynolds equations • Euler-Reynolds system • expected value • Exterior algebra • finite time interval • first material derivative • Fluid Dynamics • foliation • Fourier series • Fourier transform • frequencies • frequency energy levels • Function (mathematics) • Galilean invariance • Galilean Transformation • High Frequency • HighЈigh Interference term • HighЈigh term • HighЌow Interaction term • hilbert space • Hlder norm • Hlder regularity • h-principle • Incompressible Flow • Initial Value Problem • Integral • Integration by parts • Invertible matrix • Iteration • Lars Onsager • Lebesgue measure • Lie derivative • lifespan parameter • linear algebra • linear equation • Linear Function • Lipschitz continuity • lower indices • Main Lemma • main theorem • material derivative • matrix coefficient • mollification • Mollification term • mollifier • Momentum • moment vanishing condition • Moving Average • multi-index • Navier–Stokes equations • Newton's law • Noether's Theorem • non-negative function • nonzero solution • Notation • Onsager's conjecture • optimal regularity • orthogonality • Oscillation • oscillatory factor • oscillatory term • Parameter • parameters • Parametrix • parametrix expansion • partial differential equation • Periodic function • phase direction • phase function • phase gradient • Pointwise • Positive-definite matrix • Pressure • pressure correction • probability measure • Quantity • regularity • relative acceleration • relative velocity • Requirement • Reynolds stres • Reynolds Stress • scaling symmetry • second material derivative • Sign (mathematics) • Simultaneous Equations • Smooth function • smooth stress tensor • smooth vector field • spatial derivative • spatial frequency • Spectral density • square root • Stress • Stress equation • Stress term • Strong topology • subsequence • Summation • Support (mathematics) • Symmetric tensor • Tensor • Theorem • Third derivative • Three-dimensional space (mathematics) • time cutoff function • time derivative • Transport • transport derivative • Transport-Elliptic equation • transport equation • transport equations • transport estimate • Transport term • Upper and lower bounds • upper indices • Variable (mathematics) • vector amplitude • Vector field • Velocity • velocity correction • velocity field • viscosity • Weak convergence (Hilbert space) • weak limit • weak solution • Weak topology • Weierstrass function • Weighted arithmetic mean • Without loss of generality
• ISBN-10: 1-4008-8542-6 / 1400885426
• ISBN-13: 978-1-4008-8542-8 / 9781400885428