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Discrete & Continuous Dynamical Systems
March 2010 , Volume 28 , Issue 1
A special issue
Dedicated to Roger Temam on the Occasion of his 70th Birthday
Part II
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2010, 28(1): i-iii
doi: 10.3934/dcds.2010.28.1i
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Abstract:
This is Part II of a two-part series of Discrete and Continuous Dynamical Systems dedicated to Roger Temam on the occasion of his 70th birthday.
Born in Tunis on May 19, 1940, Roger Temam moved to Paris in 1957 to study at the University of Paris, which was at that time the only university in Paris, known as La Sorbonne. He wrote his doctoral thesis under the supervision of Professor Jacques-Louis Lions and became a professor at the University of Paris-Sud XI at Orsay in 1968. There, he founded, together with Professors Jacques Deny and Charles Goulaouic, the Laboratory of Numerical and Functional Analysis which he directed from 1972 to 1988. He was also a Maître de Conférences at the famous Ecole Polytechnique from 1968 to 1986.
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This is Part II of a two-part series of Discrete and Continuous Dynamical Systems dedicated to Roger Temam on the occasion of his 70th birthday.
Born in Tunis on May 19, 1940, Roger Temam moved to Paris in 1957 to study at the University of Paris, which was at that time the only university in Paris, known as La Sorbonne. He wrote his doctoral thesis under the supervision of Professor Jacques-Louis Lions and became a professor at the University of Paris-Sud XI at Orsay in 1968. There, he founded, together with Professors Jacques Deny and Charles Goulaouic, the Laboratory of Numerical and Functional Analysis which he directed from 1972 to 1988. He was also a Maître de Conférences at the famous Ecole Polytechnique from 1968 to 1986.
For more information please click the “Full Text” above.
2010, 28(1): 1-39
doi: 10.3934/dcds.2010.28.1
+[Abstract](2741)
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Abstract:
We consider a diffuse interface model for the evolution of an iso-thermal incompressible two-phase flow in a two-dimensional bounded domain. The model consists of the Navier-Stokes equation for the fluid velocity u coupled with a convective Allen-Cahn equation for the order (phase) parameter $\phi$, both endowed with suitable boundary conditions. We analyze the asymptotic behavior of the solutions within the theory of infinite-dimensional dissipative dynamical systems. We first prove that the initial and boundary value problem generates a strongly continuous semigroup on a suitable phase space which possesses the global attractor $ \mathcal{A}$. Then we establish the existence of an exponential attractor $ \mathcal{E}$ which entails that $\mathcal{A}$ has finite fractal dimension. This dimension is then estimated in terms of some model parameters. Moreover, assuming the potential to be real analytic, we demonstrate that, in absence of external forces, each trajectory converges to a single equilibrium by means of a Łojasiewicz-Simon inequality. We also obtain a convergence rate estimate. Finally, we discuss the case where $\phi $ is forced to take values in a bounded interval, e.g., by a so-called singular potential.
We consider a diffuse interface model for the evolution of an iso-thermal incompressible two-phase flow in a two-dimensional bounded domain. The model consists of the Navier-Stokes equation for the fluid velocity u coupled with a convective Allen-Cahn equation for the order (phase) parameter $\phi$, both endowed with suitable boundary conditions. We analyze the asymptotic behavior of the solutions within the theory of infinite-dimensional dissipative dynamical systems. We first prove that the initial and boundary value problem generates a strongly continuous semigroup on a suitable phase space which possesses the global attractor $ \mathcal{A}$. Then we establish the existence of an exponential attractor $ \mathcal{E}$ which entails that $\mathcal{A}$ has finite fractal dimension. This dimension is then estimated in terms of some model parameters. Moreover, assuming the potential to be real analytic, we demonstrate that, in absence of external forces, each trajectory converges to a single equilibrium by means of a Łojasiewicz-Simon inequality. We also obtain a convergence rate estimate. Finally, we discuss the case where $\phi $ is forced to take values in a bounded interval, e.g., by a so-called singular potential.
