Monday, 23 September 2024
In the recent resolution of the strong Onsager’s conjecture in L^3 framework, so-called wavelet-inspired Nash’s iteration has been developed. In this lecture, we will sketch this new method and provide the necessary background. The lecture will be based on our recent joint work with Matthew Novack.
We say that an ordinary or partial differential equation is regularized by noise if the addition of a suitable noise term restores well-posedness or improves regularity of the solution to the equation. Regularization by noise is by now well understood for ODEs and linear transport-type PDEs, but it is less understood for nonlinear PDEs like Euler and Navier-Stokes equations, with many open questions.
In the first part of this series of lectures, we consider the case of ODEs and associated transport equations with irregular drift: we show that the addition of an additive Brownian noise restores well-posedness of the ODE; we introduce the corresponding transport noise and show that this noise restores well-posedness for the associated transport equation. In the second part, we focus our attention on the effect of a particular transport-type noise, which is divergence-free, Gaussian, white in time and poorly correlated in space (nonsmooth Kraichnan noise). The associated linear transport model, introduced by Kraichnan, has been widely studied due to its peculiar features, like spontaneous stochasticity. In the case of incompressible 2D Euler equations, we show that the addition of nonsmoooth Kraichnan noise improves significantly the well-posedness theory, for example bringing uniqueness among finite enstrophy solutions. If time allows, we show further regularization properties of the Kraichnan noise.
Based on classical works (e.g. the works by Flandoli and coauthors) and on joint works with Marco Bagnara, Michele Coghi, Lucio Galeati, Francesco Grotto.
We start by demonstrating that numerical methods do not necessarily converge to entropy or admissible weak solutions of the Euler and Navier-Stokes equations of fluid dynamics on mesh refinement due to appearance of eddies at smaller and smaller scales. As an alternative, we revisit the concept of statistical solutions which are time-parametrized probability measures, consistent with the fluid evolution. We empirically show that the same numerical methods converge to a statistical solution and also derive verifiable sufficient conditions under which this convergence can be made rigorous. Numerical experiments illustrating interesting properties of statistical solutions are also presented. We conclude by showing how state of the art generative AI models (conditional diffusion) can significantly lower the cost of computing statistical solutions while maintaining accuracy.
There is a well-known discrepancy in mathematical fluid mechanics between phenomena that we can observe and phenomena on which we have theorems. The challenge for the mathematician is then to formulate an existence theory of solutions to the equations of hydrodynamics which is able to reflect observation. The most important such observation, forming the backbone of turbulence theory, is anomalous dissipation. In the talk, we survey some of the recent developments concerning weak solutions in this context.
Tuesday, 24 September 2024
We say that an ordinary or partial differential equation is regularized by noise if the addition of a suitable noise term restores well-posedness or improves regularity of the solution to the equation. Regularization by noise is by now well understood for ODEs and linear transport-type PDEs, but it is less understood for nonlinear PDEs like Euler and Navier-Stokes equations, with many open questions.
In the first part of this series of lectures, we consider the case of ODEs and associated transport equations with irregular drift: we show that the addition of an additive Brownian noise restores well-posedness of the ODE; we introduce the corresponding transport noise and show that this noise restores well-posedness for the associated transport equation. In the second part, we focus our attention on the effect of a particular transport-type noise, which is divergence-free, Gaussian, white in time and poorly correlated in space (nonsmooth Kraichnan noise). The associated linear transport model, introduced by Kraichnan, has been widely studied due to its peculiar features, like spontaneous stochasticity. In the case of incompressible 2D Euler equations, we show that the addition of nonsmoooth Kraichnan noise improves significantly the well-posedness theory, for example bringing uniqueness among finite enstrophy solutions. If time allows, we show further regularization properties of the Kraichnan noise.
Based on classical works (e.g. the works by Flandoli and coauthors) and on joint works with Marco Bagnara, Michele Coghi, Lucio Galeati, Francesco Grotto.
In the recent resolution of the strong Onsager’s conjecture in L^3 framework, so-called wavelet-inspired Nash’s iteration has been developed. In this lecture, we will sketch this new method and provide the necessary background. The lecture will be based on our recent joint work with Matthew Novack.
