c Russian Math . Surveys 62 :3 409–451 ⃝ 2007 RAS(DoM) and LMS Uspekhi Mat . Nauk 62 :3 5–46 DOI 10.1070/RM2007v062n03ABEH004410 Euler equations for incompressible ideal fluids C. Bardos and E. S. Titi Abstract. This article is a survey concerning the state-of-the-art mathe- matical theory of the Euler equations for an incompressible homogeneous ideal fluid. Emphasis is put on the different types of emerging instability, and how they may be related to the description of turbulence. Contents 1. Introduction 410 2. Classical existence and regularity results 413 2.1. Introduction 413 2.2. General results in 3 D 413 2.3. About the two-dimensional case 417 3. Pathological behaviour of solutions 417 4. Weak limit of solutions of the Navier–Stokes dynamics 421 4.1. Reynolds stress tensor 421 4.2. Dissipative solutions of the Euler equations 424 5. No-slip Dirichlet boundary conditions for the Navier–Stokes dynamics 427 6. Deterministic and statistical spectrum of turbulence 431 6.1. Deterministic spectrum and Wigner transform 431 6.2. The energy spectrum in the statistical theory of turbulence 433 6.3. Comparison between deterministic and statistical spectra 436 7. Prandtl and Kelvin–Helmholtz problems 438 7.1. The Prandtl boundary layer 439 7.2. The Kelvin–Helmholtz problem 441 7.2.1. Local solution 444 7.2.2. Singularities 445 7.2.3. Analyticity and pathological behaviour after the breakdown of regularity 446 Bibliography 448 This work was supported in part by NSF grant no. DMS-0504619, BSF grant no. 2004271, and ISF grant no. 120/06. AMS 2000 Mathematics Subject Classification . Primary 35Q05; Secondary 35Q30, 76Bxx, 76Dxx, 76Fxx.
410 C. Bardos and E. S. Titi 1. Introduction This contribution is mostly devoted to a time-dependent analysis of the 2 D and 3 D Euler equations ∂ t u + ∇ · ( u ⊗ u ) + ∇ p = 0 , ∇ · u = 0 (1) of an incompressible homogeneous ideal fluid. We intend to connect several known (and maybe less known) points of view concerning this very classical problem. Furthermore, we will investigate the conditions under which one can consider the above problem as the limit of the incompressible Navier–Stokes equations ∂ t u ν + ∇ · ( u ν ⊗ u ν ) − ν ∆ u ν + ∇ p ν = 0 , ∇ · u ν = 0 , (2) when the viscosity ν goes to zero, that is, as the Reynolds number goes to infinity. At the macroscopic level the Reynolds number Re corresponds to the ratio of the strength of the non-linear effects to the strength of the linear viscous effects . Therefore, with the introduction of a characteristic velocity U and a characteristic length scale L of the flow one has the dimensionless parameter Re = UL . (3) ν With the introduction of the characteristic time scale T = L/U and the dimension- less variables, u ′ = u ′ x ′ = x t ′ = t L, T , and U , the Navier–Stokes equations (2) take the non-dimensional form: ∂ t u ′ + ∇ x ′ · ( u ′ ⊗ u ′ ) − 1 Re∆ x ′ u ′ + ∇ x ′ p ′ = 0 , ∇ · u ′ = 0 . (4) These are the equations to be considered below, omitting the prime ( ′ ) and returning to the notation ν for Re − 1 . In the presence of a physical boundary the problems (1) and (2) will be considered in an open domain Ω ⊂ R d , d = 2, d = 3, with a piecewise smooth boundary ∂ Ω. There are several good reasons to focus at present on the ‘mathematical analysis’ of the Euler equations rather than the Navier–Stokes equations. 1. Turbulence applications involving the Navier–Stokes equations (4) often cor- respond to very large Reynolds numbers; and a theorem which is valid for any finite, but very large, Reynolds number is expected to be compatible with results concerning infinite Reynolds number. In fact, this is the case when Re = ∞ , which drives other results, and we will give several examples of this fact. 2. Many non-trivial and sharp results for the incompressible Navier–Stokes equa- tions rely on the smoothing effect of the Laplacian when the viscosity ν is > 0, and on the invariance of the set of solutions under the scaling u ( x, t ) �→ λu ( λx, λ 2 t ) . (5) However, simple examples with the same scalings but without an energy conser- vation law may exhibit very different behaviour concerning regularity and stability.
