# Recent Advances in Hydrodynamics (16w5102)

Arriving in Banff, Alberta Sunday, June 5 and departing Friday June 10, 2016

## Organizers

Anna Mazzucato (Pennsylvania State University)

James Kelliher (University of California, Riverside)

Magdalena Czubak (State University of New York at Binghamton)

Slim Ibrahim (University of Victoria)

John Bowman (University of Alberta)

## Objectives

The main objective of the workshop is to provide a forum for an international mix of senior and young researchers, including advanced graduate students, in the field of incompressible fluid mechanics to share their research. The second objective is to bring together participants with complementary scientific background so as to facilitate an exchange of ideas and techniques between groups that might not otherwise have opportunities to interact. The third objective is to survey the recent progress in the field, and identify new directions of research.

The main themes to be considered in the workshop are:

In boundary layer theory, one attempts to understand the behavior of a fluid near an interface, typically a solid body, at high Reynolds number (low viscosity). The major difficulty is that, classically, a viscous fluid, described by the Navier-Stokes equations, is supposed to have no velocity at the boundary (assuming the boundary is at rest). Away from the boundary, however, the fluid should very nearly satisfy the Euler equations, describing inviscid fluid flow, if the viscosity is small, and for the Euler equations only a no-penetration condition is supposed to hold at rigid boundaries. Further complicating the issue, the Navier-Stokes equations are second order while the Euler equations are first order. Hence, some transition in the behavior of the fluid should occur near the boundary: describing this transition is the province of boundary layer theory.

The first concrete step in understanding this transitional behavior was made by Ludwig Prandtl in 1904. In a seminal presentation and paper, he proposed that the flow can be viewed as having two parts: a thin layer around the body (the boundary layer) and the region outside of the boundary layer. In the boundary layer the friction is dominant, and outside of the boundary layer, the friction can be neglected, and the fluid flow is inviscid and is modeled by the Euler equations. Prandtl made the physical assumptions that the velocity in the normal direction changed more quickly than in the tangential direction near the boundary. This assumption allows certain terms in the Navier-Stokes equations to be dropped, via a formal asymptotic expansion, and leads to what are now called the Prandtl equations.

Even after 110 years, the study of the Prandtl equations is still a central topic in boundary layer theory. Indeed, two main questions surrounding the boundary layer theory are the well-posedness of the Prandtl equations and the closely related question of the convergence of the solutions of the Navier-Stokes equations to the solutions of the Euler equations in the limit of high Reynolds number. Related to this last point is the question of whether the solution to the Prandtl equations, only formally derived as it is, is actually a good corrector to the Euler solution near the boundary. The boundary layer behavior is by now fairly well understood if Navier friction boundary conditions are used for the Navier-Stokes equations instead of no-slip conditions, as they lessen, though do not eliminate, the discrepancy between the viscous and inviscid solutions near the boundary (for 3D results, see [1] and references therein). For no-slip conditions, however, the two main questions are still wide open, except when special symmetries of the initial data are assumed. We recall recent results in this direction below, though an exhaustive survey is not possible here.

The recent developments can be broadly divided into two separate groups, one concerning the Prandtl equations themselves, in particular questions of ill-posedness and blow-up of solutions, the other concerning the rigorous analysis of the boundary layers under additional, but physically motivated, assumptions on the flow such as symmetry or injection and suction at the boundary.

Regarding the Prandtl equations, a recent series [2-3-4] of papers has shown ill-posedness in the sense of Hadamard. These results are related to instabilities of boundary-layers-type functions as well (see the recent survey [5]). In the opposite direction, several recent articles deal with weakening known conditions for well-posedness, such as monotonicity or analyticity. In 2012, Masmoudi and Wong [6] obtained a new nonlinear energy estimate and recovered a classical result of Oleinik [7] of local well-posedness of the Prandtl equations in two dimensions under a monotonicity assumption of the initial data (see also [8]). Well-posedness results have also been known under analyticity assumption [9,10,11]. These results have been extended by Kukavica and Vicol in 2013, who in particular weakened the requirement of matching at the top of the boundary layer to be at a polynomial rate instead of exponential. Moreover, more recently, Kukavica, Masmoudi, Vicol, and Wong [12] presented a new type of assumption on the initial data that produces local well-posedness: initial data that is monotone on multiple regions and analytic on the complement. In addition, in 2013, Gerard-Varet and Masmoudi [13] gave a proof of local well-posedness in a Gevrey class. Cannone, Lombardo, and Sammartino [14] considered the case of incompatible initial data exhibiting initial discontinuity and subsequent high gradients in the normal variable.