There is a well-known discrepancy in mathematical fluid mechanics between phenomena that we can observe and phenomena on which we have theorems. The challenge for the mathematician is then to formulate an existence theory of solutions to the equations of hydrodynamics which is able to reflect observation. The most important such observation, forming the backbone of turbulence theory, is anomalous dissipation. In the talk, we survey some of the recent developments concerning weak solutions in this context.
Wednesday, 25 September 2024
We say that an ordinary or partial differential equation is regularized by noise if the addition of a suitable noise term restores well-posedness or improves regularity of the solution to the equation. Regularization by noise is by now well understood for ODEs and linear transport-type PDEs, but it is less understood for nonlinear PDEs like Euler and Navier-Stokes equations, with many open questions.
In the first part of this series of lectures, we consider the case of ODEs and associated transport equations with irregular drift: we show that the addition of an additive Brownian noise restores well-posedness of the ODE; we introduce the corresponding transport noise and show that this noise restores well-posedness for the associated transport equation. In the second part, we focus our attention on the effect of a particular transport-type noise, which is divergence-free, Gaussian, white in time and poorly correlated in space (nonsmooth Kraichnan noise). The associated linear transport model, introduced by Kraichnan, has been widely studied due to its peculiar features, like spontaneous stochasticity. In the case of incompressible 2D Euler equations, we show that the addition of nonsmoooth Kraichnan noise improves significantly the well-posedness theory, for example bringing uniqueness among finite enstrophy solutions. If time allows, we show further regularization properties of the Kraichnan noise.
Based on classical works (e.g. the works by Flandoli and coauthors) and on joint works with Marco Bagnara, Michele Coghi, Lucio Galeati, Francesco Grotto.
In the recent resolution of the strong Onsager’s conjecture in L^3 framework, so-called wavelet-inspired Nash’s iteration has been developed. In this lecture, we will sketch this new method and provide the necessary background. The lecture will be based on our recent joint work with Matthew Novack.
There is a well-known discrepancy in mathematical fluid mechanics between phenomena that we can observe and phenomena on which we have theorems. The challenge for the mathematician is then to formulate an existence theory of solutions to the equations of hydrodynamics which is able to reflect observation. The most important such observation, forming the backbone of turbulence theory, is anomalous dissipation. In the talk, we survey some of the recent developments concerning weak solutions in this context.
The talk is aimed to be accessible to general audience including graduate students.
In the first part of this talk, I will introduce various different equations in fluid mechanics: Navier-Stokes, Euler, Boussinesq, surface quasi-geostrophic, MHD, Hall-MHD, etc. I will describe their basic properties and some well-known results. I will also introduce some variants such as the 2D MHD system with only magnetic diffusion, 2.5D Hall-MHD system, hyperbolic Navier-Stokes equations, and share open problems and recent progress.
Thursday, 26 September 2024
In the second part, I will discuss recent developments on stochastic PDEs of fluid mechanics such as probabilistically strong solution, uniqueness path-wise or in law, non-uniqueness of Markov selection, as well as some related conjectures such as those of Onsager and Taylor.
We say that an ordinary or partial differential equation is regularized by noise if the addition of a suitable noise term restores well-posedness or improves regularity of the solution to the equation. Regularization by noise is by now well understood for ODEs and linear transport-type PDEs, but it is less understood for nonlinear PDEs like Euler and Navier-Stokes equations, with many open questions.
In the first part of this series of lectures, we consider the case of ODEs and associated transport equations with irregular drift: we show that the addition of an additive Brownian noise restores well-posedness of the ODE; we introduce the corresponding transport noise and show that this noise restores well-posedness for the associated transport equation. In the second part, we focus our attention on the effect of a particular transport-type noise, which is divergence-free, Gaussian, white in time and poorly correlated in space (nonsmooth Kraichnan noise). The associated linear transport model, introduced by Kraichnan, has been widely studied due to its peculiar features, like spontaneous stochasticity. In the case of incompressible 2D Euler equations, we show that the addition of nonsmoooth Kraichnan noise improves significantly the well-posedness theory, for example bringing uniqueness among finite enstrophy solutions. If time allows, we show further regularization properties of the Kraichnan noise.
Based on classical works (e.g. the works by Flandoli and coauthors) and on joint works with Marco Bagnara, Michele Coghi, Lucio Galeati, Francesco Grotto.
