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Discrete & Continuous Dynamical Systems - A
October 2012 , Volume 32 , Issue 10
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2012, 32(10): 3325-3377
doi: 10.3934/dcds.2012.32.3325
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Abstract:
The paper intends to lay out the first steps towards constructing a unified framework to understand the symplectic and spectral theory of finite dimensional integrable Hamiltonian systems. While it is difficult to know what the best approach to such a large classification task would be, it is possible to single out some promising directions and preliminary problems. This paper discusses them and hints at a possible path, still loosely defined, to arrive at a classification. It mainly relies on recent progress concerning integrable systems with only non-hyperbolic and non-degenerate singularities.
This work originated in an attempt to develop a theory aimed at answering some questions in quantum spectroscopy. Even though quantum integrable systems date back to the early days of quantum mechanics, such as the work of Bohr, Sommerfeld and Einstein, the theory did not blossom at the time. The development of semiclassical analysis with microlocal techniques in the last forty years now permits a constant interplay between spectral theory and symplectic geometry. A main goal of this paper is to emphasize the symplectic issues that are relevant to quantum mechanical integrable systems, and to propose a strategy to solve them.
The paper intends to lay out the first steps towards constructing a unified framework to understand the symplectic and spectral theory of finite dimensional integrable Hamiltonian systems. While it is difficult to know what the best approach to such a large classification task would be, it is possible to single out some promising directions and preliminary problems. This paper discusses them and hints at a possible path, still loosely defined, to arrive at a classification. It mainly relies on recent progress concerning integrable systems with only non-hyperbolic and non-degenerate singularities.
This work originated in an attempt to develop a theory aimed at answering some questions in quantum spectroscopy. Even though quantum integrable systems date back to the early days of quantum mechanics, such as the work of Bohr, Sommerfeld and Einstein, the theory did not blossom at the time. The development of semiclassical analysis with microlocal techniques in the last forty years now permits a constant interplay between spectral theory and symplectic geometry. A main goal of this paper is to emphasize the symplectic issues that are relevant to quantum mechanical integrable systems, and to propose a strategy to solve them.
2012, 32(10): 3379-3397
doi: 10.3934/dcds.2012.32.3379
+[Abstract](1083)
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Abstract:
This paper introduces a notion of regularity (or irregularity) of the point at infinity ($\infty$) for the unbounded open set $\Omega\subset {\mathbb R}^{N}$ concerning second order uniformly elliptic equations with bounded and measurable coefficients, according as whether the ${\mathcal A}$- harmonic measure of $\infty$ is zero (or positive). A necessary and sufficient condition for the existence of a unique bounded solution to the Dirichlet problem in an arbitrary open set of ${\mathbb R}^{N}, N\ge 3$ is established in terms of the Wiener test for the regularity of $\infty$. It coincides with the Wiener test for the regularity of $\infty$ in the case of Laplace equation. From the topological point of view, the Wiener test at $\infty$ presents thinness criteria of sets near $\infty$ in fine topology. Precisely, the open set is a deleted neigborhood of $\infty$ in fine topology if and only if $\infty$ is irregular.
This paper introduces a notion of regularity (or irregularity) of the point at infinity ($\infty$) for the unbounded open set $\Omega\subset {\mathbb R}^{N}$ concerning second order uniformly elliptic equations with bounded and measurable coefficients, according as whether the ${\mathcal A}$- harmonic measure of $\infty$ is zero (or positive). A necessary and sufficient condition for the existence of a unique bounded solution to the Dirichlet problem in an arbitrary open set of ${\mathbb R}^{N}, N\ge 3$ is established in terms of the Wiener test for the regularity of $\infty$. It coincides with the Wiener test for the regularity of $\infty$ in the case of Laplace equation. From the topological point of view, the Wiener test at $\infty$ presents thinness criteria of sets near $\infty$ in fine topology. Precisely, the open set is a deleted neigborhood of $\infty$ in fine topology if and only if $\infty$ is irregular.