A related, but separate approach to studying the vanishing viscosity limit has been to consider exact solutions of the fluid equations, but to impose physical constraints on the solutions. For example, the vanishing viscosity limit can be established rigorously if enough injection and suction are present at the boundary [15], or if symmetry is imposed on the flow (see, e.g., [16-18]), in specialized geometries such as those of straight pipes and channels. Symmetry depletes the nonlinearity and allows the control of the pressure, impeding boundary layer separation.

A similar program is being carried for the hydrostatic Euler system (also known as primitive equations for planetary and atmospheric dynamics). Indeed, the viscous (or even partially viscous) case is now known to be globally well-posed (see the works by Cao-Titi [19], Cao-Li-Titi [20]). The local well-posedness in the inviscid case was shown in [12]. The questions of boundary layers and the inviscid limit for the viscous hydrostatic Euler equations are being investigated.

The phenomenon of turbulence is closely coupled with boundary layer theory. While fluid flow is laminar at small velocities, it becomes turbulent at higher velocities. Turbulent flow is characterized by chaotic behavior: the presence of large velocity and pressure fluctuations, small momentum diffusion, and sustained rate of energy dissipation, even though the flow itself need not be random. In experiments and numerical simulations, a correlation between an increase in the Reynolds number and the transition to turbulence is observed, and turbulence is generally considered a high-Reynolds number phenomenon. Indeed, it is recognized that vorticity plays an important role in the transition from laminar to turbulent flow and in sustaining fully developed turbulence. Friction with walls produces vorticity in viscous flows and boundary layer separation can contribute to vortices in the bulk of the fluids. Modeling turbulence is a challenging problem, both from a theoretical point of view as well as from a computational standpoint: several length and time scales are coupled through inertial and diffusive effects. This workshop will bring together experts in analysis as well as in numerical analysis and scientific computing. Some aspects of turbulence in geophysical flows will also be addressed.

Since the seminal work of Onsager and Kolmogorov in the 1940s and 1950s, many phenomenological theories of turbulence have been developed. Still, a satisfactory mathematical justification of turbulence remains elusive. Some progress has been made recently in justifying some of the observed features in turbulent flows directly from the fluids equations, in particular the fact that the rate of energy dissipation becomes independent of the Reynolds number at large numbers. This observation suggests that turbulent flows may be modeled mathematically by energy dissipative solutions of the inviscid equations, if such solutions exist. Onsager [21] conjectured that weak solutions of the 3D Euler equations dissipate energy if they have Holder's regularity of order less than 1/3. This conjecture has remained open to date. However, the notion of "wild solutions" for the Euler equations, introduced in the work of Scheffer [22] and [23] and developed by De Lellis and Szekelihidi [24-25] to prove non-uniqueness of weak solutions, has provided a formidable tool that makes us now much closer to proving Onsager's conjecture.

Insight into the conjectures of Onsager and Kolmogorov can be revealed through carefully designed numerical experiments that supplement laboratory observations. Orszag's pseudospectral method [26] is the classic workhorse for simulating fully developed turbulence far away from walls. We wish to highlight recent computational advances in dealiasing pseudospectral convolutions. Furthermore, special techniques like penalty methods are required to exploit the efficiency and high spectral accuracy of the fast Fourier transform in the presence of arbitrary, nonperiodic boundaries. For homogeneous Dirichlet boundary conditions, a convergence proof of the penalty method for the Navier-Stokes equations was given by Angot [27].

The scientific program will include 45-minute talks with 15-minute breaks between the talks to give sufficient time for follow up discussions. The presentations on the first day will have survey talks on boundary layer theory and turbulence. They will be followed by talks by young and established researchers on recent progress on both topics. There will be time scheduled for informal discussions to encourage collaborations.

In summary, because of the geographic diversity of the researchers in the field, the 5-day workshop would strongly promote the dissemination of the most recent results and ideas. In addition, the environment of BIRS is optimal for stimulating formal and informal interactions that can lead to new collaborations during the workshop and especially in the years to follow. Finally, we have received a very positive response from the potential participants: 38 out of 42 have tentatively agreed to attend if the proposal is approved.

[1] Gie G.-M., Kelliher J. P., Boundary layer analysis of the Navier-Stokes equations with generalized Navier boundary conditions. J. Differential Equations 253 (2012), no. 6, 1862-1892.