There is a well-known discrepancy in mathematical fluid mechanics between phenomena that we can observe and phenomena on which we have theorems. The challenge for the mathematician is then to formulate an existence theory of solutions to the equations of hydrodynamics which is able to reflect observation. The most important such observation, forming the backbone of turbulence theory, is anomalous dissipation. In the talk, we survey some of the recent developments concerning weak solutions in this context.
In 1949 Onsager conjectured the existence of Hoelder continuous solutions of the incompressible Euler equations which do not conserve the kinetic energy. A rigorous proof of his statement has been given by Isett in 2017, crowning a decade of efforts in the subject. Onsager's original statement is however motivated by anomalous dissipation in the Navier-Stokes equations: roughly speaking it would be desirable to show that at least some dissipative Euler flow is the ``vanishing viscosity limit''. In these lectures I will review the basic ideas of the first iteration invented by La'szlo' Sze'kelyhidi Jr. and myself to produce continuous solutions which dissipate the total kinetic energy. I will then review the developments which lead Isett to solve the Onsager conjecture and touch upon the new challenges which lie ahead.
Friday, 27 September 2024
In the third part of this talk, I will discuss recent developments on singular stochastic PDEs. In short, these are SPDEs for which the random noise acting as a force is so rough that the nonlinear term becomes ill-defined as a product of distributions. We will discuss solution theory, local and global, as well as recent non-uniqueness results.
In this lecture I will explain the first work of La'szlo' and myself, which introduced for the first time ideas from differential inclusions in the study of the incompressible Euler equations. These ideas allowed to produce far-reaching generalizations of pioneering results by Scheffer and Shnirelman, showing the abundance of counterintuitive bounded solutions.
We say that an ordinary or partial differential equation is regularized by noise if the addition of a suitable noise term restores well-posedness or improves regularity of the solution to the equation. Regularization by noise is by now well understood for ODEs and linear transport-type PDEs, but it is less understood for nonlinear PDEs like Euler and Navier-Stokes equations, with many open questions.
In the first part of this series of lectures, we consider the case of ODEs and associated transport equations with irregular drift: we show that the addition of an additive Brownian noise restores well-posedness of the ODE; we introduce the corresponding transport noise and show that this noise restores well-posedness for the associated transport equation. In the second part, we focus our attention on the effect of a particular transport-type noise, which is divergence-free, Gaussian, white in time and poorly correlated in space (nonsmooth Kraichnan noise). The associated linear transport model, introduced by Kraichnan, has been widely studied due to its peculiar features, like spontaneous stochasticity. In the case of incompressible 2D Euler equations, we show that the addition of nonsmoooth Kraichnan noise improves significantly the well-posedness theory, for example bringing uniqueness among finite enstrophy solutions. If time allows, we show further regularization properties of the Kraichnan noise.
Based on classical works (e.g. the works by Flandoli and coauthors) and on joint works with Marco Bagnara, Michele Coghi, Lucio Galeati, Francesco Grotto.
We establish existence of infinitely many stationary solutions as well as ergodic stationary solutions to the three dimensional Navier--Stokes and Euler equations in the deterministic as well as stochastic setting, driven by an additive noise. The solutions belong to the regularity class $C(\mathbb{R};H^{\vartheta})\cap C^{\vartheta}(\mathbb{R};L^{2})$ for some $\vartheta>0$ and satisfy the equations in an analytically weak sense. Moreover, we are able to make conclusions regarding the vanishing viscosity limit. The result is based on a new stochastic version of the convex integration method which provides uniform moment bounds locally in the aforementioned function spaces.
Monday, 30 September 2024
In the recent resolution of the strong Onsager’s conjecture in L^3 framework, so-called wavelet-inspired Nash’s iteration has been developed. In this lecture, we will sketch this new method and provide the necessary background. The lecture will be based on our recent joint work with Matthew Novack.
In this lecture I will explain the ideas of Nash's surprising construction, in the 1950s, of many C^1 isometric embeddings of Riemannian manifolds as hypersurfaces of the Euclidean space. We will then touch upon an open problem in the area, due to to Borisov and Gromov, about the threshold Hoelder regularity for which the Nash phenomenon is possible. This problem turns out to be intimately linked to the Onsager conjecture and we will survey the results proved so far about it.
The lectures will introduce perturbations of Euler's equation by highly irregular paths where the perturbations are such that the solution preserves a range of physically relevant quantities. Using formal computations, we shall see that, when d=2, a purely Lagrangian formulation of the equation seems to be within reach.
