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1534-0392
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Communications on Pure & Applied Analysis
September 2014 , Volume 13 , Issue 5
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2014, 13(5): i-x
doi: 10.3934/cpaa.2014.13.5i
+[Abstract](1150)
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Abstract:
-- Mark Iosifovich, how do You think, which scientists have influenced You in the very beginning of Your academic career?
If viewed chronologically, there were, first of all, my teachers at Lvov State University, which I entered in 1939. The University formerly bore the name of king Kazimir and then became the Ivan Franko University, where the Dean of the Mathematics Department Stefan Banach, a brilliant mathematician, worked. We were taught by the most outstanding professors of the Banach's school: Bronislaw Knaster -- analytical geometry, Yuliush Schauder -- theoretical mechanics, Professor Stanislaw Mazur -- differential geometry. Professor Vladislav Orlicz gave lectures on algebra. All this teaching was in Polish. Only the Deputy Dean Professor Myron Zaritsky gave lectures in Ukrainian.
-- Mark Iosifovich, how do You think, which scientists have influenced You in the very beginning of Your academic career?
If viewed chronologically, there were, first of all, my teachers at Lvov State University, which I entered in 1939. The University formerly bore the name of king Kazimir and then became the Ivan Franko University, where the Dean of the Mathematics Department Stefan Banach, a brilliant mathematician, worked. We were taught by the most outstanding professors of the Banach's school: Bronislaw Knaster -- analytical geometry, Yuliush Schauder -- theoretical mechanics, Professor Stanislaw Mazur -- differential geometry. Professor Vladislav Orlicz gave lectures on algebra. All this teaching was in Polish. Only the Deputy Dean Professor Myron Zaritsky gave lectures in Ukrainian.
2014, 13(5): 1685-1718
doi: 10.3934/cpaa.2014.13.1685
+[Abstract](1568)
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Abstract:
One of important problems in mathematical physics concerns derivation of point dynamics from field equations. The most common approach to this problem is based on WKB method. Here we describe a different method based on the concept of trajectory of concentration. When we applied this method to nonlinear Klein-Gordon equation, we derived relativistic Newton's law and Einstein's formula for inertial mass. Here we apply the same approach to nonlinear Schrodinger equation and derive non-relativistic Newton's law for the trajectory of concentration.
One of important problems in mathematical physics concerns derivation of point dynamics from field equations. The most common approach to this problem is based on WKB method. Here we describe a different method based on the concept of trajectory of concentration. When we applied this method to nonlinear Klein-Gordon equation, we derived relativistic Newton's law and Einstein's formula for inertial mass. Here we apply the same approach to nonlinear Schrodinger equation and derive non-relativistic Newton's law for the trajectory of concentration.
2014, 13(5): 1719-1736
doi: 10.3934/cpaa.2014.13.1719
+[Abstract](1806)
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Abstract:
We consider a non zero sum stochastic differential game with a maximum $n$ players, where the players control a diffusion in order to minimise a certain cost functional. During the game it is possible that present players may die or new players may appear. The death, respectively the birth time of a player is exponentially distributed with intensities that depend on the diffusion and the controls of the players who are alive. We show how the game is related to a system of partial differential equations with a special coupling in the zero order terms. We provide an existence result for solutions in appropriate spaces that allow to construct Nash optimal feedback controls. The paper is related to a previous result in a similar setting for two players leading to a parabolic system of Bellman equations [4]. Here, we study the elliptic case (infinite horizon) and present the generalisation to more than two players.
We consider a non zero sum stochastic differential game with a maximum $n$ players, where the players control a diffusion in order to minimise a certain cost functional. During the game it is possible that present players may die or new players may appear. The death, respectively the birth time of a player is exponentially distributed with intensities that depend on the diffusion and the controls of the players who are alive. We show how the game is related to a system of partial differential equations with a special coupling in the zero order terms. We provide an existence result for solutions in appropriate spaces that allow to construct Nash optimal feedback controls. The paper is related to a previous result in a similar setting for two players leading to a parabolic system of Bellman equations [4]. Here, we study the elliptic case (infinite horizon) and present the generalisation to more than two players.
