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Discrete & Continuous Dynamical Systems - A
February 2014 , Volume 34 , Issue 2
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2014, 34(2): 335-366
doi: 10.3934/dcds.2014.34.335
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Abstract:
The main objective of this article is to derive new gravitational field equations and to establish a unified theory for dark energy and dark matter. The gravitational field equations with a scalar potential $\varphi$ function are derived using the Einstein-Hilbert functional, and the scalar potential $\varphi$ is a natural outcome of the divergence-free constraint of the variational elements. Gravitation is now described by the Riemannian metric $g_{\mu\nu}$, the scalar potential $\varphi$ and their interactions, unified by the new field equations. From quantum field theoretic point of view, the vector field $\Phi_\mu=D_\mu \varphi$, the gradient of the scalar function $\varphi$, is a spin-1 massless bosonic particle field. The field equations induce a natural duality between the graviton (spin-2 massless bosonic particle) and this spin-1 massless bosonic particle. Both particles can be considered as gravitational force carriers, and as they are massless, the induced forces are long-range forces. The (nonlinear) interaction between these bosonic particle fields leads to a unified theory for dark energy and dark matter. Also, associated with the scalar potential $\varphi$ is the scalar potential energy density $\frac{c^4}{8\pi G} \Phi=\frac{c^4}{8\pi G} g^{\mu\nu}D_\mu D_\nu \varphi$, which represents a new type of energy caused by the non-uniform distribution of matter in the universe. The negative part of this potential energy density produces attraction, and the positive part produces repelling force. This potential energy density is conserved with mean zero: $\int_M \Phi dM=0$. The sum of this potential energy density $\frac{c^4}{8\pi G} \Phi$ and the coupling energy between the energy-momentum tensor $T_{\mu\nu}$ and the scalar potential field $\varphi$ gives rise to a unified theory for dark matter and dark energy: The negative part of this sum represents the dark matter, which produces attraction, and the positive part represents the dark energy, which drives the acceleration of expanding galaxies. In addition, the scalar curvature of space-time obeys $R=\frac{8\pi G}{c^4} T + \Phi$. Furthermore, the proposed field equations resolve a few difficulties encountered by the classical Einstein field equations.
The main objective of this article is to derive new gravitational field equations and to establish a unified theory for dark energy and dark matter. The gravitational field equations with a scalar potential $\varphi$ function are derived using the Einstein-Hilbert functional, and the scalar potential $\varphi$ is a natural outcome of the divergence-free constraint of the variational elements. Gravitation is now described by the Riemannian metric $g_{\mu\nu}$, the scalar potential $\varphi$ and their interactions, unified by the new field equations. From quantum field theoretic point of view, the vector field $\Phi_\mu=D_\mu \varphi$, the gradient of the scalar function $\varphi$, is a spin-1 massless bosonic particle field. The field equations induce a natural duality between the graviton (spin-2 massless bosonic particle) and this spin-1 massless bosonic particle. Both particles can be considered as gravitational force carriers, and as they are massless, the induced forces are long-range forces. The (nonlinear) interaction between these bosonic particle fields leads to a unified theory for dark energy and dark matter. Also, associated with the scalar potential $\varphi$ is the scalar potential energy density $\frac{c^4}{8\pi G} \Phi=\frac{c^4}{8\pi G} g^{\mu\nu}D_\mu D_\nu \varphi$, which represents a new type of energy caused by the non-uniform distribution of matter in the universe. The negative part of this potential energy density produces attraction, and the positive part produces repelling force. This potential energy density is conserved with mean zero: $\int_M \Phi dM=0$. The sum of this potential energy density $\frac{c^4}{8\pi G} \Phi$ and the coupling energy between the energy-momentum tensor $T_{\mu\nu}$ and the scalar potential field $\varphi$ gives rise to a unified theory for dark matter and dark energy: The negative part of this sum represents the dark matter, which produces attraction, and the positive part represents the dark energy, which drives the acceleration of expanding galaxies. In addition, the scalar curvature of space-time obeys $R=\frac{8\pi G}{c^4} T + \Phi$. Furthermore, the proposed field equations resolve a few difficulties encountered by the classical Einstein field equations.
