Illposedness of incompressible fluids in supercritical Sobolev spaces

2024/04/11 by Luo, Xiaoyutao · 5 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2404.07813

Abstract

We prove that the 3D Euler and Navier-Stokes equations are strongly illposed in supercritical Sobolev spaces. In the inviscid case, for any 0 < s < (5)/(2) , we construct a C^∞_c initial velocity field with arbitrarily small H^s norm for which the unique local-in-time smooth solution of the 3D Euler equation develops large H^s norm inflation almost instantaneously. In the viscous case, the same H^s norm inflation occurs in the 3D Navier-Stokes equation for 0< s < (1)/(2) , where s = (1)/(2) is scaling critical for this equation.

Cited by

• Positive-regularity norm inflation for the 3D hypodissipative Navier--Stokes equations
• Norm Inflation For The Critical SQG Equation
• Low regularity Sobolev well-posedness for Vlasov--Poisson
• Arbitrary norm growth in the 3D Navier-Stokes equations
• Instantaneous continuous loss of Sobolev regularity for the 3D incompressible Euler equation

Related

• Strong Ill-posedness of the incompressible Euler equation in borderline Sobolev spaces
• Sharp norm inflation for 3D Navier-Stokes equations in supercritical spaces
• Norm Inflation for Generalized Navier-Stokes Equations
• Instantaneous continuous loss of Sobolev regularity for the 3D incompressible Euler equation
• Positive-regularity norm inflation for the 3D hypodissipative Navier--Stokes equations