Introduction

1. Introduction In this work I consider the Navier-Stokes-Voght (NSV) model of viscoelastic fluid, which was recently proposed as a new type of inviscid regularization for the 3D Navier-Stokes equations. I will discuss the question of the well posedness of the model and show that the global attractor of the NSV equations, driven by the analytic forcing, consists of analytic functions. Moreover, in order to provide additional evidences that the NSV model can indeed be used to study the statistical properties of the 3D Navier-Stokes equations, I will present the result of the numerical

2. study of the modified Sabra shell model of turbulence, to show that for the small value of the regularizations parameter one recovers the anomalous inertial range scaling. Navier-Stokes-Voight model We consider the following equations

3. with the periodic boundary conditions in . The represents the velocity field, is the pressure, stands for the kinematic viscosity, the forcing, and finally is a real positive regularizing length scale parameter. The NSV equations were introduced as a model of a motion of linear, viscoelastic fluid. In that case, is thought of as a lengthscale parameter characterizing the elasticity of the fluid. The NSV system was shown to be well posed both in the viscous and the inviscid regime . See, e.g., [CLT], [K], [O].

4. The long-time dynamics • It was shown previously (see In [K] and [KT]) that • The NSV equations has a finite dimensional global attractor • The global attractor of the NSV equations lies in the bounded subset of . • Questions: • Do the solutions of the NSV equations on the attractor have a dissipation range? • Can we observe the inertial range scaling of the structure functions?

5. Regularity of the attractor The addition of the changes the parabolic character of the equation, which now behaves like a damped hyperbolic system. Our goal is to construct a smooth asymptotic approximation. Stage 1: Asymptotic approximation in Let , and assume . Define functions , , satisfying

6. where • is a solution of the original NSV problem with • One can show (see [KLT]) that • is an asymptotic approximation of , i.e. • For , we have • Stage 2: Analytic asymptotic approximation • Let us assume that the forcing is analytic, and denote

7. where is a spectral projection on the low Fourier modes corresponding to wavenumbers less than . Let satisfy the equation • with zero initial condition. Than one can choose large enough, and , depending on initial data, s.t. • The function belongs to certain Gevrey class of regularity, i.e., has an exponentially decaying spectrum. • is an asymptotic approximation of , i.e.

8. The Sabra shell model Shell models are phenomenological model of turbulence retaining certain features of the original Navier-Stokes equations. I would like to demonstrate the effect of the regularization on the following Sabra shell model. Original equations describe the evolution of the complex Fourier-like components with the boundary conditions , and scalar wavenumbers satisfying

9. For a small enough viscosity, and the forcing applied to the first modes, one can observe an inertial range followed by an exponentially decaying dissipation range. Here I plot structure functions of the shell model for . Structure functions exhibit an anomalous scaling behavior in the inertial range where for

10. Below I plot structure functions of the modified Sabra shell model, with an addition of to the left hand side of the equation. We observe an inertial range, with the same scaling as the original model, whose size depends on . This dependence is a subject of a current research.

11. Conclusions • I surveyed the ongoing research concerning the possible application of the 3D Navier-Stokes-Voight model as a regularization of the 3D Navier-Stokes equations. • I show that the solutions of the 3D NSV model, which lie on the global attractor, have an exponentially decaying dissipation range. • Numerical study of the modified Sabra shell model of turbulence provide an evidence for the existence of the inertial range, with an anomalous scaling of the structure functions.

12. Bibliography [CLT] Y. Cao, E. M. Lunasin, and E. S. Titi, “Global well-posedness of the three dimensional viscous and inviscid simplified Bardina turbulence models”, Communications in Mathematical Sciences, 4, (2006), 823-884. [CLT1] P. Constantin, B. Levant, E. S. Titi, “Analytic study of the shell model of turbulence”, Physica D, 219 (2006), 120-141. [K] V. K. Kalantarov, “Attractors for some nonlinear problems of mathematical physics”, Zap. Nauchn. Sem. Inst. Steklov. (LOMI), 152 (1986), 50-54. [KLT] V. Kalantarov, B. Levant, E. S. Titi, “Gevrey regularity for the attractor of the 3D Navier-Stokes-Voight equations”, preprint. [O] A. P. Oskolkov, “The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers”, Zap. Naucn. Sem. Inst. Steklov (LOMI), 38, (1973), 98-136.

13. On a new inviscid regularization of the 3DNavier-Stokes equations Boris Levant (Weizmann Institute of Science, Israel)Joint work with V. Kalantarov (Koc University, Turkey) and E. S. Titi (U of California Irvine and Weizmann Institute of Science)