[2] Gerard-Varet D., Dormy E., On the ill-posedness of the Prandtl equation. J. Amer. Math. Soc. 23 (2010), no. 2, 591–609.

[3] Guo Y., Nguyen T., A note on Prandtl boundary layers. Comm. Pure Appl. Math. 64 (2011), no. 10, 1416--1438.

[4] Gerard-Varet D., Nguyen T., Remarks on the ill-posedness of the Prandtl equation. Asymptot. Anal. 77 (2012), no. 1-2, 71--88.

[5] Grenier E., Guo Y., Nguyen T., Spectral stability of Prandtl boundary layers: an overview. ArXiv preprint 1406.4452.

[6] Masmoudi N., Wong T. K., Local-in-Time Existence and Uniqueness of Solutions to the Prandtl Equations by Energy Methods. To appear in Comm. Pure Appl. Math.

[7] Constantin P., Kukavica I., Vicol V., On the inviscid limit of the Navier-Stokes equations. ArXiv preprint 1403.5748.

[8] Oleinik O., On the mathematical theory of boundary layer for an un- steady flow of incompressible fluid, Prikl. Mat. Meh., 30 (1966) 801–821 (Russian); translated in J. Appl. Math. Mech., 30 (1967), 951–974.

[9] Caflisch R. E., Sammartino M., 1998 Zero viscosity limit for analytic solutions, of the Navier–Stokes equation on a half-space: I. Existence for Euler and Prandtl equations Commun. Math. Phys. 192 433–61

[10] Caflisch R. E.,Sammartino M., 1997 Navier–Stokes equations on an exterior circular domain: construction of the solution and the zero viscosity limit C. R. Acad. Sci. Paris S'er. I Math. 324 861–6 MR1450439 (98f:35115)

[11] Cannone M., Lombardo M. C., Sammartino M., 2001 Existence and uniqueness for the Prandtl equations C. R. Acad. Sci. Paris S'er. I Math. 332 277–82

[12] Kukavica I., Masmoudi N., Vicol V., Wong T K, On the local well-posedness of the Prandtl and the hydrostatic Euler equations with multiple monotonicity regions. arXiv:1402.1984

[13] Gerard-Varet D., Masmoudi N., Well-posedness for the Prandtl system without analyticity or monotonicity. arXiv preprint arXiv:1305.0221, 2013.

[14] Cannone, M, Lombardo, M C, Sammartino, M, Well-posedness of Prandtl equations with non-compatible data. Nonlinearity 26 (2013), no. 12, 3077–3100.

[15] Temam R., Wang, X., Boundary layers in channel flow with injection and suction. Appl. Math. Lett. 14 (2001), no. 1, 87–91.

[16] Bona J. L., Wu J., The zero-viscosity limit of the 2D Navier-Stokes equations. Stud. Appl. Math. 109 (2002), no. 4, 265--278.

[17] Mazzucato A., Taylor M., Vanishing viscosity limits for a class of circular pipe flows. Comm. Partial Differential Equations 36 (2011), no. 2, 328--361.

[18] Han D., Mazzucato A. L., Niu D., Wang X., Boundary layer for a class of nonlinear pipe flow. J. Differential Equations 252 (2012), no. 12, 6387--6413.

[19] Cao C., Titi E.S., Global Well-posedness of the three dimensional viscous primitive Equations of large scale ocean and atmosphere dynamics. Ann. Math. 166. (2007), 245-267.

[20] Cao C., Li J., Titi E.S., Global Well-posedness of the 3D Primitive Equations with Only Horizontal Viscosity and Diffusion. ArXiv preprint 1406.1995.pdf

[21] Onsager L., Statistical hydrodynamics. Nuovo Cimento (9) 6, (1949). Supplemento, no. 2 (Convegno Internazionale di Meccanica Statistica), 279-287.

[22] Scheffer V., An inviscid flow with compact support in space-time. J. Geom. Anal. 3(4), 343-401 (1993)

[23] Shnirelman A., Weak solutions with decreasing energy of incompressible Euler equations. Commun. Math. Phys. 210(3), 541?603 (2000).

[24] De Lellis C., Szekelyhidi L. Jr., Dissipative continuous Euler flows. Invent. Math. 193 (2013), no. 2, 377-407.