2012, 32(10): 3399-3419
doi: 10.3934/dcds.2012.32.3399
+[Abstract](1222)
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Abstract:
We give an explicite formula to compute rotation numbers of piecewise linear (PL) circle homeomorphisms $f$ with the product of $f$-jumps in the break points contained in a same orbit is trivial. In particular, a simple formulas are then given for particular PL-homeomorphisms such as the PL-Herman's examples. We also deduce that if the slopes of $f$ are integral powers of an integer $m\geq 2$ and break points and their images under $f$ are $m$-adic rational numbers, then the rotation number of $f$ is rational.
We give an explicite formula to compute rotation numbers of piecewise linear (PL) circle homeomorphisms $f$ with the product of $f$-jumps in the break points contained in a same orbit is trivial. In particular, a simple formulas are then given for particular PL-homeomorphisms such as the PL-Herman's examples. We also deduce that if the slopes of $f$ are integral powers of an integer $m\geq 2$ and break points and their images under $f$ are $m$-adic rational numbers, then the rotation number of $f$ is rational.
2012, 32(10): 3421-3431
doi: 10.3934/dcds.2012.32.3421
+[Abstract](1146)
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Abstract:
Given a sequence of sets $A_n \subseteq \{0,\ldots,n-1\}$, the Furstenberg correspondence principle provides a shift-invariant measure on $2^N$ that encodes combinatorial information about infinitely many of the $A_n$'s. Here it is shown that this process can be inverted, so that for any such measure, ergodic or not, there are finite sets whose combinatorial properties approximate it arbitarily well. The finite approximations are obtained from the measure by an explicit construction, with an explicit upper bound on how large $n$ has to be to yield a sufficiently good approximation.
We draw conclusions for computable measure theory, and show, in particular, that given any computable shift-invariant measure on $2^N$, there is a computable element of $2^N$ that is generic for the measure. We also consider a generalization of the correspondence principle to countable discrete amenable groups, and once again provide an effective inverse.
Given a sequence of sets $A_n \subseteq \{0,\ldots,n-1\}$, the Furstenberg correspondence principle provides a shift-invariant measure on $2^N$ that encodes combinatorial information about infinitely many of the $A_n$'s. Here it is shown that this process can be inverted, so that for any such measure, ergodic or not, there are finite sets whose combinatorial properties approximate it arbitarily well. The finite approximations are obtained from the measure by an explicit construction, with an explicit upper bound on how large $n$ has to be to yield a sufficiently good approximation.
We draw conclusions for computable measure theory, and show, in particular, that given any computable shift-invariant measure on $2^N$, there is a computable element of $2^N$ that is generic for the measure. We also consider a generalization of the correspondence principle to countable discrete amenable groups, and once again provide an effective inverse.
2012, 32(10): 3433-3457
doi: 10.3934/dcds.2012.32.3433
+[Abstract](1275)
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Abstract:
We study relations between Rauzy classes coming from an interval exchange map and the corresponding connected components of strata of the moduli space of Abelian differentials. This gives a criterion to decide whether two permutations are in the same Rauzy class or not, without actually computing them. We prove a similar result for Rauzy classes corresponding to quadratic differentials.
We study relations between Rauzy classes coming from an interval exchange map and the corresponding connected components of strata of the moduli space of Abelian differentials. This gives a criterion to decide whether two permutations are in the same Rauzy class or not, without actually computing them. We prove a similar result for Rauzy classes corresponding to quadratic differentials.
2012, 32(10): 3459-3484
doi: 10.3934/dcds.2012.32.3459
+[Abstract](1028)
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Abstract:
Considered herein is the generalized two-component periodic Camassa-Holm system. The precise blow-up scenarios of strong solutions and several results of blow-up solutions with certain initial profiles are described in detail. The exact blow-up rates are also determined. Finally, a sufficient condition for global solutions is established.
Considered herein is the generalized two-component periodic Camassa-Holm system. The precise blow-up scenarios of strong solutions and several results of blow-up solutions with certain initial profiles are described in detail. The exact blow-up rates are also determined. Finally, a sufficient condition for global solutions is established.