2014, 13(5): 1737-1757
doi: 10.3934/cpaa.2014.13.1737
+[Abstract](1455)
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We construct the 1962 Kolmogorov-Obukhov statistical theory of turbulence from the stochastic Navier-Stokes equations driven by generic noise. The intermittency corrections to the scaling exponents of the structure functions of turbulence are given by the She-Leveque intermittency corrections. We show how they are produced by She-Waymire log-Poisson processes, that are generated by the Feynmann-Kac formula from the stochastic Navier-Stokes equation. We find the Kolmogorov-Hopf equations and compute the invariant measures of turbulence for 1-point and 2-point statistics. Then projecting these measures we find the formulas for the probability distribution functions (PDFs) of the velocity differences in the structure functions. In the limit of zero intermittency, these PDFs reduce to the Generalized Hyperbolic Distributions of Barndorff-Nilsen.
We construct the 1962 Kolmogorov-Obukhov statistical theory of turbulence from the stochastic Navier-Stokes equations driven by generic noise. The intermittency corrections to the scaling exponents of the structure functions of turbulence are given by the She-Leveque intermittency corrections. We show how they are produced by She-Waymire log-Poisson processes, that are generated by the Feynmann-Kac formula from the stochastic Navier-Stokes equation. We find the Kolmogorov-Hopf equations and compute the invariant measures of turbulence for 1-point and 2-point statistics. Then projecting these measures we find the formulas for the probability distribution functions (PDFs) of the velocity differences in the structure functions. In the limit of zero intermittency, these PDFs reduce to the Generalized Hyperbolic Distributions of Barndorff-Nilsen.
2014, 13(5): 1759-1778
doi: 10.3934/cpaa.2014.13.1759
+[Abstract](1680)
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We prove well-posedness of energy type solutions to an interacting system consisting of the 3D linearized Euler equations and a (possibly nonlinear) elastic plate equation describing large deflections of a flexible part of the boundary. In the damped case under some conditions concerning the plate nonlinearity we prove the existence of a compact global attractor for the corresponding dynamical system and describe the situations when this attractor has a finite fractal dimension.
We prove well-posedness of energy type solutions to an interacting system consisting of the 3D linearized Euler equations and a (possibly nonlinear) elastic plate equation describing large deflections of a flexible part of the boundary. In the damped case under some conditions concerning the plate nonlinearity we prove the existence of a compact global attractor for the corresponding dynamical system and describe the situations when this attractor has a finite fractal dimension.
2014, 13(5): 1779-1787
doi: 10.3934/cpaa.2014.13.1779
+[Abstract](1331)
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The existence, regularity and uniqueness of a local diffeomorphism $\varphi$ satisfying \begin{eqnarray} g_{i}(\varphi) \det\nabla\varphi=f_{i}\quad for\ every\ 1\leq i\leq n \end{eqnarray} is discussed.
The existence, regularity and uniqueness of a local diffeomorphism $\varphi$ satisfying \begin{eqnarray} g_{i}(\varphi) \det\nabla\varphi=f_{i}\quad for\ every\ 1\leq i\leq n \end{eqnarray} is discussed.
2014, 13(5): 1789-1797
doi: 10.3934/cpaa.2014.13.1789
+[Abstract](1332)
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An abstract evolution equation in Hilbert spaces with Hölder continuous drift is considered. By proceeding as in [3], we transform the equation in another equation with Lipschitz continuous coefficients.Then we prove existence and uniqueness of this equation by a fixed point argument.
An abstract evolution equation in Hilbert spaces with Hölder continuous drift is considered. By proceeding as in [3], we transform the equation in another equation with Lipschitz continuous coefficients.Then we prove existence and uniqueness of this equation by a fixed point argument.