2014, 34(2): 367-377
doi: 10.3934/dcds.2014.34.367
+[Abstract](1452)
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Abstract:
In the paper, we obtain necessary and sufficient conditions for ergodicity (with respect to the normalized Haar measure) of discrete dynamical systems $\langle f;\mathbf S_{2^-r}(a)\rangle$ on 2-adic spheres $\mathbf S_{2^-r}(a)$ of radius $2^{-r}$, $r\ge 1$, centered at some point $a$ from the ultrametric space of 2-adic integers $\mathbb Z_2$. The map $f\colon\mathbb Z_2\to\mathbb Z_2$ is assumed to be non-expanding and measure-preserving; that is, $f$ satisfies a Lipschitz condition with a constant 1 with respect to the 2-adic metric, and $f$ preserves a natural probability measure on $\mathbb Z_2$, the Haar measure $\mu_2$ on $\mathbb Z_2$ which is normalized so that $\mu_2(\mathbb Z_2)=1$.
In the paper, we obtain necessary and sufficient conditions for ergodicity (with respect to the normalized Haar measure) of discrete dynamical systems $\langle f;\mathbf S_{2^-r}(a)\rangle$ on 2-adic spheres $\mathbf S_{2^-r}(a)$ of radius $2^{-r}$, $r\ge 1$, centered at some point $a$ from the ultrametric space of 2-adic integers $\mathbb Z_2$. The map $f\colon\mathbb Z_2\to\mathbb Z_2$ is assumed to be non-expanding and measure-preserving; that is, $f$ satisfies a Lipschitz condition with a constant 1 with respect to the 2-adic metric, and $f$ preserves a natural probability measure on $\mathbb Z_2$, the Haar measure $\mu_2$ on $\mathbb Z_2$ which is normalized so that $\mu_2(\mathbb Z_2)=1$.
2014, 34(2): 379-420
doi: 10.3934/dcds.2014.34.379
+[Abstract](1528)
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Abstract:
The Vlasov equation models a group of particles moving under a potential $V$; moreover, each particle exerts a force, of potential $W$, on the other ones. We shall suppose that these particles move on the $p$-dimensional torus $T^p$ and that the interaction potential $W$ is smooth. We are going to perturb this equation by a Brownian motion on $T^p$; adapting to the viscous case methods of Gangbo, Nguyen, Tudorascu and Gomes, we study the existence of periodic solutions and the asymptotics of the Hopf-Lax semigroup.
The Vlasov equation models a group of particles moving under a potential $V$; moreover, each particle exerts a force, of potential $W$, on the other ones. We shall suppose that these particles move on the $p$-dimensional torus $T^p$ and that the interaction potential $W$ is smooth. We are going to perturb this equation by a Brownian motion on $T^p$; adapting to the viscous case methods of Gangbo, Nguyen, Tudorascu and Gomes, we study the existence of periodic solutions and the asymptotics of the Hopf-Lax semigroup.
2014, 34(2): 421-436
doi: 10.3934/dcds.2014.34.421
+[Abstract](1574)
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Abstract:
We introduce in this paper some elements for the mathematical analysis of turbulence models for oceanic surface mixing layers. We consider Richardson-number based vertical eddy diffusion models. We prove the existence of unsteady solutions if the initial condition is close to an equilibrium, via the inverse function theorem in Banach spaces. We use this result to prove the non-linear asymptotic stability of equilibrium solutions.
We introduce in this paper some elements for the mathematical analysis of turbulence models for oceanic surface mixing layers. We consider Richardson-number based vertical eddy diffusion models. We prove the existence of unsteady solutions if the initial condition is close to an equilibrium, via the inverse function theorem in Banach spaces. We use this result to prove the non-linear asymptotic stability of equilibrium solutions.
