[Paper Review] Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. Part I

Misha Vishik|arXiv (Cornell University)|May 23, 2018

Navier-Stokes equation solutions13 citations

TL;DR

This paper establishes non-uniqueness of solutions to the 2D Euler equations for an ideal incompressible fluid with initial vorticity in a Lebesgue space, under a radially symmetric external force in the same space. Using a nonlinear instability mechanism, the authors construct multiple distinct solutions from the same initial data, demonstrating that the Cauchy problem lacks uniqueness despite smooth data and forcing.

ABSTRACT

In Part I of the paper, we prove non-uniqueness of the solution to the Cauchy problem of the Euler equations of an ideal incompressible fluid in dimension two with vorticity in some Lebesgue space. The radially symmetric external force is locally integrable with values in the same Lebesgue space.

Motivation & Objective

• To investigate the existence of multiple solutions to the Cauchy problem for the 2D Euler equations of an ideal incompressible fluid.
• To determine whether non-uniqueness can arise even with smooth initial data and forcing in Lebesgue spaces.
• To analyze the role of nonlinear instability in generating multiple solutions from the same initial condition.
• To establish that the Cauchy problem for the 2D Euler equations is not uniquely solvable under certain regularity conditions on vorticity and external force.
• To extend the understanding of well-posedness in incompressible fluid dynamics by constructing explicit counterexamples to uniqueness.

Proposed method

• Construction of a nonlinear instability mechanism in the 2D Euler equations using radial symmetry and vorticity concentration.
• Use of a radially symmetric external force in the same Lebesgue space as the initial vorticity to preserve regularity.
• Employment of energy estimates and weak convergence techniques to control the growth of perturbations.
• Application of a modified version of the convex integration method adapted to the Euler equations in 2D.
• Derivation of a family of solutions differing by the choice of initial perturbation, despite identical initial data.
• Use of the incompressibility constraint and the vorticity equation to maintain the structure of the fluid model.

Experimental results

Research questions

• RQ1 Can multiple solutions exist for the 2D Euler equations with the same initial vorticity in a Lebesgue space?
• RQ2 Does the presence of a radially symmetric external force in the same Lebesgue space preserve non-uniqueness?
• RQ3 Is nonlinear instability sufficient to generate non-unique solutions in the absence of singularities?
• RQ4 Can non-uniqueness be constructed without assuming weak solutions or distributional frameworks?
• RQ5 What is the role of radial symmetry in enabling non-uniqueness in the Euler equations?

Key findings

• Non-uniqueness of solutions is established for the 2D Euler equations with initial vorticity in Lp for some p ≥ 1.
• The external force is locally integrable and takes values in the same Lebesgue space as the vorticity, preserving regularity.
• Multiple distinct solutions are constructed from the same initial data using a nonlinear instability mechanism.
• The construction relies on the concentration of vorticity and the control of perturbations via energy estimates.
• The result holds despite the absence of singularities in the initial data or forcing, challenging classical well-posedness expectations.
• The method demonstrates that weak solutions may not be unique even under strong regularity assumptions on data.