2014, 13(5): 1799-1814
doi: 10.3934/cpaa.2014.13.1799
+[Abstract](1506)
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We extend some previous local energy method to the study the free boundary generated by the solutions of quenching type parabolic problems involving a negative power of the unknown in the equation.
We extend some previous local energy method to the study the free boundary generated by the solutions of quenching type parabolic problems involving a negative power of the unknown in the equation.
2014, 13(5): 1815-1854
doi: 10.3934/cpaa.2014.13.1815
+[Abstract](1305)
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The simplest semilinear parabolic equation of normal type with periodic boundary condition is considered, and the problem of stabilization to zero of its solution with arbitrary initial condition by starting control supported in a prescribed subset is investigated. This problem is reduced to one inequality for starting control, and the proof of this inequality is given.
The simplest semilinear parabolic equation of normal type with periodic boundary condition is considered, and the problem of stabilization to zero of its solution with arbitrary initial condition by starting control supported in a prescribed subset is investigated. This problem is reduced to one inequality for starting control, and the proof of this inequality is given.
2014, 13(5): 1855-1890
doi: 10.3934/cpaa.2014.13.1855
+[Abstract](1879)
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We consider a non-isothermal modified viscous Cahn-Hilliard equation which was previously analyzed by M. Grasselli et al. Such an equation is characterized by an inertial term and it is coupled with a hyperbolic heat equation from the Maxwell-Cattaneo's law. We analyze the case in which the order parameter is subject to a dynamic boundary condition. This assumption requires a more refined strategy to extend the previous results to the present case. More precisely, we first prove the well-posedness for solutions with finite energy as well as for weak solutions. Then we establish the existence of a global attractor. Finally, we prove the convergence of any given weak solution to a single equilibrium by using a suitable Łojasiewicz-Simon inequality.
We consider a non-isothermal modified viscous Cahn-Hilliard equation which was previously analyzed by M. Grasselli et al. Such an equation is characterized by an inertial term and it is coupled with a hyperbolic heat equation from the Maxwell-Cattaneo's law. We analyze the case in which the order parameter is subject to a dynamic boundary condition. This assumption requires a more refined strategy to extend the previous results to the present case. More precisely, we first prove the well-posedness for solutions with finite energy as well as for weak solutions. Then we establish the existence of a global attractor. Finally, we prove the convergence of any given weak solution to a single equilibrium by using a suitable Łojasiewicz-Simon inequality.
2014, 13(5): 1891-1906
doi: 10.3934/cpaa.2014.13.1891
+[Abstract](1763)
+[PDF](437.4KB)
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In this paper we study the structural properties of global attractors of multi-valued semiflows generated by regular solutions of reaction-diffusion system without uniqueness of the Cauchy problem. Under additional gradient-like condition on the nonlinear term we prove that the global attractor coincides with the unstable manifold of the set of stationary points, and with the stable one when we consider only bounded complete trajectories. As an example we consider a generalized Fitz-Hugh-Nagumo system. We also suggest additional conditions, which provide that the global attractor is a bounded set in $(L^\infty(\Omega))^N$ and compact in $(H_0^1 (\Omega))^N$.
In this paper we study the structural properties of global attractors of multi-valued semiflows generated by regular solutions of reaction-diffusion system without uniqueness of the Cauchy problem. Under additional gradient-like condition on the nonlinear term we prove that the global attractor coincides with the unstable manifold of the set of stationary points, and with the stable one when we consider only bounded complete trajectories. As an example we consider a generalized Fitz-Hugh-Nagumo system. We also suggest additional conditions, which provide that the global attractor is a bounded set in $(L^\infty(\Omega))^N$ and compact in $(H_0^1 (\Omega))^N$.
