Vorticity and incompressible flow.

*(English)*Zbl 0983.76001
Cambridge Texts in Applied Mathematics. Cambridge: Cambridge University Press (ISBN 0-521-63057-6/hbk; 0-521-63948-4/pbk). xii, 545 p. (2002).

The book is a comprehensive introduction to the mathematical theory of vorticity and incompressible flows, ranging from elementary introductory material to current research topics. In the first two chapters the Euler equations (for ideal fluids) and the Navier-Stokes equations (for viscous fluids) are formulated, and many their properties are described, including exact solutions with shear, vorticity, convection and diffusion. The vorticity equation is derived to show that, for inviscid flows, the vorticity is transported and stretched along particle trajectories for three-dimensional flows and is conserved along particle paths for two-dimensional flows. The vorticity equation allows for different reformulations of Euler and Navier-Stokes equations, including vorticity-stream formulation for two-dimensional flows and an integro-differential equation for particle trajectories.

In the next two chapters the existence, uniqueness, and continuation of local-in-time smooth solutions of Euler and Navier-Stokes equations are addressed. The energy method is shown to be general enough to be applied to both equations; the particle-trajectory method is performed for the Euler equations. One of the basic results here is a precise characterization of the maximal interval \([0,T_{\ast})\) of existence of a smooth solution in terms of the accumulation of vorticity: \(T_{\ast}\) provides maximal interval if and only if the vorticity \(\omega\) accumulates so rapidly that \( \int_{0}^{T}\max_{x}|\omega (t,x)|dt\to\infty\) as \(T\to T_{\ast}\). The authors also present some other equivalent criteria. Particularly, it is proved that global existence holds for two-dimensional Euler equations and for three-dimensional Euler equations if in the latter case no vortex stretching occurs or provided that the maximum of vorticity is controlled.

A special chapter is devoted to unsolved problem on singularities developing in a finite time for the three-dimensional Euler equations. In searching for singular solutions, the authors consider several simpler analogy models that retain some of the features of the three-dimensional Euler equations. It is shown that the model vorticity equation \(\omega _t =H(\omega)\omega\), \(H\) being the Hilbert transform, has a smooth solution that blows up in finite time. Another simplified model is the two-dimensional quasi-geostrophic active scalar equations \(\theta _t +\) \(v\cdot\nabla\theta =0,\) \(v=\nabla ^{\perp}\psi ,\) \((-\Delta)^{1/2}\psi =\theta\), where \((\partial /\partial x_1 ,\partial /\partial x_2)^{\perp}=\) \((-\partial /\partial x_2 ,\partial /\partial x_1)\) . The authors prove that, if the direction field \(\nabla ^{\perp}\theta /|\nabla ^{\perp}\theta|\) remains smooth in a suitably weak sense, then no singularity is possible.

There is also a chapter devoted to computational vortex methods. The inviscid vortex method (\(\nu =0\)) approximates the vorticity by a finite number of radially symmetric blobs and then advects the blobs by their own velocity field. The random vortex method is used for viscous Navier-Stokes equations; the idea is to implement the inviscid vortex method coupled with a Brownian motion of vortex blobs to simulate diffusion. Convergence theorems are proved for both vortex and random-vortex methods, and numerical results are presented.

Next, asymptotic expansion methods are used to study the motion of slender tubes of vorticity at high Reynolds numbers, where simplified asymptotic equations allow to describe folding, wrinkling, and bending of vortex tubes. The key technical idea is to expand the Biot-Savart law determining the velocity \(v\) from the vorticity \(\omega\) in a suitable asymptotic expansion, provide that the vorticity \(\omega\) is large and confined to a narrow tube around a curve \(X(s,t)\) that gives the centerline. In this way, the authors derive simplified dynamical equations for the motion of the curve \(X(s,t)\) in various asymptotic regimes as well as equations for the interaction of many such strong vortex filaments represented by a finite collection of such curves \(X_j (s,t)\), \(1\leq j\leq N\). The simplest asymptotic theory is the self-induction approximation for the motion of a single vortex filament. By Hasimoto transformation, these simplified vortex dynamics reduce to the cubic nonlinear Schrödinger equation, a well-known completely integrable PDE with soliton behavior and heteroclinic instabilities. The case of finite number of parallel vortex filaments reduces to the interaction of point vorteces in the plane. The authors show that the interaction of nearly parallel vortex filaments has properties that incorporate features of both self-induction and mutual point-vortex interaction.