2010, 28(1): 41-65
doi: 10.3934/dcds.2010.28.41
+[Abstract](2392)
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Abstract:
Consider the Thomas-Fermi energy functional $E$ for a spin polarized atom or molecule with $N_{1} $ [resp. $N_{2}$] spin up [resp. spin down] electrons and total positive molecular charge Z. Incorporating the Fermi-Amaldi correction as Benilan, Goldstein and Goldstein did, $E$ is not convex. By replacing $E$ by a well-motivated convex minorant $ \mathcal{E}$ ,we prove that $ \mathcal{E} $ has a unique minimizing density $( \rho _{1},\rho _{2}) \ $ when $N_{1}+N_{2}\leq Z+1\ $and $N_{2}\ $is close to $N_{1}.$
Consider the Thomas-Fermi energy functional $E$ for a spin polarized atom or molecule with $N_{1} $ [resp. $N_{2}$] spin up [resp. spin down] electrons and total positive molecular charge Z. Incorporating the Fermi-Amaldi correction as Benilan, Goldstein and Goldstein did, $E$ is not convex. By replacing $E$ by a well-motivated convex minorant $ \mathcal{E}$ ,we prove that $ \mathcal{E} $ has a unique minimizing density $( \rho _{1},\rho _{2}) \ $ when $N_{1}+N_{2}\leq Z+1\ $and $N_{2}\ $is close to $N_{1}.$
2010, 28(1): 67-98
doi: 10.3934/dcds.2010.28.67
+[Abstract](3213)
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Abstract:
We study in this paper the well-posedness and the asymptotic behavior, in terms of global attractors, of the Caginalp system with coupled dynamic boundary conditions and possibly singular potentials (e.g., of logarithmic type).
We study in this paper the well-posedness and the asymptotic behavior, in terms of global attractors, of the Caginalp system with coupled dynamic boundary conditions and possibly singular potentials (e.g., of logarithmic type).
2010, 28(1): 99-130
doi: 10.3934/dcds.2010.28.99
+[Abstract](2752)
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Abstract:
The main objective of this article is to study dynamic transitions and stability for rotating incompressible flows, governed by the rotating Boussinesq equations. It is proved that there are only two types of transitions, Type-I (continuous) and Type-II (jump) transitions, as the Rayleigh number crosses the first real or complex eigenvalues. Specific criteria are given to determine the type of transitions as well.
The main objective of this article is to study dynamic transitions and stability for rotating incompressible flows, governed by the rotating Boussinesq equations. It is proved that there are only two types of transitions, Type-I (continuous) and Type-II (jump) transitions, as the Rayleigh number crosses the first real or complex eigenvalues. Specific criteria are given to determine the type of transitions as well.
2010, 28(1): 131-146
doi: 10.3934/dcds.2010.28.131
+[Abstract](2569)
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We prove Berezin-Li-Yau-type lower bounds with additional term for the eigenvalues of the Stokes operator and improve the previously known estimates for the Laplace operator. Generalizations to higher-order operators are given.
We prove Berezin-Li-Yau-type lower bounds with additional term for the eigenvalues of the Stokes operator and improve the previously known estimates for the Laplace operator. Generalizations to higher-order operators are given.
2010, 28(1): 147-159
doi: 10.3934/dcds.2010.28.147
+[Abstract](2866)
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We study uncertainty bounds and statistics of wave solutions through a random waveguide which possesses certain random inhomogeneities. The waveguide is composed of several homogeneous media with random interfaces. The main focus is on two homogeneous media which are layered randomly and periodically in space. Solutions of stochastic and deterministic problems are compared. The waveguide media parameters pertaining to the latter are the averaged values of the random parameters of the former. We investigate the eigenmodes coupling due to random inhomogeneities in media, i.e. random changes of the media parameters. We present an efficient numerical method via Legendre Polynomial Chaos expansion for obtaining output statistics including mean, variance and probability distribution of the wave solutions. Based on the statistical studies, we present uncertainty bounds and quantify the robustness of the solutions with respect to random changes of interfaces.
We study uncertainty bounds and statistics of wave solutions through a random waveguide which possesses certain random inhomogeneities. The waveguide is composed of several homogeneous media with random interfaces. The main focus is on two homogeneous media which are layered randomly and periodically in space. Solutions of stochastic and deterministic problems are compared. The waveguide media parameters pertaining to the latter are the averaged values of the random parameters of the former. We investigate the eigenmodes coupling due to random inhomogeneities in media, i.e. random changes of the media parameters. We present an efficient numerical method via Legendre Polynomial Chaos expansion for obtaining output statistics including mean, variance and probability distribution of the wave solutions. Based on the statistical studies, we present uncertainty bounds and quantify the robustness of the solutions with respect to random changes of interfaces.
