Yudovich's Existence & Uniqueness Result

• Yudovich's theorem is a foundational result ensuring uniqueness of weak solutions for the 2D incompressible Euler equations under bounded or mildly unbounded vorticity conditions.
• It leverages measure-preserving vorticity transport and an Osgood modulus of continuity to achieve a robust Lagrangian framework for proving existence and uniqueness.
• Recent extensions of the theorem expand its applicability to new function spaces, moving boundaries, and infinite-energy regimes while highlighting the precise integrability thresholds for uniqueness.

Yudovich's existence and uniqueness result is the foundational global well-posedness theory for the two-dimensional incompressible Euler equations in vorticity form at low regularity. In its classical form, it asserts uniqueness of weak solutions when the initial vorticity is bounded; in its generalized form, it admits mildly unbounded vorticities controlled through a growth function. Modern treatments present the theorem through transport of vorticity by a measure-preserving flow, an Osgood modulus of continuity for the induced velocity, and energy- or Lagrangian-based uniqueness arguments. Recent work has both unified and extended the classical theorem to new function spaces containing Yudovich spaces and $BMO$, and has also clarified the boundaries of the theory through nonuniqueness constructions and variants with moving boundaries, sources and sinks, density dependence, and infinite-energy regimes (Dominguez et al., 2023, Drivas et al., 25 Sep 2025).

1. Classical formulation in vorticity variables

For a bounded planar domain $\Omega$, or more generally in settings such as $\mathbb R^2$, $\mathbb T^2$, or a two-dimensional manifold without boundary, the 2D incompressible Euler equations are written in vorticity form as

$$\partial_t \omega + u \cdot \nabla \omega = 0, \qquad u = K_\Omega[\omega],$$

with the impermeability condition

$u\cdot n = 0 \quad \text{on } \partial\Omega$

when a boundary is present, and initial datum $\omega|_{t=0}=\omega_0$. In the bounded-domain formulation emphasized in later Lagrangian expositions, $K_\Omega$ is the Biot–Savart operator, written as $u=\nabla^\perp \Delta_\Omega^{-1}\omega$ with Dirichlet boundary conditions for $\Delta$ (Drivas et al., 25 Sep 2025).

A weak solution in the vorticity formulation is typically required to satisfy

$\Omega$0

for test functions $\Omega$1, together with the requirement that the associated flow of homeomorphisms exists and is measure-preserving (Drivas et al., 25 Sep 2025).

The classical Yudovich theorem states that if the initial vorticity belongs to $\Omega$2, then weak solutions are unique. Later formulations regard this bounded-vorticity case as the constant-growth instance of a broader Yudovich class of mildly unbounded vorticities (Dominguez et al., 2023). In the notation of the 2025 Lagrangian note, the theorem can be stated as follows: for every $\Omega$3, there exists a unique weak solution $\Omega$4, and the Lagrangian flow map is well-defined, smooth in time for fixed initial data, and continuous in the initial data and the domain (Drivas et al., 25 Sep 2025).

2. Yudovich spaces and the Osgood criterion

The functional framework introduced by Yudovich and revisited in recent work is based on admissible growth of the $\Omega$5-norms of the vorticity. Given a non-decreasing growth function $\Omega$6 and $\Omega$7, one writes

$\Omega$8

The classical bounded-vorticity regime corresponds to $\Omega$9, so that $\mathbb R^2$0 (Dominguez et al., 2023).

A parallel notation used in the Lagrangian literature is

$\mathbb R^2$1

under the growth condition

$\mathbb R^2$2

This integral divergence is the Osgood-type threshold ensuring that the modulus of continuity of the velocity is strong enough to generate a unique flow (Drivas et al., 25 Sep 2025).

In uniformly localized settings, one replaces global $\mathbb R^2$3 control by local control uniform over space. Crippa and Stefani define

$\mathbb R^2$4

and

$\mathbb R^2$5

These spaces allow no decay condition at infinity and still support global weak solutions when combined with $\mathbb R^2$6-integrability (Crippa et al., 2021).

The associated velocity modulus of continuity is explicit. In the uniformly localized Yudovich framework,

$\mathbb R^2$7

and uniqueness holds for Lagrangian weak solutions if

$\mathbb R^2$8

and $\mathbb R^2$9 is concave (Crippa et al., 2021).