The final chapters deal with mathematical issues related to non-smooth solutions of two-dimensional Euler equations. The first class of weak solutions with vorticity \(\omega (t)\) in \(L^{\infty}(\mathbb{R}^2)\cap \) \(L^1 (\mathbb{R}^2)\) is appropriate for modeling an isolated region of intense vorticity such as what one might use to model the evolution of a hurricane. The prototypical example is the vortex patch that has vorticity localized to a bounded region in the plane. The proof of unique solvability (first established by V. I. Yudovich in [U.S.S.R. Comput. Math. Math. Phys. 3, 1407-1456 (1963; Zbl 0147.44303)]) is performed, and a class of exact solutions is discussed. The particle paths are proved to be Hölder continuous with a decay in the time exponent. The authors also derive the contour dynamics equation (CDE) for the evolution of the boundary of the patch in which the vorticity is a constant multiplied by a characteristic function of a time-evolving domain. This CDE equation is used to prove that the boundary, if initially smooth, stays smooth for all time.

More singular weak solutions to Euler equations arise in modeling flows separating from rigid walls and sharp corners. In such flows the velocity changes sign discontinuously across a streamline. A velocity discontinuity in an inviscid flow is called a vortex sheet. Unlike the vortex patch, in which \(\omega\) is pointwise bounded, a vortex sheet has vorticity concentrated as a measure along a surface of codimension one. In two space dimensions, this is a curve in the plane. The authors derive a self-deforming curve equation, called Birkhoff-Rott (B-R) equation, for the evolution of the sheet. This equation is analogous to the CDE for the evolution of the boundary of vortex patch. It is shown that, unlike CDE, the B-R equation is linearly ill-posed, and can be solved in general only on short time intervals even for analytic initial data.

To attack the vortex sheet problem, the authors introduce a notion of approximate-solution sequences (AS-sequences) for the two-dimensional Euler equations, keeping in mind that the actual fluid possesses some residual viscosity, and that complex structures are usually smoothed on a very fine scale. This notion serves to capture the essence of the dynamics in a limiting process, since in order to construct a solution with initial vorticity \(\omega _0 (x)\) being a measure, one must approximate the weak solution by a family of smooth solutions and pass to the limit in approximating parameter. With \(v\) denoting the velocity, the principal requirement in the definition of AS-sequence \(v^{\varepsilon}\) is \( \int_{0}^{T}\int_{\mathbb{R}^2}(v^{\varepsilon}\Phi _t + \nabla \Phi :v^{\varepsilon}\bigotimes v^{\varepsilon}) dx dt\to 0\) as \(\varepsilon\to 0,\) where \(\Phi (t,x)\) is a test function. An AS-sequence \(v^{\varepsilon}\) is a sequence with oscillation if there is no subsequence that converges pointwise almost everywhere, but nonetheless \(v^{\varepsilon}\) converges weakly in \(L^2\). If for all subsequences strong convergence in \(L^2\) does not occur, the concentration effect occurs. The authors give several examples for which concentration or oscillation occur and yet the limit still satisfies the Euler equations. For an AS-sequence \(v^{\varepsilon}\) with \(L^1\)-vorticity control, if \(\|\omega\|_{L^{\infty}(0,T)}\) and \(\|\omega\|_{L^1 (\mathbb{R}^2)}\) are bounded, it is proved that \(v^{\varepsilon}\to v\) in \(L_{\text{loc}}^{1}\). This directly implies the existence of a weak solution to two-dimensional Euler equations with initial vorticity in \(L^1 \cap L^p\), \(p>1\). If \(\omega _0\) is a measure, an existence theorem is proved only in the case \(\omega _0 \geq 0\) or \(\omega _0\leq 0\) (Delort’s theorem).

