##
**Perfect incompressible fluids. Transl. from the French by Isabelle Gallagher and Dragos Iftimie.**
*(English)*
Zbl 0927.76002

Oxford Lecture Series in Mathematics and its Applications. 14. Oxford: Clarendon Press. x, 187 p. (1998).

The book presents recent results on the Cauchy problem for the Euler equations of a perfect incompressible fluid. A local uniqueness and existence theorem is proved in a Hölder space \(C^r ,\) \(r>1\), when the initial data are Hölder continuous. In the two-dimensional case this theorem becomes global in the following cases: the case of periodic initial data, the case when the gradient of initial velocity belongs to \(L^p\), \(p>1\), and finally, the case when initial velocity is a bounded energy perturbation of a stationary solution. All the other results of the book are also obtained in the two-dimensional case, whose advantage is the conservation of vorticity along the flow of the vector field solution.

The Yudovich global existence theorem is proved for solutions with bounded vorticity. Though the velocity field is not Lipschitzian, it is quasi-Lipschitzian, and the corresponding flow is shown to be associated with such a velocity field, with the regularity in the space variables of this flow decreasing exponentially with time. An example is given indicating that the exponential law is really achieved.

The Yudovich theorem gives rise to the vortex patch problem describing the evolution of the vortex which is initially a characteristic function of an open bounded set with a boundary of a Hölder class. A smoothness propagation theorem is proved for a closed curve encircling the vortex patch. Then the vortex sheet problem is formulated for the initial vorticity being a measure of length of a compact curve of class \(C^1\). Here, the Delort theorem is proved about the existence of a global solution when the initial vorticity is a bounded measure with positive singular part. It is shown that, at each moment, the vorticity \(w_t\) is also a bounded measure with bounded singular part, whose total mass is less than or equal to that of \(w_0\).

The Sobolev and Hölder spaces used in the book are defined on the basis of the Littlewood-Paley theory through the Fourier transform and the decomposition of the frequency space into dyadic rings. Such a technique was introduced in the study of hyperbolic nonlinear partial differential equations by J.-M. Bony [Ann. Sci. Éc. Norm. Supér. 14, 209-246 (1981)]. Particularly, the definitions give sense to a Hölder space \(C^r\) with negative real number \(r\), which finds applications in the proof of the persistence of tangential regularity of the curve encircling the vortex patch. This persistence result is obtained with the use of the J.-M. Bony concept of tangential smoothness along a system of vector fields with weak regularity; an important inequality relating to the action of the operators \(\partial _i \partial _j \Delta ^{-1}\) on the family of bounded functions is proved here also. Some results are obtained for the vortex patch problem when the boundary curve of the patch presents singularities, for example corners or cusps. It is proved that the smoothness of the curve persists exept at the singular points stemming from those existing initially. To study the regularity of the flow over time, the wave front theory is applied. To this end an elementary presentation of fundamental concepts of microlocal analysis is done, including the relationship between the wave front, the Littlewood-Paley theory, and the product of distributions.

The book is mainly devoted to nonclassical linear differential equations which describe waves in continua with strong dispersion. The study of such equations of an order greater than two goes back to Sobolev who addressed rotating flows in 1954. Applications include water waves with inertial surface, plasma, geophysics, acoustics, etc. Each equation is preceded by derivation and corresponds to waves of small amplitude. As a rule, a fundamental solution is constructed, and then a mathematical theory is developed for boundary value problems. In conclusion, a physical interpretation is given.

The first chapters concern with water waves with an inertial surface when the density of floating particles is prescribed. Here, the main result is the solution of the steady wave problem for a nonlinear equation of the Nekrasov type describing finite waves. This result is obtained by the Lyapunov-Schmidt method. The linearized equation is also studied with applications to the theory of ship waves. Particularly, it is found how the Kelvin wedge depends on the density of floating particles. One more result of the linear theory is that the frequency of steady propagating waves cannot be very high: floatation serves as a cutting of high frequencies.

The chapter about plasma deals with the equation \((\Delta u-u)_{tt}+\) \(\Delta u=0\) which describes linear ion-sound waves. It is proved that this equation permits a quasi-front, a special wave front which carries a small precursor with a set of oscillations behind. The same quasi-front phenomenon is also found for the linear equations of a stratified incompressible fluid. Nonlinear internal waves in such a fluid are studied on the basis of the Dubreil-Jacotin equation, particularly, a solution is found to describe flows with parabolic trajectories of fluid particles. A two-front phenomenon is discovered for the equations of compressible stratified fluids. The first front is acoustic and the second one is a gravitational quasi-front. In the last chapter, the complete mathematical study is also given of nonclassical equations governing planetary waves, capillary waves on shallow water, and waves in rotating fluids.