2010, 28(1): 161-179
doi: 10.3934/dcds.2010.28.161
+[Abstract](3091)
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The Kneser theorem for ordinary differential equations without uniqueness says that the attainability set is compact and connected at each instant of time. We establish corresponding results for the attainability set of weak solutions for the 3D Navier-Stokes equations satisfying an energy inequality. First, we present a simplified proof of our earlier result with respect to the weak topology in the space $H$. Then we prove that this result also holds with respect to the strong topology on $H$ provided that the weak solutions satisfying the weak version of the energy inequality are continuous. Finally, using these results, we show the connectedness of the global attractor of a family of setvalued semiflows generated by the weak solutions of the NSE satisfying suitable properties.
The Kneser theorem for ordinary differential equations without uniqueness says that the attainability set is compact and connected at each instant of time. We establish corresponding results for the attainability set of weak solutions for the 3D Navier-Stokes equations satisfying an energy inequality. First, we present a simplified proof of our earlier result with respect to the weak topology in the space $H$. Then we prove that this result also holds with respect to the strong topology on $H$ provided that the weak solutions satisfying the weak version of the energy inequality are continuous. Finally, using these results, we show the connectedness of the global attractor of a family of setvalued semiflows generated by the weak solutions of the NSE satisfying suitable properties.
2010, 28(1): 181-197
doi: 10.3934/dcds.2010.28.181
+[Abstract](2123)
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We consider in this paper a mathematical model for the incompressible flows with a surfactant monolayer. The presence of surfactant gives rise to coupling surface terms which make the analysis and simulation challenging. We study the well-posedness of this coupled system of PDEs with physically relevant boundary conditions, as well as the stability of a numerical scheme. We also preform numerical simulations by a fast-spectral method and use it to study the effect of surfactant concentration on the motion of an incompressible fluid in a cylinder.
We consider in this paper a mathematical model for the incompressible flows with a surfactant monolayer. The presence of surfactant gives rise to coupling surface terms which make the analysis and simulation challenging. We study the well-posedness of this coupled system of PDEs with physically relevant boundary conditions, as well as the stability of a numerical scheme. We also preform numerical simulations by a fast-spectral method and use it to study the effect of surfactant concentration on the motion of an incompressible fluid in a cylinder.
2010, 28(1): 199-226
doi: 10.3934/dcds.2010.28.199
+[Abstract](3701)
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The simulation of dynamical systems involving random coefficients by means of stochastic spectral methods (Polynomial Chaos or other types of orthogonal stochastic expansions) is faced with well known computational difficulties, arising in particular due to the broadening of the solution spectrum as time evolves. The simulation of such systems thus requires increasing the basis dimension and computational resources for long time integration. This paper deals with systems having almost surely a stable limit cycles. It is proposed to reformulate the problem at hand in a rescaled time framework such that the spectrum of the rescaled solution remains narrow-banded. Two variants of this approach are considered and evaluated. The first relies on an explicit expression of a time-dependent, uncertain, time scale related to some distance between the corresponding solution and a reference deterministic system. The time scale is adjusted at each time step so that the distance from the reference system solution remains small, mimicking "in phase'' behavior. The second variant achieves the same objective by borrowing concepts from optimal control theory, and yields more precise time-scale estimates at the price of a higher CPU cost. It is thus more appropriate for uncertain systems exhibiting a stiff behavior and complex limit cycles. The method is applied to the case of a linear oscillator with uncertain properties, and to a stiff nonlinear chemical system involving uncertain reaction constants. The tests demonstrate the effectiveness of the proposed approaches, at least in situations where the topology of the limit cycle does not change when the uncertain system parameters vary.