However, special care is needed to give rigorous meaning to the noisy terms of the equation and in these lectures, we will consider the framework of rough paths. We will see how the so-called 'Sewing Lemma' can be used to define integrals as Riemann sums w.r.t paths of low regularity and how to use this result to construct rough path integrals. Then we will derive very precise a priori estimates for differential equations driven by rough paths and these estimates will be used to prove well-posedness of equations where the drift term satisfies an Osgood regularity. Moreover, we will study flows generated by the differential equations and see that the flows are volume preserving under natural assumptions on the noise and vector fields.
Returning to fluid equations, we shall use the above results to prove well-posedness of the roughly perturbed Euler equation in Lagrangian form when d=2. We will consider well-posedness in the class of equations with bounded and integrable vorticity, also known as Yudovich theory.
We consider the complete Navier--Stokes--Fourier system governing the time evolution of a general compressible, viscous, and heat conducting fluid. The fluid motion is driven by non--conservative boundary conditions. We show that, unlike the conservative systems, the non--conservative ones admit in general a bounded absorbing set. Asymptotic compactness of global in time solutions on this set is then established. The result has several corollaries including the existence of a stationary statistical solutions, convergence of ergodic averages and convergence to equailibrium solutions for small perturbations.
Tuesday, 01 October 2024
In the recent resolution of the strong Onsager’s conjecture in L^3 framework, so-called wavelet-inspired Nash’s iteration has been developed. In this lecture, we will sketch this new method and provide the necessary background. The lecture will be based on our recent joint work with Matthew Novack.
In this lecture I will explain a second type of iteration, introduced by La'szlo' and myself, which produces continuous solutions of incompressible Euler in a fashion which shares a lot of similarities with the theorem of Nash explained in Lecture 3.
The lectures will introduce perturbations of Euler's equation by highly irregular paths where the perturbations are such that the solution preserves a range of physically relevant quantities. Using formal computations, we shall see that, when d=2, a purely Lagrangian formulation of the equation seems to be within reach.
However, special care is needed to give rigorous meaning to the noisy terms of the equation and in these lectures, we will consider the framework of rough paths. We will see how the so-called 'Sewing Lemma' can be used to define integrals as Riemann sums w.r.t paths of low regularity and how to use this result to construct rough path integrals. Then we will derive very precise a priori estimates for differential equations driven by rough paths and these estimates will be used to prove well-posedness of equations where the drift term satisfies an Osgood regularity. Moreover, we will study flows generated by the differential equations and see that the flows are volume preserving under natural assumptions on the noise and vector fields.
Returning to fluid equations, we shall use the above results to prove well-posedness of the roughly perturbed Euler equation in Lagrangian form when d=2. We will consider well-posedness in the class of equations with bounded and integrable vorticity, also known as Yudovich theory.
Many physical phenomena may be modeled by first order symmetric hyperbolic equations with degenerate dissipative or diffusive terms. This is the case in gas dynamics, where the mass is conserved during the evolution, but the momentum balance includes a diffusion (viscosity) or damping (relaxation) term.
Such so-called partially dissipative or diffusive systems have been extensively studied by S .Kawashima in his PhD thesis from 1984. For a rather general class of systems he pointed out a simple necessary condition for the global existence of solutions in the vicinity of constant solutions.
This condition that is nowadays named (SK) condition has been revisited in a number of research works. In particular, K. Beauchard and E. Zuazua proposed in 2010 an explicit method for constructing a Lyapunov functional allowing to refine Kawashima’s results and to establish global existence results in some situations that were not covered before. Very recently, advances have been made by T. Crin-Barat in his PhD thesis dedicated to the partially dissipative case, and in J.-P. Adogbo’s thesis dedicated to the partially diffusive case. Compared to the previous works, the authors adopted a `critical’ Besov framework that allows not only to consider a larger class of data but also to prove by very simple scaling arguments optimal time-decay estimates and to justify the strong relaxation or diffusive limit.
In this course, we will provide the basics of the Fourier analysis and nonlinear analysis allowing to introduce the functional framework that is needed to establish the aforementioned results, then we will investigate partially dissipative or diffusive systems. Some examples of applications in fluid dynamics will be given.
