scholarly journals Onsager's Conjecture for Admissible Weak Solutions

2018 ◽  
Vol 72 (2) ◽  
pp. 229-274 ◽  
Author(s):  
Tristan Buckmaster ◽  
Camillo De Lellis ◽  
László Székelyhidi ◽  
Vlad Vicol
Keyword(s):  
Weak Solutions ◽  
Onsager’S Conjecture

scholarly journals Energy Conservation for 3D Tropical Climate Model in Periodic Domains

2021 ◽  
Vol 2021 ◽  
pp. 1-4
Author(s):  
Hui Zhang
Keyword(s):  
Energy Conservation ◽  
Euler Equations ◽  
Weak Solutions ◽  
Climate Model ◽  
Sufficient Conditions ◽  
Tropical Climate ◽  
Tropical Climate Model ◽  
Periodic Domains ◽  
Onsager’S Conjecture

In this paper, we prove the energy conservation for the weak solutions of the 3D tropical climate model under some sufficient conditions. Our results are similar to Onsager’s conjecture which is on energy conservation for weak solutions of Euler equations.

On Onsager's Conjecture

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Euler Equations ◽  
Weak Solutions ◽  
Time Derivative ◽  
Space Time ◽  
Spatial Derivative ◽  
Time Intervals ◽  
Higher Regularity ◽  
Onsager’S Conjecture

This chapter deals with Onsager's conjecture, which would be implied by a stronger form of Lemma (10.1). It considers what could be proven assuming Conjecture (10.1) by turning to Theorem 13.1, which states that for every δ‎ > 0, there exist nontrivial weak solutions (v, p) to the Euler equations on ℝ x ³. Here the energy will increase or decrease in certain time intervals. In order to determine which Hölder norms stay under control during the iteration, the chapter observes that the bound for the spatial derivative of the corrections V and P also controls their full space-time derivative. The chapter also discusses higher regularity for the energy, written as a sum of energy increments.

scholarly journals Onsager's conjecture in bounded domains for the conservation of entropy and other companion laws

2019 ◽  
Vol 475 (2230) ◽  
pp. 20190289
Author(s):  
C. Bardos ◽  
P. Gwiazda ◽  
A. Świerczewska-Gwiazda ◽  
E. S. Titi ◽  
E. Wiedemann
Keyword(s):  
Conservation Laws ◽  
Weak Solutions ◽  
Normal Component ◽  
Bounded Domains ◽  
Generalized Entropy ◽  
Fractional Differentiability ◽  
General Conservation ◽  
Onsager’S Conjecture

We show that weak solutions of general conservation laws in bounded domains conserve their generalized entropy, and other respective companion laws, if they possess a certain fractional differentiability of order one-third in the interior of the domain, and if the normal component of the corresponding fluxes tend to zero as one approaches the boundary. This extends various recent results of the authors.

scholarly journals Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Weak Solutions ◽  
Compact Support ◽  
Main Lemma ◽  
Three Dimensions ◽  
Nonlinear Pdes ◽  
Hölder Continuous ◽  
Nonlinear Phase ◽  
Nonuniqueness Of Solutions ◽  
Fluid Dynamics Equations ◽  
Onsager’S Conjecture

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. This book 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.

Introduction

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Euler Equations ◽  
Ideal Fluid ◽  
Weak Solutions ◽  
Modern Language ◽  
Incompressible Euler Equations ◽  
Spatial Dimensions ◽  
Periodic Weak Solutions ◽  
Incompressible Euler ◽  
Onsager’S Conjecture

In the paper [DLS13], De Lellis and Székelyhidi introduce a method for constructing periodic weak solutions to the incompressible Euler equations{∂tv+div v⊗v+∇p=0                       div v=0in three spatial dimensions that are continuous but do not conserve energy. The motivation for constructing such solutions comes from a conjecture of Lars Onsager [Ons49] on the theory of turbulence in an ideal fluid. In the modern language of PDE, Onsager's conjecture can be translated as follows....

scholarly journals Local versus nonlocal elliptic equations: short-long range field interactions

2020 ◽  
Vol 10 (1) ◽  
pp. 895-921
Author(s):  
Daniele Cassani ◽  
Luca Vilasi ◽  
Youjun Wang
Keyword(s):  
Asymptotic Behavior ◽  
Maximum Principle ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Spectral Properties ◽  
Fractional Laplacian ◽  
Coupling Parameter ◽  
Nonlocal Operators ◽  
The Maximum Principle ◽  
Regularity Results

Abstract In this paper we study a class of one-parameter family of elliptic equations which combines local and nonlocal operators, namely the Laplacian and the fractional Laplacian. We analyze spectral properties, establish the validity of the maximum principle, prove existence, nonexistence, symmetry and regularity results for weak solutions. The asymptotic behavior of weak solutions as the coupling parameter vanishes (which turns the problem into a purely nonlocal one) or goes to infinity (reducing the problem to the classical semilinear Laplace equation) is also investigated.

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