2012, 32(10): 3485-3499
doi: 10.3934/dcds.2012.32.3485
+[Abstract](870)
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Abstract:
We extend Sharkovsky's Theorem to several new classes of spaces, which include some well-known examples of non-locally connected continua, such as the topologist's sine curve and the Warsaw circle. In some of these examples the theorem applies directly (with the same ordering), and in other examples the theorem requires an altered partial ordering on the integers. In the latter case, we describe all possible sets of periods for functions on such spaces, which are based on multiples of Sharkovsky's order.
We extend Sharkovsky's Theorem to several new classes of spaces, which include some well-known examples of non-locally connected continua, such as the topologist's sine curve and the Warsaw circle. In some of these examples the theorem applies directly (with the same ordering), and in other examples the theorem requires an altered partial ordering on the integers. In the latter case, we describe all possible sets of periods for functions on such spaces, which are based on multiples of Sharkovsky's order.
2012, 32(10): 3501-3524
doi: 10.3934/dcds.2012.32.3501
+[Abstract](1204)
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In this paper, we investigate the formation of singularities of the classical solution to the Cauchy problem of quasi-linear hyperbolic system and give a sharp limit formula for the lifespan of the classical solution. It is important that we only require that the initial data are sufficiently small in the $L^1$ sense and the BV sense.
In this paper, we investigate the formation of singularities of the classical solution to the Cauchy problem of quasi-linear hyperbolic system and give a sharp limit formula for the lifespan of the classical solution. It is important that we only require that the initial data are sufficiently small in the $L^1$ sense and the BV sense.
2012, 32(10): 3525-3537
doi: 10.3934/dcds.2012.32.3525
+[Abstract](1258)
+[PDF](447.6KB)
Abstract:
We consider a generalisation of the baker's transformation, consisting of a skew-product of contractions and a $\beta$-transformation. The Hausdorff dimension and Lebesgue measure of the attractor is calculated for a set of parameters with positive measure. The proofs use a new transverality lemma similar to Solomyak's [12]. This transversality, which is applicable to the considered class of maps holds for a larger set of parameters than Solomyak's transversality.
We consider a generalisation of the baker's transformation, consisting of a skew-product of contractions and a $\beta$-transformation. The Hausdorff dimension and Lebesgue measure of the attractor is calculated for a set of parameters with positive measure. The proofs use a new transverality lemma similar to Solomyak's [12]. This transversality, which is applicable to the considered class of maps holds for a larger set of parameters than Solomyak's transversality.
2012, 32(10): 3539-3565
doi: 10.3934/dcds.2012.32.3539
+[Abstract](1438)
+[PDF](1016.8KB)
Abstract:
Lyapunov functions are an important tool to determine the basin of attraction of exponentially stable equilibria in dynamical systems. In Marinósson (2002), a method to construct Lyapunov functions was presented, using finite differences on finite elements and thus transforming the construction problem into a linear programming problem. In Hafstein (2004), it was shown that this method always succeeds in constructing a Lyapunov function, except for a small, given neighbourhood of the equilibrium.
For two-dimensional systems, this local problem was overcome by choosing a fan-like triangulation around the equilibrium. In Giesl/Hafstein (2010) the existence of a piecewise linear Lyapunov function was shown, and in Giesl/Hafstein (2012) it was shown that the above method with a fan-like triangulation always succeeds in constructing a Lyapunov function, without any local exception. However, the previous papers only considered two-dimensional systems. This paper generalises the existence of piecewise linear Lyapunov functions to arbitrary dimensions.
Lyapunov functions are an important tool to determine the basin of attraction of exponentially stable equilibria in dynamical systems. In Marinósson (2002), a method to construct Lyapunov functions was presented, using finite differences on finite elements and thus transforming the construction problem into a linear programming problem. In Hafstein (2004), it was shown that this method always succeeds in constructing a Lyapunov function, except for a small, given neighbourhood of the equilibrium.