To tackle the three-dimensional Euler equations, the authors apply the notion of measure solution. The existence of such a solution is proved even when concentrations and oscillations do take place. The last chapter involves a theoretical and computational study of one-dimensional Vlasov-Poisson equations, which serve as a simplified model in which many of the unresolved issues for weak solutions of Euler equations can be answered in an explicit fashion.

In the next two chapters the existence, uniqueness, and continuation of local-in-time smooth solutions of Euler and Navier-Stokes equations are addressed. The energy method is shown to be general enough to be applied to both equations; the particle-trajectory method is performed for the Euler equations. One of the basic results here is a precise characterization of the maximal interval \([0,T_{\ast})\) of existence of a smooth solution in terms of the accumulation of vorticity: \(T_{\ast}\) provides maximal interval if and only if the vorticity \(\omega\) accumulates so rapidly that \( \int_{0}^{T}\max_{x}|\omega (t,x)|dt\to\infty\) as \(T\to T_{\ast}\). The authors also present some other equivalent criteria. Particularly, it is proved that global existence holds for two-dimensional Euler equations and for three-dimensional Euler equations if in the latter case no vortex stretching occurs or provided that the maximum of vorticity is controlled.

A special chapter is devoted to unsolved problem on singularities developing in a finite time for the three-dimensional Euler equations. In searching for singular solutions, the authors consider several simpler analogy models that retain some of the features of the three-dimensional Euler equations. It is shown that the model vorticity equation \(\omega _t =H(\omega)\omega\), \(H\) being the Hilbert transform, has a smooth solution that blows up in finite time. Another simplified model is the two-dimensional quasi-geostrophic active scalar equations \(\theta _t +\) \(v\cdot\nabla\theta =0,\) \(v=\nabla ^{\perp}\psi ,\) \((-\Delta)^{1/2}\psi =\theta\), where \((\partial /\partial x_1 ,\partial /\partial x_2)^{\perp}=\) \((-\partial /\partial x_2 ,\partial /\partial x_1)\) . The authors prove that, if the direction field \(\nabla ^{\perp}\theta /|\nabla ^{\perp}\theta|\) remains smooth in a suitably weak sense, then no singularity is possible.

There is also a chapter devoted to computational vortex methods. The inviscid vortex method (\(\nu =0\)) approximates the vorticity by a finite number of radially symmetric blobs and then advects the blobs by their own velocity field. The random vortex method is used for viscous Navier-Stokes equations; the idea is to implement the inviscid vortex method coupled with a Brownian motion of vortex blobs to simulate diffusion. Convergence theorems are proved for both vortex and random-vortex methods, and numerical results are presented.

Next, asymptotic expansion methods are used to study the motion of slender tubes of vorticity at high Reynolds numbers, where simplified asymptotic equations allow to describe folding, wrinkling, and bending of vortex tubes. The key technical idea is to expand the Biot-Savart law determining the velocity \(v\) from the vorticity \(\omega\) in a suitable asymptotic expansion, provide that the vorticity \(\omega\) is large and confined to a narrow tube around a curve \(X(s,t)\) that gives the centerline. In this way, the authors derive simplified dynamical equations for the motion of the curve \(X(s,t)\) in various asymptotic regimes as well as equations for the interaction of many such strong vortex filaments represented by a finite collection of such curves \(X_j (s,t)\), \(1\leq j\leq N\). The simplest asymptotic theory is the self-induction approximation for the motion of a single vortex filament. By Hasimoto transformation, these simplified vortex dynamics reduce to the cubic nonlinear Schrödinger equation, a well-known completely integrable PDE with soliton behavior and heteroclinic instabilities. The case of finite number of parallel vortex filaments reduces to the interaction of point vorteces in the plane. The authors show that the interaction of nearly parallel vortex filaments has properties that incorporate features of both self-induction and mutual point-vortex interaction.