3. Lagrangian mechanism of existence and uniqueness

The structural core of Yudovich theory is the observation that vorticity is transported by the fluid flow. If $\mathbb T^2$0 denotes the particle trajectory map,

$\mathbb T^2$1

then

$\mathbb T^2$2

This turns the Euler dynamics into an ODE on the space of measure-preserving homeomorphisms $\mathbb T^2$3,

$\mathbb T^2$4

and makes the regularity of the velocity modulus decisive for uniqueness (Drivas et al., 25 Sep 2025).

The proof scheme in recent Lagrangian reconstructions has several stable components. First, the Biot–Savart law maps vorticities in $\mathbb T^2$5 to velocity fields with Osgood modulus of continuity; in the bounded-vorticity case, the modulus is log-Lipschitz. Second, one sets up a Picard scheme in $\mathbb T^2$6 with distance $\mathbb T^2$7. Third, kernel estimates for the Biot–Savart operator yield a nonlinear Grönwall inequality controlled by an Osgood modulus. Finally, Osgood’s lemma implies that two flows that coincide initially must coincide for all later times, and therefore the transported vorticities coincide as well (Drivas et al., 25 Sep 2025).

This approach also yields quantitative continuity of the solution map. For two initial vorticities $\mathbb T^2$8 with associated flows $\mathbb T^2$9 and $$\partial_t \omega + u \cdot \nabla \omega = 0, \qquad u = K_\Omega[\omega],$$0, and any $$\partial_t \omega + u \cdot \nabla \omega = 0, \qquad u = K_\Omega[\omega],$$1,

$$\partial_t \omega + u \cdot \nabla \omega = 0, \qquad u = K_\Omega[\omega],$$2

where $$\partial_t \omega + u \cdot \nabla \omega = 0, \qquad u = K_\Omega[\omega],$$3 is defined from the Osgood modulus $$\partial_t \omega + u \cdot \nabla \omega = 0, \qquad u = K_\Omega[\omega],$$4 (Drivas et al., 25 Sep 2025). In the bounded-vorticity case, the same note states that the solution map is Hölder continuous in the initial data, with a time-deteriorating modulus.

A recurrent misconception is that the theorem is purely an $$\partial_t \omega + u \cdot \nabla \omega = 0, \qquad u = K_\Omega[\omega],$$5-estimate for vorticity. Modern formulations show instead that the decisive object is the modulus of continuity of the induced velocity field and its Osgood character. This is why the theorem persists in generalized Yudovich spaces and in several nonclassical geometries.

4. Extrapolation theory, $$\partial_t \omega + u \cdot \nabla \omega = 0, \qquad u = K_\Omega[\omega],$$6, and sharp Yudovich scales

A major modern reformulation of Yudovich theory is given by the extrapolation framework of “Uniqueness for 2D Euler and transport equations via extrapolation” (Dominguez et al., 2023). The paper replaces the classical interpolation pair $$\partial_t \omega + u \cdot \nabla \omega = 0, \qquad u = K_\Omega[\omega],$$7 by $$\partial_t \omega + u \cdot \nabla \omega = 0, \qquad u = K_\Omega[\omega],$$8 and introduces the sharp Yudovich spaces

$$\partial_t \omega + u \cdot \nabla \omega = 0, \qquad u = K_\Omega[\omega],$$9

where $u\cdot n = 0 \quad \text{on } \partial\Omega$0 is the Strömberg–Jawerth–Torchinsky maximal function norm, with $u\cdot n = 0 \quad \text{on } \partial\Omega$1 for $u\cdot n = 0 \quad \text{on } \partial\Omega$2 and $u\cdot n = 0 \quad \text{on } \partial\Omega$3 (Dominguez et al., 2023).

This construction unifies and extends the classical Yudovich and Vishik results. The inclusions

$u\cdot n = 0 \quad \text{on } \partial\Omega$4

hold, and for $u\cdot n = 0 \quad \text{on } \partial\Omega$5,

$u\cdot n = 0 \quad \text{on } \partial\Omega$6

The norm admits the characterization

$u\cdot n = 0 \quad \text{on } \partial\Omega$7

where $u\cdot n = 0 \quad \text{on } \partial\Omega$8 is the sharp maximal operator (Dominguez et al., 2023).

The corresponding uniqueness theorem is stated as follows: if $u\cdot n = 0 \quad \text{on } \partial\Omega$9 or $\omega|_{t=0}=\omega_0$0, and if $\omega|_{t=0}=\omega_0$1 satisfies the Osgood-type condition

$\omega|_{t=0}=\omega_0$2

then a Lagrangian weak solution of the Euler equations with initial vorticity

$\omega|_{t=0}=\omega_0$3

is uniquely determined by its initial value (Dominguez et al., 2023). The same paper gives a velocity modulus estimate of the form

$\omega|_{t=0}=\omega_0$4

which supplies the Osgood continuity needed for the Lagrangian argument.