The simulation of dynamical systems involving random coefficients by means of stochastic spectral methods (Polynomial Chaos or other types of orthogonal stochastic expansions) is faced with well known computational difficulties, arising in particular due to the broadening of the solution spectrum as time evolves. The simulation of such systems thus requires increasing the basis dimension and computational resources for long time integration. This paper deals with systems having almost surely a stable limit cycles. It is proposed to reformulate the problem at hand in a rescaled time framework such that the spectrum of the rescaled solution remains narrow-banded. Two variants of this approach are considered and evaluated. The first relies on an explicit expression of a time-dependent, uncertain, time scale related to some distance between the corresponding solution and a reference deterministic system. The time scale is adjusted at each time step so that the distance from the reference system solution remains small, mimicking "in phase'' behavior. The second variant achieves the same objective by borrowing concepts from optimal control theory, and yields more precise time-scale estimates at the price of a higher CPU cost. It is thus more appropriate for uncertain systems exhibiting a stiff behavior and complex limit cycles. The method is applied to the case of a linear oscillator with uncertain properties, and to a stiff nonlinear chemical system involving uncertain reaction constants. The tests demonstrate the effectiveness of the proposed approaches, at least in situations where the topology of the limit cycle does not change when the uncertain system parameters vary.
2010, 28(1): 227-241
doi: 10.3934/dcds.2010.28.227
+[Abstract](2484)
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This is the second of two articles dedicated to Roger Temam and Claude-Michel Brauner on turbulence large-eddy simulations using stochastic two-point closures. The first paper [31] has dealt with applications to isotropic turbulence. It has also discussed personal memories of Roger and Claude-Michel, and how we have collaborated on turbulence two-point closures applied to Burgers equation by studying the so-called Burgers-MRCM model. The present paper is basically a review of what can be done with the same models for LES of inhomogeneous turbulence (free-shear and wall-bounded flows) both in incompressible and compressible situations. It borrows results taken from Lesieur and colleagues [28][29][30]. We discuss also simulations obtained with the aid of dynamic multilevel methods (DML) of Dubois, Jauberteau and Temam [19]. Afterwards we consider the incompressible free-shear flows: temporal mixing layers, for which we review the helical-pairing phenomenon both numerically and experimentally, and mixing of a passive scalar in coaxial jets. We study the plane channel with LES and DML calculations. We look at passive control using longitudinal riblets and optimal control as developed by Temam and coworkers. We calculate with DNS and LES channels and mixing layers rotating about a spanwise axis, and demonstrate a universal character of the local Rossby number in anticyclonic regions. Finally we discuss compressible turbulence LES with applications to a subsonic (Mach 0.7) and supersonic (Mach 1.4) round jet.
This is the second of two articles dedicated to Roger Temam and Claude-Michel Brauner on turbulence large-eddy simulations using stochastic two-point closures. The first paper [31] has dealt with applications to isotropic turbulence. It has also discussed personal memories of Roger and Claude-Michel, and how we have collaborated on turbulence two-point closures applied to Burgers equation by studying the so-called Burgers-MRCM model. The present paper is basically a review of what can be done with the same models for LES of inhomogeneous turbulence (free-shear and wall-bounded flows) both in incompressible and compressible situations. It borrows results taken from Lesieur and colleagues [28][29][30]. We discuss also simulations obtained with the aid of dynamic multilevel methods (DML) of Dubois, Jauberteau and Temam [19]. Afterwards we consider the incompressible free-shear flows: temporal mixing layers, for which we review the helical-pairing phenomenon both numerically and experimentally, and mixing of a passive scalar in coaxial jets. We study the plane channel with LES and DML calculations. We look at passive control using longitudinal riblets and optimal control as developed by Temam and coworkers. We calculate with DNS and LES channels and mixing layers rotating about a spanwise axis, and demonstrate a universal character of the local Rossby number in anticyclonic regions. Finally we discuss compressible turbulence LES with applications to a subsonic (Mach 0.7) and supersonic (Mach 1.4) round jet.
2010, 28(1): 243-257
doi: 10.3934/dcds.2010.28.243
+[Abstract](2944)
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In this paper we establish the theory on the semiglobal classical solution to first order quasilinear hyperbolic systems with a kind of nonlocal boundary conditions, and based on this, the corresponding exact boundary controllability and observability are obtained by a constructive method. Moreover, with the linearized Saint-Venant system and the 1-D linear wave equation as examples, we show that the number of both boundary controls and boundary observations can not be reduced, and consequently, we conclude that the exact boundary controllability for a hyperbolic system in a network with loop can not be realized generically.
In this paper we establish the theory on the semiglobal classical solution to first order quasilinear hyperbolic systems with a kind of nonlocal boundary conditions, and based on this, the corresponding exact boundary controllability and observability are obtained by a constructive method. Moreover, with the linearized Saint-Venant system and the 1-D linear wave equation as examples, we show that the number of both boundary controls and boundary observations can not be reduced, and consequently, we conclude that the exact boundary controllability for a hyperbolic system in a network with loop can not be realized generically.