[25] Buckmaster T., De Lellis C., Szekelyhidi, L. Jr., Dissipative Euler flows with Onsager-critical spatial regularity, arXiv:1404.6915 [math.AP] [26] Orszag S., Numerical methods for the simulation of turbulence, Phys. Fluids Supp. II, 12, 250-257 (1969).

[27] Angot P., Bruneau C.-H., Fabrie P., Numerische Mathematik, 81:497, 1999.

The main themes to be considered in the workshop are:

**Boundary Layer Theory**In boundary layer theory, one attempts to understand the behavior of a fluid near an interface, typically a solid body, at high Reynolds number (low viscosity). The major difficulty is that, classically, a viscous fluid, described by the Navier-Stokes equations, is supposed to have no velocity at the boundary (assuming the boundary is at rest). Away from the boundary, however, the fluid should very nearly satisfy the Euler equations, describing inviscid fluid flow, if the viscosity is small, and for the Euler equations only a no-penetration condition is supposed to hold at rigid boundaries. Further complicating the issue, the Navier-Stokes equations are second order while the Euler equations are first order. Hence, some transition in the behavior of the fluid should occur near the boundary: describing this transition is the province of boundary layer theory.

The first concrete step in understanding this transitional behavior was made by Ludwig Prandtl in 1904. In a seminal presentation and paper, he proposed that the flow can be viewed as having two parts: a thin layer around the body (the boundary layer) and the region outside of the boundary layer. In the boundary layer the friction is dominant, and outside of the boundary layer, the friction can be neglected, and the fluid flow is inviscid and is modeled by the Euler equations. Prandtl made the physical assumptions that the velocity in the normal direction changed more quickly than in the tangential direction near the boundary. This assumption allows certain terms in the Navier-Stokes equations to be dropped, via a formal asymptotic expansion, and leads to what are now called the Prandtl equations.

Even after 110 years, the study of the Prandtl equations is still a central topic in boundary layer theory. Indeed, two main questions surrounding the boundary layer theory are the well-posedness of the Prandtl equations and the closely related question of the convergence of the solutions of the Navier-Stokes equations to the solutions of the Euler equations in the limit of high Reynolds number. Related to this last point is the question of whether the solution to the Prandtl equations, only formally derived as it is, is actually a good corrector to the Euler solution near the boundary. The boundary layer behavior is by now fairly well understood if Navier friction boundary conditions are used for the Navier-Stokes equations instead of no-slip conditions, as they lessen, though do not eliminate, the discrepancy between the viscous and inviscid solutions near the boundary (for 3D results, see [1] and references therein). For no-slip conditions, however, the two main questions are still wide open, except when special symmetries of the initial data are assumed. We recall recent results in this direction below, though an exhaustive survey is not possible here.

The recent developments can be broadly divided into two separate groups, one concerning the Prandtl equations themselves, in particular questions of ill-posedness and blow-up of solutions, the other concerning the rigorous analysis of the boundary layers under additional, but physically motivated, assumptions on the flow such as symmetry or injection and suction at the boundary.

Regarding the Prandtl equations, a recent series [2-3-4] of papers has shown ill-posedness in the sense of Hadamard. These results are related to instabilities of boundary-layers-type functions as well (see the recent survey [5]). In the opposite direction, several recent articles deal with weakening known conditions for well-posedness, such as monotonicity or analyticity. In 2012, Masmoudi and Wong [6] obtained a new nonlinear energy estimate and recovered a classical result of Oleinik [7] of local well-posedness of the Prandtl equations in two dimensions under a monotonicity assumption of the initial data (see also [8]). Well-posedness results have also been known under analyticity assumption [9,10,11]. These results have been extended by Kukavica and Vicol in 2013, who in particular weakened the requirement of matching at the top of the boundary layer to be at a polynomial rate instead of exponential. Moreover, more recently, Kukavica, Masmoudi, Vicol, and Wong [12] presented a new type of assumption on the initial data that produces local well-posedness: initial data that is monotone on multiple regions and analytic on the complement. In addition, in 2013, Gerard-Varet and Masmoudi [13] gave a proof of local well-posedness in a Gevrey class. Cannone, Lombardo, and Sammartino [14] considered the case of incompatible initial data exhibiting initial discontinuity and subsequent high gradients in the normal variable.