For two-dimensional systems, this local problem was overcome by choosing a fan-like triangulation around the equilibrium. In Giesl/Hafstein (2010) the existence of a piecewise linear Lyapunov function was shown, and in Giesl/Hafstein (2012) it was shown that the above method with a fan-like triangulation always succeeds in constructing a Lyapunov function, without any local exception. However, the previous papers only considered two-dimensional systems. This paper generalises the existence of piecewise linear Lyapunov functions to arbitrary dimensions.
2012, 32(10): 3567-3585
doi: 10.3934/dcds.2012.32.3567
+[Abstract](1116)
+[PDF](450.9KB)
Abstract:
In this paper, arbitrarily many solutions, in particular arbitrarily many nodal solutions, are proved to exist for perturbed elliptic equations of the form \begin{equation*}\label{} \left\{ \begin{array}{ll} \displaystyle -\Delta_p u+|u|^{p-2}u = Q(x)(f(u)+\varepsilon g(u)),\ \ \ x\in \mathbb R^N, \\ u\in W^{1,p}(\mathbb R^N), \end{array} \right. (P_\varepsilon) \end{equation*} where $\Delta_p$ is the $p$-Laplacian operator defined by $\Delta_p u=\text{div}(|\nabla u|^{p-2}\nabla u)$, $p>1$, $Q\in \mathcal{C}(\mathbb R^N,\mathbb R)$ is a positive function, $f\in\mathcal{C}(\mathbb R, \mathbb R)$ oscillates either near the origin or near the infinity, and $\epsilon$ is a real number. For $g$ it is only required that $g\in\mathcal{C}(\mathbb R, \mathbb R)$. Under appropriate assumptions on $Q$ and $f$ the following results which are special cases of more general ones are proved: the unperturbed problem $(P_0)$ has infinitely many nodal solutions, and for any $n\in\mathbb N$ the perturbed problem $(P_\varepsilon)$ has at least $n$ nodal solutions provided that $|\epsilon|$ is sufficiently small.
In this paper, arbitrarily many solutions, in particular arbitrarily many nodal solutions, are proved to exist for perturbed elliptic equations of the form \begin{equation*}\label{} \left\{ \begin{array}{ll} \displaystyle -\Delta_p u+|u|^{p-2}u = Q(x)(f(u)+\varepsilon g(u)),\ \ \ x\in \mathbb R^N, \\ u\in W^{1,p}(\mathbb R^N), \end{array} \right. (P_\varepsilon) \end{equation*} where $\Delta_p$ is the $p$-Laplacian operator defined by $\Delta_p u=\text{div}(|\nabla u|^{p-2}\nabla u)$, $p>1$, $Q\in \mathcal{C}(\mathbb R^N,\mathbb R)$ is a positive function, $f\in\mathcal{C}(\mathbb R, \mathbb R)$ oscillates either near the origin or near the infinity, and $\epsilon$ is a real number. For $g$ it is only required that $g\in\mathcal{C}(\mathbb R, \mathbb R)$. Under appropriate assumptions on $Q$ and $f$ the following results which are special cases of more general ones are proved: the unperturbed problem $(P_0)$ has infinitely many nodal solutions, and for any $n\in\mathbb N$ the perturbed problem $(P_\varepsilon)$ has at least $n$ nodal solutions provided that $|\epsilon|$ is sufficiently small.
2012, 32(10): 3587-3620
doi: 10.3934/dcds.2012.32.3587
+[Abstract](1106)
+[PDF](753.7KB)
Abstract:
We point out an interesting relation between transport in Hamiltonian dynamics and Floer homology. We generalize homoclinic Floer homology from $\mathbb{R}^2$ and closed surfaces to two-dimensional cylinders. The relative symplectic action of two homoclinic points is identified with the flux through a turnstile (as defined in MacKay & Meiss & Percival [19]) and Mather's [20] difference in action $\Delta W$. The Floer boundary operator is shown to annihilate turnstiles and we prove that the rank of certain filtered homology groups and the flux grow linearly with the number of iterations of the underlying symplectomorphism.