The Yudovich global existence theorem is proved for solutions with bounded vorticity. Though the velocity field is not Lipschitzian, it is quasi-Lipschitzian, and the corresponding flow is shown to be associated with such a velocity field, with the regularity in the space variables of this flow decreasing exponentially with time. An example is given indicating that the exponential law is really achieved.

The Yudovich theorem gives rise to the vortex patch problem describing the evolution of the vortex which is initially a characteristic function of an open bounded set with a boundary of a Hölder class. A smoothness propagation theorem is proved for a closed curve encircling the vortex patch. Then the vortex sheet problem is formulated for the initial vorticity being a measure of length of a compact curve of class \(C^1\). Here, the Delort theorem is proved about the existence of a global solution when the initial vorticity is a bounded measure with positive singular part. It is shown that, at each moment, the vorticity \(w_t\) is also a bounded measure with bounded singular part, whose total mass is less than or equal to that of \(w_0\).

The Sobolev and Hölder spaces used in the book are defined on the basis of the Littlewood-Paley theory through the Fourier transform and the decomposition of the frequency space into dyadic rings. Such a technique was introduced in the study of hyperbolic nonlinear partial differential equations by J.-M. Bony [Ann. Sci. Éc. Norm. Supér. 14, 209-246 (1981)]. Particularly, the definitions give sense to a Hölder space \(C^r\) with negative real number \(r\), which finds applications in the proof of the persistence of tangential regularity of the curve encircling the vortex patch. This persistence result is obtained with the use of the J.-M. Bony concept of tangential smoothness along a system of vector fields with weak regularity; an important inequality relating to the action of the operators \(\partial _i \partial _j \Delta ^{-1}\) on the family of bounded functions is proved here also. Some results are obtained for the vortex patch problem when the boundary curve of the patch presents singularities, for example corners or cusps. It is proved that the smoothness of the curve persists exept at the singular points stemming from those existing initially. To study the regularity of the flow over time, the wave front theory is applied. To this end an elementary presentation of fundamental concepts of microlocal analysis is done, including the relationship between the wave front, the Littlewood-Paley theory, and the product of distributions.

The book is mainly devoted to nonclassical linear differential equations which describe waves in continua with strong dispersion. The study of such equations of an order greater than two goes back to Sobolev who addressed rotating flows in 1954. Applications include water waves with inertial surface, plasma, geophysics, acoustics, etc. Each equation is preceded by derivation and corresponds to waves of small amplitude. As a rule, a fundamental solution is constructed, and then a mathematical theory is developed for boundary value problems. In conclusion, a physical interpretation is given.

The first chapters concern with water waves with an inertial surface when the density of floating particles is prescribed. Here, the main result is the solution of the steady wave problem for a nonlinear equation of the Nekrasov type describing finite waves. This result is obtained by the Lyapunov-Schmidt method. The linearized equation is also studied with applications to the theory of ship waves. Particularly, it is found how the Kelvin wedge depends on the density of floating particles. One more result of the linear theory is that the frequency of steady propagating waves cannot be very high: floatation serves as a cutting of high frequencies.

The chapter about plasma deals with the equation \((\Delta u-u)_{tt}+\) \(\Delta u=0\) which describes linear ion-sound waves. It is proved that this equation permits a quasi-front, a special wave front which carries a small precursor with a set of oscillations behind. The same quasi-front phenomenon is also found for the linear equations of a stratified incompressible fluid. Nonlinear internal waves in such a fluid are studied on the basis of the Dubreil-Jacotin equation, particularly, a solution is found to describe flows with parabolic trajectories of fluid particles. A two-front phenomenon is discovered for the equations of compressible stratified fluids. The first front is acoustic and the second one is a gravitational quasi-front. In the last chapter, the complete mathematical study is also given of nonclassical equations governing planetary waves, capillary waves on shallow water, and waves in rotating fluids.

Reviewer: Vladimir Shelukhin (Rio de Janeiro)

### MSC:

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

76Bxx | Incompressible inviscid fluids |

35Q35 | PDEs in connection with fluid mechanics |

76X05 | Ionized gas flow in electromagnetic fields; plasmic flow |