The final chapters deal with mathematical issues related to non-smooth solutions of two-dimensional Euler equations. The first class of weak solutions with vorticity \(\omega (t)\) in \(L^{\infty}(\mathbb{R}^2)\cap \) \(L^1 (\mathbb{R}^2)\) is appropriate for modeling an isolated region of intense vorticity such as what one might use to model the evolution of a hurricane. The prototypical example is the vortex patch that has vorticity localized to a bounded region in the plane. The proof of unique solvability (first established by V. I. Yudovich in [U.S.S.R. Comput. Math. Math. Phys. 3, 1407-1456 (1963; Zbl 0147.44303)]) is performed, and a class of exact solutions is discussed. The particle paths are proved to be Hölder continuous with a decay in the time exponent. The authors also derive the contour dynamics equation (CDE) for the evolution of the boundary of the patch in which the vorticity is a constant multiplied by a characteristic function of a time-evolving domain. This CDE equation is used to prove that the boundary, if initially smooth, stays smooth for all time.

More singular weak solutions to Euler equations arise in modeling flows separating from rigid walls and sharp corners. In such flows the velocity changes sign discontinuously across a streamline. A velocity discontinuity in an inviscid flow is called a vortex sheet. Unlike the vortex patch, in which \(\omega\) is pointwise bounded, a vortex sheet has vorticity concentrated as a measure along a surface of codimension one. In two space dimensions, this is a curve in the plane. The authors derive a self-deforming curve equation, called Birkhoff-Rott (B-R) equation, for the evolution of the sheet. This equation is analogous to the CDE for the evolution of the boundary of vortex patch. It is shown that, unlike CDE, the B-R equation is linearly ill-posed, and can be solved in general only on short time intervals even for analytic initial data.

To attack the vortex sheet problem, the authors introduce a notion of approximate-solution sequences (AS-sequences) for the two-dimensional Euler equations, keeping in mind that the actual fluid possesses some residual viscosity, and that complex structures are usually smoothed on a very fine scale. This notion serves to capture the essence of the dynamics in a limiting process, since in order to construct a solution with initial vorticity \(\omega _0 (x)\) being a measure, one must approximate the weak solution by a family of smooth solutions and pass to the limit in approximating parameter. With \(v\) denoting the velocity, the principal requirement in the definition of AS-sequence \(v^{\varepsilon}\) is \( \int_{0}^{T}\int_{\mathbb{R}^2}(v^{\varepsilon}\Phi _t + \nabla \Phi :v^{\varepsilon}\bigotimes v^{\varepsilon}) dx dt\to 0\) as \(\varepsilon\to 0,\) where \(\Phi (t,x)\) is a test function. An AS-sequence \(v^{\varepsilon}\) is a sequence with oscillation if there is no subsequence that converges pointwise almost everywhere, but nonetheless \(v^{\varepsilon}\) converges weakly in \(L^2\). If for all subsequences strong convergence in \(L^2\) does not occur, the concentration effect occurs. The authors give several examples for which concentration or oscillation occur and yet the limit still satisfies the Euler equations. For an AS-sequence \(v^{\varepsilon}\) with \(L^1\)-vorticity control, if \(\|\omega\|_{L^{\infty}(0,T)}\) and \(\|\omega\|_{L^1 (\mathbb{R}^2)}\) are bounded, it is proved that \(v^{\varepsilon}\to v\) in \(L_{\text{loc}}^{1}\). This directly implies the existence of a weak solution to two-dimensional Euler equations with initial vorticity in \(L^1 \cap L^p\), \(p>1\). If \(\omega _0\) is a measure, an existence theorem is proved only in the case \(\omega _0 \geq 0\) or \(\omega _0\leq 0\) (Delort’s theorem).

To tackle the three-dimensional Euler equations, the authors apply the notion of measure solution. The existence of such a solution is proved even when concentrations and oscillations do take place. The last chapter involves a theoretical and computational study of one-dimensional Vlasov-Poisson equations, which serve as a simplified model in which many of the unresolved issues for weak solutions of Euler equations can be answered in an explicit fashion.

Reviewer: Vladimir Shelukhin (Novosibirsk)

##### MSC:

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

76B03 | Existence, uniqueness, and regularity theory for incompressible inviscid fluids |

76B47 | Vortex flows for incompressible inviscid fluids |

76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |

76M23 | Vortex methods applied to problems in fluid mechanics |

35R35 | Free boundary problems for PDEs |

35Q30 | Navier-Stokes equations |