We point out an interesting relation between transport in Hamiltonian dynamics and Floer homology. We generalize homoclinic Floer homology from $\mathbb{R}^2$ and closed surfaces to two-dimensional cylinders. The relative symplectic action of two homoclinic points is identified with the flux through a turnstile (as defined in MacKay & Meiss & Percival [19]) and Mather's [20] difference in action $\Delta W$. The Floer boundary operator is shown to annihilate turnstiles and we prove that the rank of certain filtered homology groups and the flux grow linearly with the number of iterations of the underlying symplectomorphism.
2012, 32(10): 3621-3649
doi: 10.3934/dcds.2012.32.3621
+[Abstract](1299)
+[PDF](589.0KB)
Abstract:
In this paper, we study a class of nonlocal dispersion equation with monostable nonlinearity in $n$-dimensional space \[ \begin{cases} u_t - J\ast u +u+d(u(t,x))= \displaystyle \int_{\mathbb{R}^n} f_\beta (y) b(u(t-\tau,x-y)) dy, \\ u(s,x)=u_0(s,x), \ \ s\in[-\tau,0], \ x\in \mathbb{R}^n, \end{cases} \] where the nonlinear functions $d(u)$ and $b(u)$ possess the monostable characters like Fisher-KPP type, $f_\beta(x)$ is the heat kernel, and the kernel $J(x)$ satisfies ${\hat J}(\xi)=1-\mathcal{K}|\xi|^\alpha+o(|\xi|^\alpha)$ for $0<\alpha\le 2$ and $\mathcal{K}>0$. After establishing the existence for both the planar traveling waves $\phi(x\cdot{\bf e}+ct)$ for $c\ge c_*$ ($c_*$ is the critical wave speed) and the solution $u(t,x)$ for the Cauchy problem, as well as the comparison principles, we prove that, all noncritical planar wavefronts $\phi(x\cdot{\bf e}+ct)$ are globally stable with the exponential convergence rate $t^{-n/\alpha}e^{-\mu_\tau t}$ for $\mu_\tau>0$, and the critical wavefronts $\phi(x\cdot{\bf e}+c_*t)$ are globally stable in the algebraic form $t^{-n/\alpha}$, and these rates are optimal. As application,we also automatically obtain the stability of traveling wavefronts to the classical Fisher-KPP dispersion equations. The adopted approach is Fourier transform and the weighted energy method with a suitably selected weight function.
In this paper, we study a class of nonlocal dispersion equation with monostable nonlinearity in $n$-dimensional space \[ \begin{cases} u_t - J\ast u +u+d(u(t,x))= \displaystyle \int_{\mathbb{R}^n} f_\beta (y) b(u(t-\tau,x-y)) dy, \\ u(s,x)=u_0(s,x), \ \ s\in[-\tau,0], \ x\in \mathbb{R}^n, \end{cases} \] where the nonlinear functions $d(u)$ and $b(u)$ possess the monostable characters like Fisher-KPP type, $f_\beta(x)$ is the heat kernel, and the kernel $J(x)$ satisfies ${\hat J}(\xi)=1-\mathcal{K}|\xi|^\alpha+o(|\xi|^\alpha)$ for $0<\alpha\le 2$ and $\mathcal{K}>0$. After establishing the existence for both the planar traveling waves $\phi(x\cdot{\bf e}+ct)$ for $c\ge c_*$ ($c_*$ is the critical wave speed) and the solution $u(t,x)$ for the Cauchy problem, as well as the comparison principles, we prove that, all noncritical planar wavefronts $\phi(x\cdot{\bf e}+ct)$ are globally stable with the exponential convergence rate $t^{-n/\alpha}e^{-\mu_\tau t}$ for $\mu_\tau>0$, and the critical wavefronts $\phi(x\cdot{\bf e}+c_*t)$ are globally stable in the algebraic form $t^{-n/\alpha}$, and these rates are optimal. As application,we also automatically obtain the stability of traveling wavefronts to the classical Fisher-KPP dispersion equations. The adopted approach is Fourier transform and the weighted energy method with a suitably selected weight function.