2014, 34(2): 437-459
doi: 10.3934/dcds.2014.34.437
+[Abstract](1727)
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Abstract:
In this paper, we study the Cauchy problem of the 3-dimensional (3D) generalized Navier-Stokes equations (gNS) in the Triebel-Lizorkin spaces $\dot{F}^{-\alpha,r}_{q_\alpha}$ with $(\alpha,r)\in(1,\frac{5}{4})\times[1,\infty]$ and $q_\alpha=\frac{3}{\alpha-1}$. Our work establishes a dichotomy of well-posedness and ill-posedness depending on $r$. Specifically, by combining the new endpoint bilinear estimates in $L^{\!q_\alpha}_x\!L^2_T$ and $L^\infty_T\dot{F}^{-\alpha,1}_{q_\alpha}$ and characterization of the Triebel-Lizorkin spaces via fractional semigroup, we prove well-posedness of the gNS in $\dot{F}^{-\alpha,r}_{q_\alpha}$ for $r\in[1,2]$. Meanwhile, for any $r\in(2,\infty]$, we show that the solution to the gNS can develop norm inflation in the sense that arbitrarily small initial data in $\dot{F}^{-\alpha,r}_{q_\alpha}$ can produce arbitrarily large solution after arbitrarily short time.
In this paper, we study the Cauchy problem of the 3-dimensional (3D) generalized Navier-Stokes equations (gNS) in the Triebel-Lizorkin spaces $\dot{F}^{-\alpha,r}_{q_\alpha}$ with $(\alpha,r)\in(1,\frac{5}{4})\times[1,\infty]$ and $q_\alpha=\frac{3}{\alpha-1}$. Our work establishes a dichotomy of well-posedness and ill-posedness depending on $r$. Specifically, by combining the new endpoint bilinear estimates in $L^{\!q_\alpha}_x\!L^2_T$ and $L^\infty_T\dot{F}^{-\alpha,1}_{q_\alpha}$ and characterization of the Triebel-Lizorkin spaces via fractional semigroup, we prove well-posedness of the gNS in $\dot{F}^{-\alpha,r}_{q_\alpha}$ for $r\in[1,2]$. Meanwhile, for any $r\in(2,\infty]$, we show that the solution to the gNS can develop norm inflation in the sense that arbitrarily small initial data in $\dot{F}^{-\alpha,r}_{q_\alpha}$ can produce arbitrarily large solution after arbitrarily short time.
2014, 34(2): 461-475
doi: 10.3934/dcds.2014.34.461
+[Abstract](1970)
+[PDF](428.0KB)
Abstract:
In this paper, by an approximating argument, we obtain infinitely many radial solutions for the following elliptic systems with critical Sobolev growth $$ \left\lbrace\begin{array}{l} -\Delta u=|u|^{2^*-2}u + \frac{η \alpha}{\alpha+β}|u|^{\alpha-2}u |v|^β + \frac{σ p}{p+q} |u|^{p-2}u|v|^q , \ \ x ∈ B , \\ -\Delta v = |v|^{2^*-2}v + \frac{η β}{\alpha+ β } |u|^{\alpha }|v|^{β-2}v + \frac{σ q}{p+q} |u|^{p}|v|^{q-2}v , \ \ x ∈ B , \\ u = v = 0, \ \ &x \in \partial B, \end{array}\right. $$ where $N > \frac{2(p + q + 1) }{p + q - 1}, η, σ > 0, \alpha,β > 1$ and $\alpha + β = 2^* = : \frac{2N}{N-2} ,$ $p,\,q\ge 1$, $2\le p +q<2^*$ and $B\subset \mathbb{R}^N$ is an open ball centered at the origin.
In this paper, by an approximating argument, we obtain infinitely many radial solutions for the following elliptic systems with critical Sobolev growth $$ \left\lbrace\begin{array}{l} -\Delta u=|u|^{2^*-2}u + \frac{η \alpha}{\alpha+β}|u|^{\alpha-2}u |v|^β + \frac{σ p}{p+q} |u|^{p-2}u|v|^q , \ \ x ∈ B , \\ -\Delta v = |v|^{2^*-2}v + \frac{η β}{\alpha+ β } |u|^{\alpha }|v|^{β-2}v + \frac{σ q}{p+q} |u|^{p}|v|^{q-2}v , \ \ x ∈ B , \\ u = v = 0, \ \ &x \in \partial B, \end{array}\right. $$ where $N > \frac{2(p + q + 1) }{p + q - 1}, η, σ > 0, \alpha,β > 1$ and $\alpha + β = 2^* = : \frac{2N}{N-2} ,$ $p,\,q\ge 1$, $2\le p +q<2^*$ and $B\subset \mathbb{R}^N$ is an open ball centered at the origin.