Wednesday, 02 October 2024
In the recent resolution of the strong Onsager’s conjecture in L^3 framework, so-called wavelet-inspired Nash’s iteration has been developed. In this lecture, we will sketch this new method and provide the necessary background. The lecture will be based on our recent joint work with Matthew Novack.
Many physical phenomena may be modeled by first order symmetric hyperbolic equations with degenerate dissipative or diffusive terms. This is the case in gas dynamics, where the mass is conserved during the evolution, but the momentum balance includes a diffusion (viscosity) or damping (relaxation) term.
Such so-called partially dissipative or diffusive systems have been extensively studied by S .Kawashima in his PhD thesis from 1984. For a rather general class of systems he pointed out a simple necessary condition for the global existence of solutions in the vicinity of constant solutions.
This condition that is nowadays named (SK) condition has been revisited in a number of research works. In particular, K. Beauchard and E. Zuazua proposed in 2010 an explicit method for constructing a Lyapunov functional allowing to refine Kawashima’s results and to establish global existence results in some situations that were not covered before. Very recently, advances have been made by T. Crin-Barat in his PhD thesis dedicated to the partially dissipative case, and in J.-P. Adogbo’s thesis dedicated to the partially diffusive case. Compared to the previous works, the authors adopted a `critical’ Besov framework that allows not only to consider a larger class of data but also to prove by very simple scaling arguments optimal time-decay estimates and to justify the strong relaxation or diffusive limit.
In this course, we will provide the basics of the Fourier analysis and nonlinear analysis allowing to introduce the functional framework that is needed to establish the aforementioned results, then we will investigate partially dissipative or diffusive systems. Some examples of applications in fluid dynamics will be given.
The lectures will introduce perturbations of Euler's equation by highly irregular paths where the perturbations are such that the solution preserves a range of physically relevant quantities. Using formal computations, we shall see that, when d=2, a purely Lagrangian formulation of the equation seems to be within reach.
However, special care is needed to give rigorous meaning to the noisy terms of the equation and in these lectures, we will consider the framework of rough paths. We will see how the so-called 'Sewing Lemma' can be used to define integrals as Riemann sums w.r.t paths of low regularity and how to use this result to construct rough path integrals. Then we will derive very precise a priori estimates for differential equations driven by rough paths and these estimates will be used to prove well-posedness of equations where the drift term satisfies an Osgood regularity. Moreover, we will study flows generated by the differential equations and see that the flows are volume preserving under natural assumptions on the noise and vector fields.
Returning to fluid equations, we shall use the above results to prove well-posedness of the roughly perturbed Euler equation in Lagrangian form when d=2. We will consider well-posedness in the class of equations with bounded and integrable vorticity, also known as Yudovich theory.
Thursday, 03 October 2024
Many physical phenomena may be modeled by first order symmetric hyperbolic equations with degenerate dissipative or diffusive terms. This is the case in gas dynamics, where the mass is conserved during the evolution, but the momentum balance includes a diffusion (viscosity) or damping (relaxation) term.
Such so-called partially dissipative or diffusive systems have been extensively studied by S .Kawashima in his PhD thesis from 1984. For a rather general class of systems he pointed out a simple necessary condition for the global existence of solutions in the vicinity of constant solutions.
This condition that is nowadays named (SK) condition has been revisited in a number of research works. In particular, K. Beauchard and E. Zuazua proposed in 2010 an explicit method for constructing a Lyapunov functional allowing to refine Kawashima’s results and to establish global existence results in some situations that were not covered before. Very recently, advances have been made by T. Crin-Barat in his PhD thesis dedicated to the partially dissipative case, and in J.-P. Adogbo’s thesis dedicated to the partially diffusive case. Compared to the previous works, the authors adopted a `critical’ Besov framework that allows not only to consider a larger class of data but also to prove by very simple scaling arguments optimal time-decay estimates and to justify the strong relaxation or diffusive limit.
In this course, we will provide the basics of the Fourier analysis and nonlinear analysis allowing to introduce the functional framework that is needed to establish the aforementioned results, then we will investigate partially dissipative or diffusive systems. Some examples of applications in fluid dynamics will be given.
In this lecture I will cover the developments in the area which followed our work and finally lead Isett to the resolution of the Onsager conjecture. If time allows I will give a glimpse of the challenges that lie ahead if one would like to give a rigorous justification that some dissipative solutions of the Euler equations are indeed limit of classical solutions of Navier-Stokes.