A related, but separate approach to studying the vanishing viscosity limit has been to consider exact solutions of the fluid equations, but to impose physical constraints on the solutions. For example, the vanishing viscosity limit can be established rigorously if enough injection and suction are present at the boundary [15], or if symmetry is imposed on the flow (see, e.g., [16-18]), in specialized geometries such as those of straight pipes and channels. Symmetry depletes the nonlinearity and allows the control of the pressure, impeding boundary layer separation.

A similar program is being carried for the hydrostatic Euler system (also known as primitive equations for planetary and atmospheric dynamics). Indeed, the viscous (or even partially viscous) case is now known to be globally well-posed (see the works by Cao-Titi [19], Cao-Li-Titi [20]). The local well-posedness in the inviscid case was shown in [12]. The questions of boundary layers and the inviscid limit for the viscous hydrostatic Euler equations are being investigated.

**Turbulence**The phenomenon of turbulence is closely coupled with boundary layer theory. While fluid flow is laminar at small velocities, it becomes turbulent at higher velocities. Turbulent flow is characterized by chaotic behavior: the presence of large velocity and pressure fluctuations, small momentum diffusion, and sustained rate of energy dissipation, even though the flow itself need not be random. In experiments and numerical simulations, a correlation between an increase in the Reynolds number and the transition to turbulence is observed, and turbulence is generally considered a high-Reynolds number phenomenon. Indeed, it is recognized that vorticity plays an important role in the transition from laminar to turbulent flow and in sustaining fully developed turbulence. Friction with walls produces vorticity in viscous flows and boundary layer separation can contribute to vortices in the bulk of the fluids. Modeling turbulence is a challenging problem, both from a theoretical point of view as well as from a computational standpoint: several length and time scales are coupled through inertial and diffusive effects. This workshop will bring together experts in analysis as well as in numerical analysis and scientific computing. Some aspects of turbulence in geophysical flows will also be addressed.

Since the seminal work of Onsager and Kolmogorov in the 1940s and 1950s, many phenomenological theories of turbulence have been developed. Still, a satisfactory mathematical justification of turbulence remains elusive. Some progress has been made recently in justifying some of the observed features in turbulent flows directly from the fluids equations, in particular the fact that the rate of energy dissipation becomes independent of the Reynolds number at large numbers. This observation suggests that turbulent flows may be modeled mathematically by energy dissipative solutions of the inviscid equations, if such solutions exist. Onsager [21] conjectured that weak solutions of the 3D Euler equations dissipate energy if they have Holder's regularity of order less than 1/3. This conjecture has remained open to date. However, the notion of "wild solutions" for the Euler equations, introduced in the work of Scheffer [22] and [23] and developed by De Lellis and Szekelihidi [24-25] to prove non-uniqueness of weak solutions, has provided a formidable tool that makes us now much closer to proving Onsager's conjecture.

Insight into the conjectures of Onsager and Kolmogorov can be revealed through carefully designed numerical experiments that supplement laboratory observations. Orszag's pseudospectral method [26] is the classic workhorse for simulating fully developed turbulence far away from walls. We wish to highlight recent computational advances in dealiasing pseudospectral convolutions. Furthermore, special techniques like penalty methods are required to exploit the efficiency and high spectral accuracy of the fast Fourier transform in the presence of arbitrary, nonperiodic boundaries. For homogeneous Dirichlet boundary conditions, a convergence proof of the penalty method for the Navier-Stokes equations was given by Angot [27].

**Scientific Program**The scientific program will include 45-minute talks with 15-minute breaks between the talks to give sufficient time for follow up discussions. The presentations on the first day will have survey talks on boundary layer theory and turbulence. They will be followed by talks by young and established researchers on recent progress on both topics. There will be time scheduled for informal discussions to encourage collaborations.

In summary, because of the geographic diversity of the researchers in the field, the 5-day workshop would strongly promote the dissemination of the most recent results and ideas. In addition, the environment of BIRS is optimal for stimulating formal and informal interactions that can lead to new collaborations during the workshop and especially in the years to follow. Finally, we have received a very positive response from the potential participants: 38 out of 42 have tentatively agreed to attend if the proposal is approved.

**References:**[1] Gie G.-M., Kelliher J. P., Boundary layer analysis of the Navier-Stokes equations with generalized Navier boundary conditions. J. Differential Equations 253 (2012), no. 6, 1862-1892.

[2] Gerard-Varet D., Dormy E., On the ill-posedness of the Prandtl equation. J. Amer. Math. Soc. 23 (2010), no. 2, 591–609.

[3] Guo Y., Nguyen T., A note on Prandtl boundary layers. Comm. Pure Appl. Math. 64 (2011), no. 10, 1416--1438.