2014, 34(2): 477-509
doi: 10.3934/dcds.2014.34.477
+[Abstract](1729)
+[PDF](4891.7KB)
Abstract:
In this paper we develop and test a structure-preserving discretization scheme for rotating and/or stratified fluid dynamics. The numerical scheme is based on a finite dimensional approximation of the group of volume preserving diffeomorphisms recently proposed in [25,9] and is derived via a discrete version of the Euler-Poincaré variational formulation of rotating stratified fluids. The resulting variational integrator allows for a discrete version of Kelvin circulation theorem, is applicable to irregular meshes and, being symplectic, exhibits excellent long term energy behavior. We then report a series of preliminary tests for rotating stratified flows in configurations that are symmetric with respect to translation along one of the spatial directions. In the benchmark processes of hydrostatic and/or geostrophic adjustments, these tests show that the slow and fast component of the flow are correctly reproduced. The harder test of inertial instability is in full agreement with the common knowledge of the process of development and saturation of this instability, while preserving energy nearly perfectly and respecting conservation laws.
In this paper we develop and test a structure-preserving discretization scheme for rotating and/or stratified fluid dynamics. The numerical scheme is based on a finite dimensional approximation of the group of volume preserving diffeomorphisms recently proposed in [25,9] and is derived via a discrete version of the Euler-Poincaré variational formulation of rotating stratified fluids. The resulting variational integrator allows for a discrete version of Kelvin circulation theorem, is applicable to irregular meshes and, being symplectic, exhibits excellent long term energy behavior. We then report a series of preliminary tests for rotating stratified flows in configurations that are symmetric with respect to translation along one of the spatial directions. In the benchmark processes of hydrostatic and/or geostrophic adjustments, these tests show that the slow and fast component of the flow are correctly reproduced. The harder test of inertial instability is in full agreement with the common knowledge of the process of development and saturation of this instability, while preserving energy nearly perfectly and respecting conservation laws.
2014, 34(2): 511-529
doi: 10.3934/dcds.2014.34.511
+[Abstract](1869)
+[PDF](454.7KB)
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We derive the fundamental solution of the linearized problem of the motion of a viscous fluid around a rotating body when the axis of rotation of the body is not parallel to the velocity of the fluid at infinity.
We derive the fundamental solution of the linearized problem of the motion of a viscous fluid around a rotating body when the axis of rotation of the body is not parallel to the velocity of the fluid at infinity.
2014, 34(2): 531-556
doi: 10.3934/dcds.2014.34.531
+[Abstract](1590)
+[PDF](522.8KB)
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We consider the collection of uniformly discrete point sets in Euclidean space equipped with the vague topology. For a point set in this collection, we characterise minimality of an associated dynamical system by almost repetitivity of the point set. We also provide linear versions of almost repetitivity which lead to uniquely ergodic systems. Apart from linearly repetitive point sets, examples are given by periodic point sets with almost periodic modulations, and by point sets derived from primitive substitution tilings of finite local complexity with respect to the Euclidean group with dense tile orientations.
We consider the collection of uniformly discrete point sets in Euclidean space equipped with the vague topology. For a point set in this collection, we characterise minimality of an associated dynamical system by almost repetitivity of the point set. We also provide linear versions of almost repetitivity which lead to uniquely ergodic systems. Apart from linearly repetitive point sets, examples are given by periodic point sets with almost periodic modulations, and by point sets derived from primitive substitution tilings of finite local complexity with respect to the Euclidean group with dense tile orientations.
2014, 34(2): 557-566
doi: 10.3934/dcds.2014.34.557
+[Abstract](1760)
+[PDF](366.7KB)
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We use a novel transformation of the reduced Ostrovsky equation to the integrable Tzitzéica equation and prove global existence of small-norm solutions in Sobolev space $H^3(\mathbb{R})$. This scenario is an alternative to finite-time wave breaking of large-norm solutions of the reduced Ostrovsky equation. We also discuss a sharp sufficient condition for the finite-time wave breaking.