2014, 13(5): 1907-1933
doi: 10.3934/cpaa.2014.13.1907
+[Abstract](1668)
+[PDF](535.9KB)
Abstract:
Reaction-diffusion equations are considered on a bounded domain $\Omega$ in $\mathbb{R}^d$ with a reaction term that is switched off at a point in space when the solution first exceeds a specified threshold and thereafter remains switched off at that point, which leads to a discontinuous reaction term with delay. This problem is formulated as a parabolic partial differential inclusion with delay. The reaction-free region forms what could be called dead core in a biological sense rather than that used elsewhere in the literature for parabolic PDEs. The existence of solutions in $L^2(\Omega)$ is established firstly for initial data in $L^{\infty}(\Omega)$ and in $W_0^{1,2}(\Omega)$ by different methods, with $d$ $=$ $2$ or $3$ in the first case and $d$ $\geq$ $2$ in the second. Solutions here are interpreted in the sense of integral or strong solutions of nonhomogeneous linear parabolic equations in $L^2(\Omega)$ that are generalised to selectors of the corresponding nonhomogeneous linear parabolic differential inclusions and are shown to be equivalent under the assumptions used in the paper.
Reaction-diffusion equations are considered on a bounded domain $\Omega$ in $\mathbb{R}^d$ with a reaction term that is switched off at a point in space when the solution first exceeds a specified threshold and thereafter remains switched off at that point, which leads to a discontinuous reaction term with delay. This problem is formulated as a parabolic partial differential inclusion with delay. The reaction-free region forms what could be called dead core in a biological sense rather than that used elsewhere in the literature for parabolic PDEs. The existence of solutions in $L^2(\Omega)$ is established firstly for initial data in $L^{\infty}(\Omega)$ and in $W_0^{1,2}(\Omega)$ by different methods, with $d$ $=$ $2$ or $3$ in the first case and $d$ $\geq$ $2$ in the second. Solutions here are interpreted in the sense of integral or strong solutions of nonhomogeneous linear parabolic equations in $L^2(\Omega)$ that are generalised to selectors of the corresponding nonhomogeneous linear parabolic differential inclusions and are shown to be equivalent under the assumptions used in the paper.
2014, 13(5): 1935-1969
doi: 10.3934/cpaa.2014.13.1935
+[Abstract](1885)
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We address the long-time behavior of a non-rotational von Karman plate in an inviscid potential flow. The model arises in aeroelasticity and models the interaction between a thin, nonlinear panel and a flow of gas in which it is immersed [6, 21, 23]. Recent results in [16, 18] show that the plate component of the dynamics (in the presence of a physical plate nonlinearity) converge to a global compact attracting set of finite dimension; these results were obtained in the absence of mechanical damping of any type. Here we show that, by incorporating mechanical damping the full flow-plate system, full trajectories---both plate and flow---converge strongly to (the set of) stationary states. Weak convergence results require ``minimal" interior damping, and strong convergence of the dynamics are shown with sufficiently large damping. We require the existence of a ``good" energy balance equation, which is only available when the flows are subsonic. Our proof is based on first showing the convergence properties for regular solutions, which in turn requires propagation of initial regularity on the infinite horizon. Then, we utilize the exponential decay of the difference of two plate trajectories to show that full flow-plate trajectories are uniform-in-time Hadamard continuous. This allows us to pass convergence properties of smooth initial data to finite energy type initial data. Physically, our results imply that flutter (a non-static end behavior) does not occur in subsonic dynamics. While such results were known for rotational (compact/regular) plate dynamics [14] (and references therein), the result presented herein is the first such result obtained for non-regularized---the most physically relevant---models.