2010, 28(1): 259-274
doi: 10.3934/dcds.2010.28.259
+[Abstract](2978)
+[PDF](315.6KB)
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Mixing a passive scalar field by stirring can be measured in a variety of ways including tracer particle dispersion, via the flux-gradient relationship, or by suppression of scalar concentration variations in the presence of inhomogeneous sources and sinks. The mixing efficiency or efficacy of a particular flow is often expressed in terms of enhanced diffusivity and quantified as an effective diffusion coefficient. In this work we compare and contrast several notions of effective diffusivity. We thoroughly examine the fundamental case of a steady sinusoidal shear flow mixing a scalar sustained by a steady sinusoidal source-sink distribution to explore apparent quantitative inconsistencies among the measures. Ultimately the conflicts are attributed to the noncommutative asymptotic limits of large Péclet number and large length-scale separation. We then propose another approach, a generalization of Batchelor's 1949 theory of diffusion in homogeneous turbulence, that helps unify the particle dispersion and concentration variance suppression measures.
Mixing a passive scalar field by stirring can be measured in a variety of ways including tracer particle dispersion, via the flux-gradient relationship, or by suppression of scalar concentration variations in the presence of inhomogeneous sources and sinks. The mixing efficiency or efficacy of a particular flow is often expressed in terms of enhanced diffusivity and quantified as an effective diffusion coefficient. In this work we compare and contrast several notions of effective diffusivity. We thoroughly examine the fundamental case of a steady sinusoidal shear flow mixing a scalar sustained by a steady sinusoidal source-sink distribution to explore apparent quantitative inconsistencies among the measures. Ultimately the conflicts are attributed to the noncommutative asymptotic limits of large Péclet number and large length-scale separation. We then propose another approach, a generalization of Batchelor's 1949 theory of diffusion in homogeneous turbulence, that helps unify the particle dispersion and concentration variance suppression measures.
2010, 28(1): 275-310
doi: 10.3934/dcds.2010.28.275
+[Abstract](3148)
+[PDF](434.3KB)
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Our aim in this paper is to study the Cahn-Hilliard equation with singular potentials and dynamic boundary conditions. In particular, we prove, owing to proper approximations of the singular potential and a suitable notion of variational solutions, the existence and uniqueness of solutions. We also discuss the separation of the solutions from the singularities of the potential. Finally, we prove the existence of global and exponential attractors.
Our aim in this paper is to study the Cahn-Hilliard equation with singular potentials and dynamic boundary conditions. In particular, we prove, owing to proper approximations of the singular potential and a suitable notion of variational solutions, the existence and uniqueness of solutions. We also discuss the separation of the solutions from the singularities of the potential. Finally, we prove the existence of global and exponential attractors.
2010, 28(1): 311-341
doi: 10.3934/dcds.2010.28.311
+[Abstract](2916)
+[PDF](394.8KB)
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Two theorems concerning strong wellposedness are established for the complex Ginzburg-Landau equation. One of them is concerned with strong $L^{2}$-wellposedness, that is, strong wellposedness for $L^{2}$-initial data. The other deals with $H_{0}^{1}$-initial data as a partial extension. By a technical innovation it becomes possible to prove the convergence of approximate solutions without compactness. This type of convergence is known with accretivity methods when the argument of the complex coefficient is small. The new device yields the generation of a class of non-contraction semigroups even when the argument is large. The results are both obtained as application of abstract theory of semilinear evolution equations with subdifferential operators.
Two theorems concerning strong wellposedness are established for the complex Ginzburg-Landau equation. One of them is concerned with strong $L^{2}$-wellposedness, that is, strong wellposedness for $L^{2}$-initial data. The other deals with $H_{0}^{1}$-initial data as a partial extension. By a technical innovation it becomes possible to prove the convergence of approximate solutions without compactness. This type of convergence is known with accretivity methods when the argument of the complex coefficient is small. The new device yields the generation of a class of non-contraction semigroups even when the argument is large. The results are both obtained as application of abstract theory of semilinear evolution equations with subdifferential operators.
