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book: Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time
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Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time

  • Philip Isett and Philip Isett
Language: English
Published/Copyright: 2017

About this book

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.

Author / Editor information

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

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    Frontmatter

    i
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    Contents

    v
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    Preface

    ix
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    Part I. Introduction

    1
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    Part II. General Considerations of the Scheme

    11
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    Part III. Basic Construction of the Correction

    19
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    Part IV. Obtaining Solutions from the Construction

    56
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    Part V. Construction of Regular Weak Solutions: Preliminaries

    97
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    Part VI Construction of Regular Weak Solutions: Estimating the Correction

    135
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    Part VII. Construction of Regular Weak Solutions: Estimating the New Stress

    152
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    Acknowledgments

    183
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    Appendices

    185
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    References

    197
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    Index

    201
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Publishing information
Pages and Images/Illustrations in book
eBook published on:
February 6, 2017
eBook ISBN:
9781400885428
Pages and Images/Illustrations in book
View content
https://doi.org/10.1515/9781400885428
eBook ISBN: 9781400885428
Keywords for this book
Euler equations (fluid dynamics); Equation; Material derivative; Iteration; Weak solution; Theorem; Derivative; Vector field; Estimation; Parameter; Energy level; Support (mathematics); Conjecture; Amplitude; Error term; Momentum; C0; Parametrix; Weak topology; Mollifier; Pointwise; Continuous function; Convection–diffusion equation; Symmetric tensor; Relative velocity; Quantity; Commutator; Sign (mathematics); Third derivative; Time derivative; Lipschitz continuity; Bilinear form; Dimensional analysis; Linear equation; Requirement; Dimension; Calculation; Absolute value; Initial value problem; Degrees of freedom (statistics); Weierstrass function; High frequency; Galilean invariance; Computation; Eigenfunction; Probability measure; Invertible matrix; Simultaneous equations; Continuous function (set theory); Galilean transformation; Orthogonality; Linear function; Integration by parts; Einstein notation; Three-dimensional space (mathematics); Dissipation; Partial differential equation; Approximation; Exterior algebra; Summation; Spatial frequency; Oscillation; Baire category theorem; Square root; Boundedness; Weak convergence (Hilbert space); Hilbert space; Energy density; Lebesgue measure; Navier–Stokes equations
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