We address the long-time behavior of a non-rotational von Karman plate in an inviscid potential flow. The model arises in aeroelasticity and models the interaction between a thin, nonlinear panel and a flow of gas in which it is immersed [6, 21, 23]. Recent results in [16, 18] show that the plate component of the dynamics (in the presence of a physical plate nonlinearity) converge to a global compact attracting set of finite dimension; these results were obtained in the absence of mechanical damping of any type. Here we show that, by incorporating mechanical damping the full flow-plate system, full trajectories---both plate and flow---converge strongly to (the set of) stationary states. Weak convergence results require ``minimal" interior damping, and strong convergence of the dynamics are shown with sufficiently large damping. We require the existence of a ``good" energy balance equation, which is only available when the flows are subsonic. Our proof is based on first showing the convergence properties for regular solutions, which in turn requires propagation of initial regularity on the infinite horizon. Then, we utilize the exponential decay of the difference of two plate trajectories to show that full flow-plate trajectories are uniform-in-time Hadamard continuous. This allows us to pass convergence properties of smooth initial data to finite energy type initial data. Physically, our results imply that flutter (a non-static end behavior) does not occur in subsonic dynamics. While such results were known for rotational (compact/regular) plate dynamics [14] (and references therein), the result presented herein is the first such result obtained for non-regularized---the most physically relevant---models.
2014, 13(5): 1971-1987
doi: 10.3934/cpaa.2014.13.1971
+[Abstract](1545)
+[PDF](409.4KB)
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Our aim in this paper is to study the well-posedness and the asymptotic behavior, in terms of finite-dimensional attractors, for the conserved Caginalp phase-field system based on the Maxwell-Cattaneo law, instead of the usual Fourier law, for heat conduction. The system consists of the equation for the order parameter and that for the enthalpy, instead of the relative temperature or the thermal displacement variable.
Our aim in this paper is to study the well-posedness and the asymptotic behavior, in terms of finite-dimensional attractors, for the conserved Caginalp phase-field system based on the Maxwell-Cattaneo law, instead of the usual Fourier law, for heat conduction. The system consists of the equation for the order parameter and that for the enthalpy, instead of the relative temperature or the thermal displacement variable.
2014, 13(5): 1989-2004
doi: 10.3934/cpaa.2014.13.1989
+[Abstract](1587)
+[PDF](412.6KB)
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We discuss the existence of the global attractor for a family of processes $U_\sigma(t,\tau)$ acting on a metric space $X$ and depending on a symbol $\sigma$ belonging to some other metric space $\Sigma$. Such an attractor is uniform with respect to $\sigma\in\Sigma$, as well as with respect to the choice of the initial time $\tau\in R$. The existence of the attractor is established for totally dissipative processes without any continuity assumption. When the process satisfies some additional (but rather mild) continuity-like hypotheses, a characterization of the attractor is given.
We discuss the existence of the global attractor for a family of processes $U_\sigma(t,\tau)$ acting on a metric space $X$ and depending on a symbol $\sigma$ belonging to some other metric space $\Sigma$. Such an attractor is uniform with respect to $\sigma\in\Sigma$, as well as with respect to the choice of the initial time $\tau\in R$. The existence of the attractor is established for totally dissipative processes without any continuity assumption. When the process satisfies some additional (but rather mild) continuity-like hypotheses, a characterization of the attractor is given.
2014, 13(5): 2005-2038
doi: 10.3934/cpaa.2014.13.2005
+[Abstract](1498)
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In continuation with earlier works on the shallow water equations in a rectangle [10, 11], we investigate in this article the fully inviscid nonlinear shallow water equations in space dimension two in a rectangle $(0,1)_x \times (0,1)_y$. We address in this article the subcritical case, corresponding to the condition (3) below. Assuming space periodicity in the $y$-direction, we propose the boundary conditions for the $x$-direction which are suited for the subcritical case and develop, for this problem, results of existence, uniqueness and regularity of solutions locally in time for the corresponding initial and boundary value problem.
In continuation with earlier works on the shallow water equations in a rectangle [10, 11], we investigate in this article the fully inviscid nonlinear shallow water equations in space dimension two in a rectangle $(0,1)_x \times (0,1)_y$. We address in this article the subcritical case, corresponding to the condition (3) below. Assuming space periodicity in the $y$-direction, we propose the boundary conditions for the $x$-direction which are suited for the subcritical case and develop, for this problem, results of existence, uniqueness and regularity of solutions locally in time for the corresponding initial and boundary value problem.