Euler equations for incompressible ideal fluids 411 1. With φ a scalar function, the viscous Hamilton–Jacobi type or Burgers equation ∂ t φ − ν ∆ φ + 1 2 |∇ φ | 2 = 0 Ω × R + in t , (6) φ ( · , 0) = φ 0 ( · ) ∈ L ∞ (Ω) , φ ( x, t ) = 0 for x ∈ ∂ Ω , and has (because of the maximum principle) a global smooth solution for any ν > 0. However, for ν = 0 it is well known that certain solutions of the inviscid Burgers equation (6) will become singular (with shocks) in finite time. 2. Denote by |∇| the square root of the operator − ∆, defined in Ω and with Dirichlet homogeneous boundary conditions. Let us consider the solution u ( x, t ) of the equation ∂ t u − ν ∆ u + 1 Ω × R + 2 |∇| ( u 2 ) = 0 in t , (7) u ( · , 0) = u 0 ( · ) ∈ L ∞ (Ω) . u ( x, t ) = 0 for x ∈ ∂ Ω , and (8) Then one has the following proposition. Proposition 1.1. Assume that the initial data u 0 satisfies the relation � u 0 ( x ) φ 1 ( x ) dx = − M < 0 , (9) Ω where φ 1 ( x ) � 0 denotes the first eigenfunction of the operator − ∆ ( with Dirichlet boundary condition ), − ∆ φ 1 = λ 1 φ 1 . If M is sufficiently large , then the correspond- ing solution u ( x, t ) of the system (7), (8) blows up in a finite time . Proof . The L 2 scalar product of the equation (7) with φ 1 ( x ) gives √ λ 1 � � � d u ( x, t ) 2 φ 1 ( x ) dx. u ( x, t ) φ 1 ( x ) dx + νλ 1 u ( x, t ) φ 1 ( x ) dx = − dt 2 Ω Ω Ω Since φ 1 ( x ) � 0, the Cauchy–Schwarz inequality implies that � � � 2 � � �� � � u ( x, t ) 2 φ 1 ( x ) dx u ( x, t ) φ 1 ( x ) dx φ 1 ( x ) dx . � Ω Ω � As a result, the quantity m ( t ) = − u ( x, t ) φ 1 ( x ) dx satisfies the relation Ω � √ λ 1 φ 1 ( x ) dx dm m 2 Ω dt + λ 1 m � with m (0) = M, 2 and the conclusion of the proposition follows. Remark 1.1. The above example was given with Ω = R 3 by Montgomery-Smith [1] (under the name ‘cheap Navier–Stokes equations’) with the purpose of revealing the role of the conservation of energy (which is not present in the above examples)
412 C. Bardos and E. S. Titi in the Navier–Stokes dynamics. His proof shows that the same blowup property may appear in any space dimension for the solution of the ‘cheap hyperviscosity equations’ ∂ t u + ν ( − ∆) m u + 1 2 |∇| ( u 2 ) = 0 . On the other hand, one should observe that the above argument does not apply to the Kuramoto–Sivashinsky-like equations ∂ t φ + ν ( − ∆) m φ + α ∆ φ + 1 2 |∇ φ | 2 = 0 (10) for m � 2. Without a maximum principle or without a control of some sort on the energy, the question of global existence of a smooth solution or finite-time blowup of some solution of the above equation is an open problem in Ω = R n for n � 2 and for m � 2. However, if in (10) the term |∇ φ | 2 is replaced by |∇ φ | 2+ γ , γ > 0, then one can prove the blowup of some solutions (cf. [2] and references therein). In conclusion, the above examples indicate that the conservation of some sort of energy, which is guaranteed by the structure of the equation, is essential in the analysis of the dynamics of the underlying problem. In particular, this very basic fact plays an essential role in the dynamics of the Euler equations. Taking into account the above simple examples, the rest of the paper is organized as follows. In § 2 classical existence and regularity results for the time-dependent Euler equations are presented. § 3 provides more examples concerning the patho- logical behaviour of solutions of the Euler equations. The fact that the solutions of the Euler equations may exhibit oscillatory behaviour implies similar behaviour for the solutions of the Navier–Stokes equations as the viscosity tends to zero. The existence of (or lack thereof) strong convergence is analyzed in § 4 with the introduction of the Reynolds stress tensor and the notion of dissipative solution . A standard and very important problem for both theoretical study and applications is the vanishing-viscosity limit of solutions of the Navier–Stokes equations subject to the no-slip Dirichlet boundary condition in domains with physical boundaries. Very few mathematical results are available for this very unstable situation. One of the most striking results is a theorem of Kato [3], which is presented in § 5. § 6 is again devoted to the Reynolds stress tensor. We show that with the intro- duction of the Wigner measure the notion of Reynolds stress tensor, deduced from the defect in strong convergence as the viscosity tends to zero, plays the same role as the one originally introduced in the statistical theory of turbulence. When the zero viscosity limit of solutions of the Navier–Stokes equations is compared with the solution of the Euler equations, the main difference appears in a boundary layer which is described by the Prandtl equations. These equations are briefly described in § 7. There it is also recalled how the mathematical results are in agree- ment with the instability of the physical problem. The Kelvin–Helmholtz problem also exhibits some similar basic instabilities, but it is in some sense simpler. This is explained at the end of § 7, where it is also shown that some recent results of [4], [5], [6] on the regularity of the vortex sheet (interface) do contribute to an understanding of the instabilities of the original problem.
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