In the recent resolution of the strong Onsager’s conjecture in L^3 framework, so-called wavelet-inspired Nash’s iteration has been developed. In this lecture, we will sketch this new method and provide the necessary background. The lecture will be based on our recent joint work with Matthew Novack.
The flow associated to a given smooth vector field $u(t,x)$ is well-defined, it is smooth, and it can be used to solve the classical linear PDEs associated to $u$, namely the transport and the continuity equations. The situation is substantially different when $u$ is not smooth (e.g. when $u$ is not Lipschitz continuous in the space variable, but only Sobolev or BV). Goal of the lecture is to provide an overview on classical and recent results about flow maps and PDEs associated to non-smooth vector fields.
Friday, 04 October 2024
Many physical phenomena may be modeled by first order symmetric hyperbolic equations with degenerate dissipative or diffusive terms. This is the case in gas dynamics, where the mass is conserved during the evolution, but the momentum balance includes a diffusion (viscosity) or damping (relaxation) term.
Such so-called partially dissipative or diffusive systems have been extensively studied by S .Kawashima in his PhD thesis from 1984. For a rather general class of systems he pointed out a simple necessary condition for the global existence of solutions in the vicinity of constant solutions.
This condition that is nowadays named (SK) condition has been revisited in a number of research works. In particular, K. Beauchard and E. Zuazua proposed in 2010 an explicit method for constructing a Lyapunov functional allowing to refine Kawashima’s results and to establish global existence results in some situations that were not covered before. Very recently, advances have been made by T. Crin-Barat in his PhD thesis dedicated to the partially dissipative case, and in J.-P. Adogbo’s thesis dedicated to the partially diffusive case. Compared to the previous works, the authors adopted a `critical’ Besov framework that allows not only to consider a larger class of data but also to prove by very simple scaling arguments optimal time-decay estimates and to justify the strong relaxation or diffusive limit.
In this course, we will provide the basics of the Fourier analysis and nonlinear analysis allowing to introduce the functional framework that is needed to establish the aforementioned results, then we will investigate partially dissipative or diffusive systems. Some examples of applications in fluid dynamics will be given.
The lectures will introduce perturbations of Euler's equation by highly irregular paths where the perturbations are such that the solution preserves a range of physically relevant quantities. Using formal computations, we shall see that, when d=2, a purely Lagrangian formulation of the equation seems to be within reach.
However, special care is needed to give rigorous meaning to the noisy terms of the equation and in these lectures, we will consider the framework of rough paths. We will see how the so-called 'Sewing Lemma' can be used to define integrals as Riemann sums w.r.t paths of low regularity and how to use this result to construct rough path integrals. Then we will derive very precise a priori estimates for differential equations driven by rough paths and these estimates will be used to prove well-posedness of equations where the drift term satisfies an Osgood regularity. Moreover, we will study flows generated by the differential equations and see that the flows are volume preserving under natural assumptions on the noise and vector fields.
Returning to fluid equations, we shall use the above results to prove well-posedness of the roughly perturbed Euler equation in Lagrangian form when d=2. We will consider well-posedness in the class of equations with bounded and integrable vorticity, also known as Yudovich theory.
The lectures will introduce perturbations of Euler's equation by highly irregular paths where the perturbations are such that the solution preserves a range of physically relevant quantities. Using formal computations, we shall see that, when d=2, a purely Lagrangian formulation of the equation seems to be within reach.
However, special care is needed to give rigorous meaning to the noisy terms of the equation and in these lectures, we will consider the framework of rough paths. We will see how the so-called 'Sewing Lemma' can be used to define integrals as Riemann sums w.r.t paths of low regularity and how to use this result to construct rough path integrals. Then we will derive very precise a priori estimates for differential equations driven by rough paths and these estimates will be used to prove well-posedness of equations where the drift term satisfies an Osgood regularity. Moreover, we will study flows generated by the differential equations and see that the flows are volume preserving under natural assumptions on the noise and vector fields.
Returning to fluid equations, we shall use the above results to prove well-posedness of the roughly perturbed Euler equation in Lagrangian form when d=2. We will consider well-posedness in the class of equations with bounded and integrable vorticity, also known as Yudovich theory.