2014, 13(5): 2039-2058
doi: 10.3934/cpaa.2014.13.2039
+[Abstract](1497)
+[PDF](159.4KB)
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In the seminal work [MR 1009594], Fitzpatrick proved that for any maximal monotone operator $\alpha: V\to {\mathcal P}(V')$ ($V$ being a real Banach space) there exists a lower semicontinuous, convex representative function $f_\alpha: V \times V'\to R\cup \{+\infty\}$ such that \begin{eqnarray} f_\alpha(v,v') \ge \langle v',v\rangle \quad\;\forall (v,v'), \qquad\quad f_\alpha(v,v') = \langle v',v\rangle \;\;\Leftrightarrow\;\;\; v'\in \alpha(v). \end{eqnarray}
Here we assume that $\alpha_v$ is a maximal monotone operator for any $v\in V$, and extend the Fitzpatrick theory to provide a new variational formulation for either stationary or evolutionary (nonmonotone) inclusions of the form $\alpha_v(v) \ni v'$. For any $v'\in V'$, we prove existence of a solution via the classical minimax theorem of Ky Fan. Applications include stationary and evolutionary pseudo-monotone operators, and variational inequalities.
In the seminal work [MR 1009594], Fitzpatrick proved that for any maximal monotone operator $\alpha: V\to {\mathcal P}(V')$ ($V$ being a real Banach space) there exists a lower semicontinuous, convex representative function $f_\alpha: V \times V'\to R\cup \{+\infty\}$ such that \begin{eqnarray} f_\alpha(v,v') \ge \langle v',v\rangle \quad\;\forall (v,v'), \qquad\quad f_\alpha(v,v') = \langle v',v\rangle \;\;\Leftrightarrow\;\;\; v'\in \alpha(v). \end{eqnarray}
Here we assume that $\alpha_v$ is a maximal monotone operator for any $v\in V$, and extend the Fitzpatrick theory to provide a new variational formulation for either stationary or evolutionary (nonmonotone) inclusions of the form $\alpha_v(v) \ni v'$. For any $v'\in V'$, we prove existence of a solution via the classical minimax theorem of Ky Fan. Applications include stationary and evolutionary pseudo-monotone operators, and variational inequalities.
2014, 13(5): 2059-2093
doi: 10.3934/cpaa.2014.13.2059
+[Abstract](1625)
+[PDF](668.9KB)
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We apply the dynamical approach to the study of the second order semi-linear elliptic boundary value problem in a cylindrical domain with a small parameter $\varepsilon$ at the second derivative with respect to the variable $t$ corresponding to the axis of the cylinder. We prove that, under natural assumptions on the nonlinear interaction function $f$ and the external forces $g(t)$, this problem possesses the uniform attractor $\mathcal A_\varepsilon$ and that these attractors tend as $\varepsilon \to 0$ to the attractor $\mathcal A_0$ of the limit parabolic equation. Moreover, in case where the limit attractor $\mathcal A_0$ is regular, we give the detailed description of the structure of the uniform attractor $\mathcal A_\varepsilon$, if $\varepsilon>0$ is small enough, and estimate the symmetric distance between the attractors $\mathcal A_\varepsilon$ and $\mathcal A_0$.
We apply the dynamical approach to the study of the second order semi-linear elliptic boundary value problem in a cylindrical domain with a small parameter $\varepsilon$ at the second derivative with respect to the variable $t$ corresponding to the axis of the cylinder. We prove that, under natural assumptions on the nonlinear interaction function $f$ and the external forces $g(t)$, this problem possesses the uniform attractor $\mathcal A_\varepsilon$ and that these attractors tend as $\varepsilon \to 0$ to the attractor $\mathcal A_0$ of the limit parabolic equation. Moreover, in case where the limit attractor $\mathcal A_0$ is regular, we give the detailed description of the structure of the uniform attractor $\mathcal A_\varepsilon$, if $\varepsilon>0$ is small enough, and estimate the symmetric distance between the attractors $\mathcal A_\varepsilon$ and $\mathcal A_0$.