2012, 32(10): 3651-3664
doi: 10.3934/dcds.2012.32.3651
+[Abstract](1128)
+[PDF](453.0KB)
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As is well-known, the existence of a cone-field with constant orbit core dimension is, roughly speaking, equivalent to hyperbolicity, and consequently guarantees expansivity and shadowing. In this paper we study the case when the given cone-field does not have the constant orbit core dimension. It occurs that we still obtain expansivity even in general metric spaces.
Main Result. Let $X$ be a metric space and let $f:X \rightharpoonup X$ be a given partial map. If there exists a uniform cone-field on $X$ such that $f$ is cone-hyperbolic, then $f$ is uniformly expansive, i.e. there exists $N \in \mathbb{N}$, $\lambda \in [0,1)$ and $\epsilon > 0$ such that for all orbits $\mathrm{x},\mathrm{v}:{-N,\ldots,N} \to X$ \[ d_{\sup}(\mathrm{x},\mathrm{v}) \leq \epsilon \Longrightarrow d(\mathrm{x}_0,\mathrm{v}_0) \leq \lambda d_{\sup}(\mathrm{x},\mathrm{v}). \] } We also show a simple example of a cone hyperbolic orbit in $\mathbb{R}^3$ which does not have the shadowing property.
As is well-known, the existence of a cone-field with constant orbit core dimension is, roughly speaking, equivalent to hyperbolicity, and consequently guarantees expansivity and shadowing. In this paper we study the case when the given cone-field does not have the constant orbit core dimension. It occurs that we still obtain expansivity even in general metric spaces.
Main Result. Let $X$ be a metric space and let $f:X \rightharpoonup X$ be a given partial map. If there exists a uniform cone-field on $X$ such that $f$ is cone-hyperbolic, then $f$ is uniformly expansive, i.e. there exists $N \in \mathbb{N}$, $\lambda \in [0,1)$ and $\epsilon > 0$ such that for all orbits $\mathrm{x},\mathrm{v}:{-N,\ldots,N} \to X$ \[ d_{\sup}(\mathrm{x},\mathrm{v}) \leq \epsilon \Longrightarrow d(\mathrm{x}_0,\mathrm{v}_0) \leq \lambda d_{\sup}(\mathrm{x},\mathrm{v}). \] } We also show a simple example of a cone hyperbolic orbit in $\mathbb{R}^3$ which does not have the shadowing property.
2012, 32(10): 3665-3690
doi: 10.3934/dcds.2012.32.3665
+[Abstract](1477)
+[PDF](499.2KB)
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In this article, we consider a non-autonomous diffuse interface model for an isothermal incompressible two-phase flow in a two-dimensional bounded domain. We assume that the external force is singularly oscillating and depends on a small parameter $ \epsilon. $ We prove the existence of the uniform global attractor $A^{\epsilon}. $ Furthermore, using the method of [13] in the case of the two-dimensional Navier-Stokes systems, we study the convergence of $A^{\epsilon} $ as $ \epsilon $ goes to zero. Let us mention that the nonlinearity involved in the model considered in this article is slightly stronger than the one in the two-dimensional Navier-Stokes system studied in [13].
In this article, we consider a non-autonomous diffuse interface model for an isothermal incompressible two-phase flow in a two-dimensional bounded domain. We assume that the external force is singularly oscillating and depends on a small parameter $ \epsilon. $ We prove the existence of the uniform global attractor $A^{\epsilon}. $ Furthermore, using the method of [13] in the case of the two-dimensional Navier-Stokes systems, we study the convergence of $A^{\epsilon} $ as $ \epsilon $ goes to zero. Let us mention that the nonlinearity involved in the model considered in this article is slightly stronger than the one in the two-dimensional Navier-Stokes system studied in [13].
2012, 32(10): 3691-3713
doi: 10.3934/dcds.2012.32.3691
+[Abstract](1288)
+[PDF](426.8KB)
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We consider a parabolic-elliptic system of equations that arises in modelling the chemotaxis in bacteria and the evolution of self-attracting clusters. In the case space dimension $3 \leq N \leq 9$, we will derive criteria of the blow-up rate of solutions, and identify an explicit class of initial data for which the blow-up is of self-similar rate. Our argument is based on the study of the asymptotic properties of backward self-similar solutions to the system together with the intersection comparison principle.