2010, 28(1): 343-373
doi: 10.3934/dcds.2010.28.343
+[Abstract](2596)
+[PDF](332.8KB)
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We introduce the notion of →p -multivoque Leray-Lions operator
We introduce the notion of →p -multivoque Leray-Lions operator
-div→p(δφ i(x,δu/δxi))=Au
on a Banach-Sobolev function space V→p and we study the generalized eigenvalue problem Au=λδj(u). Here δφ i (resp. δj) denotes the subdifferential in the sense of convex analysis or more generally in the sense of H. Clarke.
2010, 28(1): 375-403
doi: 10.3934/dcds.2010.28.375
+[Abstract](3092)
+[PDF](350.8KB)
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The Navier-Stokes-Voigt model of viscoelastic incompressible fluid has been recently proposed as a regularization of the three-dimensional Navier-Stokes equations for the purpose of direct numerical simulations. Besides the kinematic viscosity parameter, $\nu>0$, this model possesses a regularizing parameter, $\alpha> 0$, a given length scale parameter, so that $\frac{\alpha^2}{\nu}$ is the relaxation time of the viscoelastic fluid. In this work, we derive several statistical properties of the invariant measures associated with the solutions of the three-dimensional Navier-Stokes-Voigt equations. Moreover, we prove that, for fixed viscosity , $\nu>0$, as the regularizing parameter $\alpha$ tends to zero, there exists a subsequence of probability invariant measures converging, in a suitable sense, to a strong stationary statistical solution of the three-dimensional Navier-Stokes equations, which is a regularized version of the notion of stationary statistical solutions - a generalization of the concept of invariant measure introduced and investigated by Foias. This fact supports earlier numerical observations, and provides an additional evidence that, for small values of the regularization parameter $\alpha$, the Navier-Stokes-Voigt model can indeed be considered as a model to study the statistical properties of the three-dimensional Navier-Stokes equations and turbulent flows via direct numerical simulations.
The Navier-Stokes-Voigt model of viscoelastic incompressible fluid has been recently proposed as a regularization of the three-dimensional Navier-Stokes equations for the purpose of direct numerical simulations. Besides the kinematic viscosity parameter, $\nu>0$, this model possesses a regularizing parameter, $\alpha> 0$, a given length scale parameter, so that $\frac{\alpha^2}{\nu}$ is the relaxation time of the viscoelastic fluid. In this work, we derive several statistical properties of the invariant measures associated with the solutions of the three-dimensional Navier-Stokes-Voigt equations. Moreover, we prove that, for fixed viscosity , $\nu>0$, as the regularizing parameter $\alpha$ tends to zero, there exists a subsequence of probability invariant measures converging, in a suitable sense, to a strong stationary statistical solution of the three-dimensional Navier-Stokes equations, which is a regularized version of the notion of stationary statistical solutions - a generalization of the concept of invariant measure introduced and investigated by Foias. This fact supports earlier numerical observations, and provides an additional evidence that, for small values of the regularization parameter $\alpha$, the Navier-Stokes-Voigt model can indeed be considered as a model to study the statistical properties of the three-dimensional Navier-Stokes equations and turbulent flows via direct numerical simulations.
2010, 28(1): 405-423
doi: 10.3934/dcds.2010.28.405
+[Abstract](3186)
+[PDF](2510.7KB)
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We present unconditionally stable and convergent numerical sche- mes for gradient flows with energy of the form $ \int_\Omega( F(\nabla\phi(\x)) + \frac{\epsilon^2}{2}|\Delta\phi(\x)|^2 )$dx. The construction of the schemes involves an appropriate extension of Eyre's idea of convex-concave decomposition of the energy functional. As an application, we derive unconditionally stable and convergent schemes for epitaxial film growth models with slope selection (F(y)= 1/4(|y|2-1)2) and without slope selection (F(y)= -1/2ln(1+|y|2)). We conclude the paper with some preliminary computations that employ the proposed schemes.
We present unconditionally stable and convergent numerical sche- mes for gradient flows with energy of the form $ \int_\Omega( F(\nabla\phi(\x)) + \frac{\epsilon^2}{2}|\Delta\phi(\x)|^2 )$dx. The construction of the schemes involves an appropriate extension of Eyre's idea of convex-concave decomposition of the energy functional. As an application, we derive unconditionally stable and convergent schemes for epitaxial film growth models with slope selection (F(y)= 1/4(|y|2-1)2) and without slope selection (F(y)= -1/2ln(1+|y|2)). We conclude the paper with some preliminary computations that employ the proposed schemes.
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