[Paper Review] Lagrangian solutions to the 2D euler system with L^1 vorticity and infinite energy

Anna Bohun, François Bouchut+1 more|arXiv (Cornell University)|Aug 18, 2015

Navier-Stokes equation solutions4 citations

TL;DR

This paper establishes the existence and strong stability of Lagrangian solutions to the 2D incompressible Euler equations with initial vorticity in $L^1(\mathbb{R}^2)$, a setting where kinetic energy may be infinite. By leveraging recent theory on regular Lagrangian flows for vector fields with singular integrals, the authors prove that $L^1$ weak convergence of initial vorticity implies strong convergence of the associated flows and solutions, thereby extending well-posedness to infinite-energy regimes.

ABSTRACT

We consider solutions to the two-dimensional incompressible Euler system with only integrable vorticity, thus with possibly locally infinite energy. With such regularity, we use the recently developed theory of Lagrangian flows associated to vector fields with gradient given by a singular integral in order to define Lagrangian solutions, for which the vorticity is transported by the flow. We prove strong stability of these solutions via strong convergence of the flow, under only the assumption of L^1 weak convergence of the initial vorticity. The existence of Lagrangian solutions to the Euler system follows for arbitrary L^1 vorticity. Relations with previously known notions of solutions are established.

Motivation & Objective

• To establish a rigorous existence theory for solutions to the 2D Euler equations when initial vorticity is in $L^1(\mathbb{R}^2)$, a regime where kinetic energy may be locally or globally infinite.
• To define and study Lagrangian solutions in this low-regularity setting, where the vorticity is transported by a flow associated with the velocity field.
• To prove strong stability of these solutions under $L^1$ weak convergence of initial vorticity, ensuring convergence of the flow and the solution.
• To clarify the relationship between Lagrangian solutions and previously known solution concepts such as renormalized and symmetrized solutions.
• To extend the classical well-posedness framework beyond $L^\infty$ or $L^p$ vorticity assumptions to the full $L^1$ class, including non-compact and infinite-energy initial data.

Proposed method

• Define Lagrangian solutions via the theory of regular Lagrangian flows associated to velocity fields generated by the Biot-Savart law from $L^1$ vorticity.
• Use a priori error estimates for regular Lagrangian flows with gradient given by singular integrals, as developed in [8], to control the flow and velocity behavior.
• Construct approximate solutions by mollifying initial data in $L^1$, ensuring the mollified vorticity remains in $L^1$ and the velocity field is well-defined via convolution with the Biot-Savart kernel.
• Prove strong convergence of the mollified flows and velocities in $C([0,T];L^1_{\rm{loc}}(\mathbb{R}^2))$ under $L^1$ weak convergence of initial vorticity.
• Establish the symmetrized vorticity formulation as a key tool to pass to the limit and verify that the limit solution satisfies the symmetrized weak formulation.
• Introduce the concept of Lagrangian symmetrized solutions—solutions that are both Lagrangian and satisfy the symmetrized vorticity formulation—thereby unifying multiple solution notions.

Experimental results

Research questions

• RQ1 Can Lagrangian solutions be rigorously defined and constructed for the 2D Euler equations when initial vorticity is only in $L^1(\mathbb{R}^2)$, without assuming finite kinetic energy?
• RQ2 Does $L^1$ weak convergence of initial vorticity imply strong convergence of the associated Lagrangian flows and solutions?
• RQ3 Are Lagrangian solutions equivalent to or strictly stronger than previously known solution concepts such as renormalized or symmetrized solutions in this low-regularity setting?
• RQ4 Can the existence of solutions be established in the $L^1$ vorticity regime without relying on $L^2$ or $L^p$ bounds on velocity or vorticity?
• RQ5 What is the role of the symmetrized vorticity formulation in ensuring compactness and stability of the solution sequence in the $L^1$ setting?

Key findings

• Lagrangian solutions exist for all initial vorticity in $L^1(\mathbb{R}^2)$, even when the initial kinetic energy is infinite.
• Strong stability of these solutions is proven: if initial vorticity converges weakly in $L^1$, then the associated flows and solutions converge strongly in $C([0,T];L^1_{\rm{loc}}(\mathbb{R}^2))$.
• The limit of mollified solutions is a Lagrangian symmetrized solution, which satisfies both the Lagrangian transport and the symmetrized vorticity formulation.
• Under the additional assumption $v^0 \in L^2_{\rm{loc}}(\mathbb{R}^2)$, the limit velocity field belongs to $L^\infty((0,T);L^2_{\rm{loc}}(\mathbb{R}^2))$, and the solution satisfies the weak velocity formulation.
• The symmetrized vorticity formulation is preserved in the limit, and the nonlinear term in the formulation converges due to weak convergence of $\omega_n(t,x)\omega_n(t,y)$ in $L^1(\mathbb{R}^2 \times \mathbb{R}^2)$.
• The theory unifies Lagrangian, renormalized, and symmetrized solution concepts: every Lagrangian solution is automatically renormalized, and under equi-integrability, it is also symmetrized.