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Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models. (English) Zbl 1127.35034

The authors study a variant of the Navier-Stokes equations (NSE) ie. \[ \begin{aligned} & \partial_t v+\nu\Delta v-(u\nabla)u-\nabla p=f,\quad \text{div}\,u=\text{div}\,v=0,\\ & 0=u-\alpha^2\Delta u,\quad u(x,0)=u_0(x)\end{aligned}\tag{1} \] with \(u, v, f\) periodic in the periodic box \(\Omega=[0,2\pi L]^3\) and with \(f=f(x)\) time independent. The motivation to study (1) stems from its connection with turbulence theory. In fact (1) is known as the so called simplified Bardina model of turbulence. This relationship is discussed at some length in the introduction, but the focus of the paper is on (1). In order to handle (1) in a way similar to the NSE, (1) is cast into a functional frame, what necessitates the introduction of various familiar spaces and operators. Thus letting \(T\) be the linear space of vector-valued, \(2\pi L\)-periodic polynomials, one sets \[ V_0=\biggl\{v\in T;\;\text{div\,}v=0\;\&\int_\Omega v \,dx^3=0 \biggr\} \] and let \(V\) and \(H\) be the closures of \(V_0\) with respect to the norms in \(W^{1,2}_{\text{per}}(\Omega)^3\) and \(L^2(\Omega)^3\), respectively. One denotes by \(P:L^2(\Omega)^3\to H\) the orthogonal projection onto \(H\) and by \(A=-P\Delta\), \((\text{dom}(A)=W^{2,2}_{\text{per}}(\Omega)^3\cap H)\) the Stokes operator. Finally one defines the nonlinearity \(B\) via \[ B(u,v)=P(u\nabla)v,\quad u,v\in V \] which is continuous from \(V\times V\) into the dual \(V'\) of \(V\).
Based on this stipulations one puts (1) into abstract form, i.e.
\[ \partial_t v=\nu Av+B(u,u)=f,\quad v=u+\alpha^2 Au,\quad v_0=u_0+\alpha^2Au_0.\tag{2} \]
A notion of weak solution for (2) is then introduced. After a discussion of the relationship between (2) and turbulent channel flow the authors proceed to the proof of Thm. 4.1 which asserts global existence and uniqueness of weak solutions to (2). The proof is based on a Galerkin method. Although it proceeds essentially along established lines, it uses quite a series of tricky estimates. Theorem 4.1 guarantees the existence of the solution semigroup \(S(t)\), \(t\geq 0\) which associates with \(u_0\) the solution \(u(t)=S(t)u_0\), \(t\geq 0\) of (2). The authors now prove Theorem 5.3 which asserts the existence of a global attractor \({\mathcal A}\subseteq V\) for \(S(t)\), \(t\geq 0\).
Moreover they prove the estimates \[ d_H({\mathcal A})\leq d_F({\mathcal A})\leq c\,G^{6/5}(L/\alpha)^{18/5} \] where \(d_H\) and \(d_F\) are Hausdorff- and fractal dimension, while \(G\) is the Grashoff number.
Further topics discussed are energy spectra (Section 6) and finally (Section 7) the Euler version of (1), resp. (2), i.e. \[ \partial_t v+B(u,u)=f,\quad v=u+\alpha^2 Au,\quad v_0=u_0+\alpha^2Au_0.\tag{3} \] For (3), a local existence and uniqueness result is proved.

MSC:

35Q30 Navier-Stokes equations
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35D05 Existence of generalized solutions of PDE (MSC2000)
76F20 Dynamical systems approach to turbulence
76F55 Statistical turbulence modeling
76F65 Direct numerical and large eddy simulation of turbulence
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
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