We consider a parabolic-elliptic system of equations that arises in modelling the chemotaxis in bacteria and the evolution of self-attracting clusters. In the case space dimension $3 \leq N \leq 9$, we will derive criteria of the blow-up rate of solutions, and identify an explicit class of initial data for which the blow-up is of self-similar rate. Our argument is based on the study of the asymptotic properties of backward self-similar solutions to the system together with the intersection comparison principle.
2012, 32(10): 3715-3732
doi: 10.3934/dcds.2012.32.3715
+[Abstract](1280)
+[PDF](1393.6KB)
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In this work we look for central configurations of the planar $1+n$ body problem such that, after the addition of one or two satellites, we have a new planar central configuration. We determine all such configurations in two cases: the first, the addition of two satellites considering that all satellites have equal infinitesimal masses and the second case where one satellite is added but the infinitesimal masses are not necessarily equal.
In this work we look for central configurations of the planar $1+n$ body problem such that, after the addition of one or two satellites, we have a new planar central configuration. We determine all such configurations in two cases: the first, the addition of two satellites considering that all satellites have equal infinitesimal masses and the second case where one satellite is added but the infinitesimal masses are not necessarily equal.
2012, 32(10): 3733-3771
doi: 10.3934/dcds.2012.32.3733
+[Abstract](1136)
+[PDF](632.3KB)
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In this paper, we consider Hartree-type equations on the two-dimensional torus and on the plane. We prove polynomial bounds on the growth of high Sobolev norms of solutions to these equations. The proofs of our results are based on the adaptation to two dimensions of the techniques we had previously used in [49, 50] to study the analogous problem in one dimension. Since we are working in two dimensions, a more detailed analysis of the resonant frequencies is needed, as was previously used in the work of Colliander-Keel-Staffilani-Takaoka-Tao [19].
In this paper, we consider Hartree-type equations on the two-dimensional torus and on the plane. We prove polynomial bounds on the growth of high Sobolev norms of solutions to these equations. The proofs of our results are based on the adaptation to two dimensions of the techniques we had previously used in [49, 50] to study the analogous problem in one dimension. Since we are working in two dimensions, a more detailed analysis of the resonant frequencies is needed, as was previously used in the work of Colliander-Keel-Staffilani-Takaoka-Tao [19].
2012, 32(10): 3773-3785
doi: 10.3934/dcds.2012.32.3773
+[Abstract](1115)
+[PDF](329.2KB)
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We study the exponential rate of decay of Lebesgue numbers of open covers in topological dynamical systems. We show that topological entropy is bounded by this rate multiplied by dimension. Some corollaries and examples are discussed.
We study the exponential rate of decay of Lebesgue numbers of open covers in topological dynamical systems. We show that topological entropy is bounded by this rate multiplied by dimension. Some corollaries and examples are discussed.
2012, 32(10): 3787-3800
doi: 10.3934/dcds.2012.32.3787
+[Abstract](1321)
+[PDF](416.7KB)
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The longtime dynamics of the three dimensional (3D) Brinkman-Forchheimer equations with time-dependent forcing term is investigated. It is proved that there exists a uniform attractor for this nonautonomous 3D Brinkman-Forchheimer equations in the space $\mathbb{H}^1(\Omega)$. When the Darcy coefficient $\alpha$ is properly large and $L^2_b$-norm of the forcing term is properly small, it is shown that there exists a unique bounded and asymptotically stable solution with interesting corollaries.
The longtime dynamics of the three dimensional (3D) Brinkman-Forchheimer equations with time-dependent forcing term is investigated. It is proved that there exists a uniform attractor for this nonautonomous 3D Brinkman-Forchheimer equations in the space $\mathbb{H}^1(\Omega)$. When the Darcy coefficient $\alpha$ is properly large and $L^2_b$-norm of the forcing term is properly small, it is shown that there exists a unique bounded and asymptotically stable solution with interesting corollaries.
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