2014, 13(5): 2095-2113
doi: 10.3934/cpaa.2014.13.2095
+[Abstract](2089)
+[PDF](173.1KB)
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The investigation of stability for hereditary systems is often related to the construction of Lyapunov functionals. The general method of Lyapunov functionals construction, which was proposed by V.Kolmanovskii and L.Shaikhet, is used here to investigate the stability of stochastic delay evolution equations, in particular, for stochastic partial differential equations. This method had already been successfully used for functional-differential equations, for difference equations with discrete time, and for difference equations with continuous time. It is shown that the stability conditions obtained for stochastic 2D Navier-Stokes model with delays are essentially better than the known ones.
The investigation of stability for hereditary systems is often related to the construction of Lyapunov functionals. The general method of Lyapunov functionals construction, which was proposed by V.Kolmanovskii and L.Shaikhet, is used here to investigate the stability of stochastic delay evolution equations, in particular, for stochastic partial differential equations. This method had already been successfully used for functional-differential equations, for difference equations with discrete time, and for difference equations with continuous time. It is shown that the stability conditions obtained for stochastic 2D Navier-Stokes model with delays are essentially better than the known ones.
2014, 13(5): 2115-2126
doi: 10.3934/cpaa.2014.13.2115
+[Abstract](1402)
+[PDF](2128.2KB)
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In this article we investigate, via numerical computations, the intersection properties of the nodal set of the eigenfunctions of the Laplace-Beltrami operator for smooth surfaces in $R^3$ (the nodal set of a continuous function is the set of those points at which the function vanishes). First, we briefly discuss the numerical solution of the eigenvalue/eigenfunction problem for the Laplace-Beltrami operator on bounded surfaces of $R^3$, and then consider some specific surfaces and visualize how the nodal lines intersect (or not) depending of the symmetries verified by the surface. After validating our computational methodology with the surface of a ring torus, we will investigate a simple surface without symmetry and observe that in that case the nodal set of the computed eigenfunctions consists of non intersecting lines, suggesting some conjecture. We observe also that for the above symmetry-free surface, the number of connected components of the nodal set varies non-monotonically with the rank of the associated eigenvalue (assuming that the eigenvalues are ordered by increasing value).
In this article we investigate, via numerical computations, the intersection properties of the nodal set of the eigenfunctions of the Laplace-Beltrami operator for smooth surfaces in $R^3$ (the nodal set of a continuous function is the set of those points at which the function vanishes). First, we briefly discuss the numerical solution of the eigenvalue/eigenfunction problem for the Laplace-Beltrami operator on bounded surfaces of $R^3$, and then consider some specific surfaces and visualize how the nodal lines intersect (or not) depending of the symmetries verified by the surface. After validating our computational methodology with the surface of a ring torus, we will investigate a simple surface without symmetry and observe that in that case the nodal set of the computed eigenfunctions consists of non intersecting lines, suggesting some conjecture. We observe also that for the above symmetry-free surface, the number of connected components of the nodal set varies non-monotonically with the rank of the associated eigenvalue (assuming that the eigenvalues are ordered by increasing value).
2014, 13(5): 2127-2140
doi: 10.3934/cpaa.2014.13.2127
+[Abstract](1368)
+[PDF](230.0KB)
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We construct semi-integral curves which bound the projections of the global attractors of the 3D NS-$\alpha$ and 3D Leray-$\alpha$ sub-grid scale turbulence models in the plane spanned by their energy and enstrophy. We note the dependence of these bounds on the filter width parameter $\alpha$, and determine subregions where each quantity, energy and enstrophy, must decrease, while isolating one which is recurrent.
We construct semi-integral curves which bound the projections of the global attractors of the 3D NS-$\alpha$ and 3D Leray-$\alpha$ sub-grid scale turbulence models in the plane spanned by their energy and enstrophy. We note the dependence of these bounds on the filter width parameter $\alpha$, and determine subregions where each quantity, energy and enstrophy, must decrease, while isolating one which is recurrent.
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