We use a novel transformation of the reduced Ostrovsky equation to the integrable Tzitzéica equation and prove global existence of small-norm solutions in Sobolev space $H^3(\mathbb{R})$. This scenario is an alternative to finite-time wave breaking of large-norm solutions of the reduced Ostrovsky equation. We also discuss a sharp sufficient condition for the finite-time wave breaking.
2014, 34(2): 567-587
doi: 10.3934/dcds.2014.34.567
+[Abstract](1902)
+[PDF](486.2KB)
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We prove the global existence of weak solutions to the Navier-Stokes equations of compressible heat-conducting fluids in two spatial dimensions with initial data and external forces which are large and spherically symmetric. The solutions will be obtained as the limit of the approximate solutions in an annular domain. We first derive a number of regularity results on the approximate physical quantities in the ``fluid region'', as well as the new uniform integrability of the velocity and temperature in the entire space-time domain by exploiting the theory of the Orlicz spaces. By virtue of these a priori estimates we then argue in a manner similar to that in [Arch. Rational Mech. Anal. 173 (2004), 297-343] to pass to the limit and show that the limiting functions are indeed a weak solution which satisfies the mass and momentum equations in the entire space-time domain in the sense of distributions, and the energy equation in any compact subset of the ``fluid region''.
We prove the global existence of weak solutions to the Navier-Stokes equations of compressible heat-conducting fluids in two spatial dimensions with initial data and external forces which are large and spherically symmetric. The solutions will be obtained as the limit of the approximate solutions in an annular domain. We first derive a number of regularity results on the approximate physical quantities in the ``fluid region'', as well as the new uniform integrability of the velocity and temperature in the entire space-time domain by exploiting the theory of the Orlicz spaces. By virtue of these a priori estimates we then argue in a manner similar to that in [Arch. Rational Mech. Anal. 173 (2004), 297-343] to pass to the limit and show that the limiting functions are indeed a weak solution which satisfies the mass and momentum equations in the entire space-time domain in the sense of distributions, and the energy equation in any compact subset of the ``fluid region''.
2014, 34(2): 589-597
doi: 10.3934/dcds.2014.34.589
+[Abstract](1400)
+[PDF](319.4KB)
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Let $g_{\alpha}$ be a one-parameter family of one-dimensional maps with a cascade of period doubling bifurcations. Between each of these bifurcations, a superstable periodic orbit is known to exist. An example of such a family is the well-known logistic map. In this paper we deal with the effect of a quasi-periodic perturbation (with only one frequency) on this cascade. Let us call $\varepsilon$ the perturbing parameter. It is known that, if $\varepsilon$ is small enough, the superstable periodic orbits of the unperturbed map become attracting invariant curves (depending on $\alpha$ and $\varepsilon$) of the perturbed system. In this article we focus on the reducibility of these invariant curves.
The paper shows that, under generic conditions, there are both reducible and non-reducible invariant curves depending on the values of $\alpha$ and $\varepsilon$. The curves in the space $(\alpha,\varepsilon)$ separating the reducible (or the non-reducible) regions are called reducibility loss bifurcation curves. If the map satifies an extra condition (condition satisfied by the quasi-periodically forced logistic map) then we show that, from each superattracting point of the unperturbed map, two reducibility loss bifurcation curves are born. This means that these curves are present for all the cascade.
Let $g_{\alpha}$ be a one-parameter family of one-dimensional maps with a cascade of period doubling bifurcations. Between each of these bifurcations, a superstable periodic orbit is known to exist. An example of such a family is the well-known logistic map. In this paper we deal with the effect of a quasi-periodic perturbation (with only one frequency) on this cascade. Let us call $\varepsilon$ the perturbing parameter. It is known that, if $\varepsilon$ is small enough, the superstable periodic orbits of the unperturbed map become attracting invariant curves (depending on $\alpha$ and $\varepsilon$) of the perturbed system. In this article we focus on the reducibility of these invariant curves.