[4] Gerard-Varet D., Nguyen T., Remarks on the ill-posedness of the Prandtl equation. Asymptot. Anal. 77 (2012), no. 1-2, 71--88.

[5] Grenier E., Guo Y., Nguyen T., Spectral stability of Prandtl boundary layers: an overview. ArXiv preprint 1406.4452.

[6] Masmoudi N., Wong T. K., Local-in-Time Existence and Uniqueness of Solutions to the Prandtl Equations by Energy Methods. To appear in Comm. Pure Appl. Math.

[7] Constantin P., Kukavica I., Vicol V., On the inviscid limit of the Navier-Stokes equations. ArXiv preprint 1403.5748.

[8] Oleinik O., On the mathematical theory of boundary layer for an un- steady flow of incompressible fluid, Prikl. Mat. Meh., 30 (1966) 801–821 (Russian); translated in J. Appl. Math. Mech., 30 (1967), 951–974.

[9] Caflisch R. E., Sammartino M., 1998 Zero viscosity limit for analytic solutions, of the Navier–Stokes equation on a half-space: I. Existence for Euler and Prandtl equations Commun. Math. Phys. 192 433–61

[10] Caflisch R. E.,Sammartino M., 1997 Navier–Stokes equations on an exterior circular domain: construction of the solution and the zero viscosity limit C. R. Acad. Sci. Paris S'er. I Math. 324 861–6 MR1450439 (98f:35115)

[11] Cannone M., Lombardo M. C., Sammartino M., 2001 Existence and uniqueness for the Prandtl equations C. R. Acad. Sci. Paris S'er. I Math. 332 277–82

[12] Kukavica I., Masmoudi N., Vicol V., Wong T K, On the local well-posedness of the Prandtl and the hydrostatic Euler equations with multiple monotonicity regions. arXiv:1402.1984

[13] Gerard-Varet D., Masmoudi N., Well-posedness for the Prandtl system without analyticity or monotonicity. arXiv preprint arXiv:1305.0221, 2013.

[14] Cannone, M, Lombardo, M C, Sammartino, M, Well-posedness of Prandtl equations with non-compatible data. Nonlinearity 26 (2013), no. 12, 3077–3100.

[15] Temam R., Wang, X., Boundary layers in channel flow with injection and suction. Appl. Math. Lett. 14 (2001), no. 1, 87–91.

[16] Bona J. L., Wu J., The zero-viscosity limit of the 2D Navier-Stokes equations. Stud. Appl. Math. 109 (2002), no. 4, 265--278.

[17] Mazzucato A., Taylor M., Vanishing viscosity limits for a class of circular pipe flows. Comm. Partial Differential Equations 36 (2011), no. 2, 328--361.

[18] Han D., Mazzucato A. L., Niu D., Wang X., Boundary layer for a class of nonlinear pipe flow. J. Differential Equations 252 (2012), no. 12, 6387--6413.

[19] Cao C., Titi E.S., Global Well-posedness of the three dimensional viscous primitive Equations of large scale ocean and atmosphere dynamics. Ann. Math. 166. (2007), 245-267.

[20] Cao C., Li J., Titi E.S., Global Well-posedness of the 3D Primitive Equations with Only Horizontal Viscosity and Diffusion. ArXiv preprint 1406.1995.pdf

[21] Onsager L., Statistical hydrodynamics. Nuovo Cimento (9) 6, (1949). Supplemento, no. 2 (Convegno Internazionale di Meccanica Statistica), 279-287.

[22] Scheffer V., An inviscid flow with compact support in space-time. J. Geom. Anal. 3(4), 343-401 (1993)

[23] Shnirelman A., Weak solutions with decreasing energy of incompressible Euler equations. Commun. Math. Phys. 210(3), 541?603 (2000).

[24] De Lellis C., Szekelyhidi L. Jr., Dissipative continuous Euler flows. Invent. Math. 193 (2013), no. 2, 377-407.

[25] Buckmaster T., De Lellis C., Szekelyhidi, L. Jr., Dissipative Euler flows with Onsager-critical spatial regularity, arXiv:1404.6915 [math.AP] [26] Orszag S., Numerical methods for the simulation of turbulence, Phys. Fluids Supp. II, 12, 250-257 (1969).

[27] Angot P., Bruneau C.-H., Fabrie P., Numerische Mathematik, 81:497, 1999.