Construction of solutions to the 3D Euler equations with initial data in H^β for β>0

2022/07/28 by Calvin Khor, Changxing Miao, Khor, Calvin +1
Mathematics · #35Q35 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations #Primary: 35F50 #Secondary: 35A02

paper · pdf · doi:10.48550/arxiv.2207.14041

openalex publication_date 2022/07/28 · openalex created_date 2022/07/30 · openalex updated_date 2026/07/28

Abstract

In this paper, we use the method of convex integration to construct infinitely many distributional solutions in H^β for 0<β≪1 to the initial value problem for the three-dimensional incompressible Euler equations. We show that if the initial data has any small fractional derivative in L², then we can construct solutions with some regularity, so that the corresponding L² energy is continuous in time. This is distinct from the L² existence result of E. Wiedemann, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 28, No. 5, 727--730 (2011; Zbl 1228.35172), where the energy is discontinuous at 0.

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