The extrapolation perspective is significant because it shows that the classical bounded-vorticity theorem is not the endpoint of the theory. In particular, the framework covers vorticity classes larger than both $\omega|_{t=0}=\omega_0$5 and $\omega|_{t=0}=\omega_0$6, including examples described in the paper that are in neither of those spaces but do belong to $\omega|_{t=0}=\omega_0$7 (Dominguez et al., 2023).

5. Geometric and physical variants

Yudovich-type well-posedness has been extended to settings where the classical impermeable-boundary, fixed-domain formulation is no longer adequate.

For flows with prescribed entering and exiting normal velocity on the boundary, Yudovich’s 1962 result established existence and uniqueness of classical solutions under Hölder vorticity. The bounded-vorticity uniqueness problem for permeable boundaries remained open until Weigant and Papin treated a rectangle, and was later extended to smooth bounded domains with multiple sources and sinks. The 2021 paper proves uniqueness of weak solutions in the Yudovich class $\omega|_{t=0}=\omega_0$8 for arbitrary smooth domains with several internal sources and sinks by combining two energy estimates and an Osgood argument; one estimate controls the usual kinetic energy, while the second uses a harmonic test function to compensate the sign-indefinite boundary term produced by inflow and outflow (Noisette et al., 2021).

In fluid–solid interaction, Glass and Sueur establish uniqueness of weak solutions for a rigid body immersed in a 2D incompressible inviscid fluid in a bounded domain, “a la Yudovich,” as long as no collision occurs. The initial vorticity is assumed in $\omega|_{t=0}=\omega_0$9, the fluid velocity is log-Lipschitz, and the only obstruction to continuation is collision of the moving solid with the container boundary (Glass et al., 2012).

In the density-dependent incompressible Euler system, a direct adaptation of the homogeneous Yudovich theory fails because the vorticity equation contains pressure–density coupling terms. The 2025 paper “Yudovich theory under geometric regularity for density-dependent incompressible fluids” introduces the transported tangent field

$K_\Omega$0

and the directional derivative

$K_\Omega$1

It proves conditional existence and uniqueness of Yudovich-type solutions under the a priori bound

$K_\Omega$2

Existence is obtained as a stability-by-approximation statement for smooth solutions with uniform geometric control, and uniqueness holds among Yudovich-type solutions satisfying the same geometric bound (Fanelli, 29 Jun 2025). A plausible implication is that, in the non-homogeneous setting, the classical Osgood mechanism survives only after being supplemented by anisotropic geometric regularity.

6. Infinite-energy extensions and sharpness of the theory

Recent work has expanded the theorem beyond finite-energy regimes while also demonstrating genuine failure below the Yudovich threshold.

The paper “Unbounded Yudovich Solutions of the Euler Equations” studies velocity fields with square-root growth $K_\Omega$3 and bounded vorticity. It replaces global $K_\Omega$4 control by local Morrey spaces

$K_\Omega$5

and defines

$K_\Omega$6

For $K_\Omega$7, the paper proves global existence and uniqueness, continuity of the solution map on bounded-vorticity balls, and a canonical pressure selection under the far-field condition

$K_\Omega$8

for all $K_\Omega$9 (Cobb et al., 2024). This extends Yudovich theory to infinite-energy classes compatible with Galilean invariance.

By contrast, the paper “Nonuniqueness of solutions to the Euler equations with vorticity in a Lorentz space” proves that the classical uniqueness theorem cannot be extended to vorticity merely in the Lorentz space $u=\nabla^\perp \Delta_\Omega^{-1}\omega$0. It constructs a nontrivial solution

$u=\nabla^\perp \Delta_\Omega^{-1}\omega$1

with

$u=\nabla^\perp \Delta_\Omega^{-1}\omega$2

showing that distinct solutions can emerge from the same initial data in this class (Brué et al., 2021). The construction uses convex integration and intermittent-jet-type building blocks.

Taken together, these results indicate that Yudovich’s theorem is both robust and delicate. It is robust under substantial enlargement of the admissible phase space—localized spaces, extrapolation scales, certain permeable boundaries, moving rigid bodies, and some infinite-energy regimes—but delicate with respect to endpoint integrability. This suggests that the decisive issue is not boundedness of vorticity in isolation, but whether the induced velocity retains an Osgood-continuous Lagrangian flow in the relevant geometry.