The paper shows that, under generic conditions, there are both reducible and non-reducible invariant curves depending on the values of $\alpha$ and $\varepsilon$. The curves in the space $(\alpha,\varepsilon)$ separating the reducible (or the non-reducible) regions are called reducibility loss bifurcation curves. If the map satifies an extra condition (condition satisfied by the quasi-periodically forced logistic map) then we show that, from each superattracting point of the unperturbed map, two reducibility loss bifurcation curves are born. This means that these curves are present for all the cascade.
2014, 34(2): 599-611
doi: 10.3934/dcds.2014.34.599
+[Abstract](1227)
+[PDF](340.8KB)
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We use Wagner's algorithm to estimate the number of periodic points of certain selfmaps on compact surfaces with boundary. When counting according to homotopy classes, we can use the asymptotic density to measure the size of sets of selfmaps. In this sense, we show that ``almost all'' such selfmaps have periodic points of every period, and that in fact the number of periodic points of period $n$ grows exponentially in $n$. We further discuss this exponential growth rate and the topological and fundamental-group entropies of these maps.
Since our approach is via the Nielsen number, which is homotopy and homotopy-type invariant, our results hold for selfmaps of any space which has the homotopy type of a compact surface with boundary.
We use Wagner's algorithm to estimate the number of periodic points of certain selfmaps on compact surfaces with boundary. When counting according to homotopy classes, we can use the asymptotic density to measure the size of sets of selfmaps. In this sense, we show that ``almost all'' such selfmaps have periodic points of every period, and that in fact the number of periodic points of period $n$ grows exponentially in $n$. We further discuss this exponential growth rate and the topological and fundamental-group entropies of these maps.
Since our approach is via the Nielsen number, which is homotopy and homotopy-type invariant, our results hold for selfmaps of any space which has the homotopy type of a compact surface with boundary.
2014, 34(2): 613-633
doi: 10.3934/dcds.2014.34.613
+[Abstract](1918)
+[PDF](566.8KB)
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In this article we study the existence the existence of nonconstant steady state solutions for the following relaxed cross-diffusion system $$ \left\lbrace\begin{array}{l} \partial_t u-\Delta[a(\tilde v)u]=0,\;\text{ in } (0,\infty)\times\Omega,\\ \partial_t v-\Delta[b(\tilde u)v]=0,\;\text{ in } (0,\infty)\times\Omega,\\ -\delta\Delta \tilde u+\tilde u=u,\;\text{ in }\Omega,\\ -\delta\Delta \tilde v+\tilde v=v,\;\text{ in }\Omega,\\ \partial_n u=\partial_n v=\partial\tilde u=\partial_n\tilde u=0,\;\text{ on } (0,\infty) \times \partial\Omega, \end{array}\right. $$ with $\Omega$ a bounded smooth domain, $n$ the outer unit normal to $\partial\Omega$, $\delta>0$ denotes the relaxation parameter. The functions $a(\tilde v)$, $b(\tilde u)$ account for nonlinear cross-diffusion, being $a(\tilde v)=1+{\tilde v}^\gamma$, $b(\tilde u)=1+{\tilde u}^\eta$ with $\gamma, \eta >1$ a model example. We give conditions for the stability of constant steady state solutions and we prove that under suitable conditions Turing patterns arise considering $\delta$ as a bifurcation parameter.
In this article we study the existence the existence of nonconstant steady state solutions for the following relaxed cross-diffusion system $$ \left\lbrace\begin{array}{l} \partial_t u-\Delta[a(\tilde v)u]=0,\;\text{ in } (0,\infty)\times\Omega,\\ \partial_t v-\Delta[b(\tilde u)v]=0,\;\text{ in } (0,\infty)\times\Omega,\\ -\delta\Delta \tilde u+\tilde u=u,\;\text{ in }\Omega,\\ -\delta\Delta \tilde v+\tilde v=v,\;\text{ in }\Omega,\\ \partial_n u=\partial_n v=\partial\tilde u=\partial_n\tilde u=0,\;\text{ on } (0,\infty) \times \partial\Omega, \end{array}\right. $$ with $\Omega$ a bounded smooth domain, $n$ the outer unit normal to $\partial\Omega$, $\delta>0$ denotes the relaxation parameter. The functions $a(\tilde v)$, $b(\tilde u)$ account for nonlinear cross-diffusion, being $a(\tilde v)=1+{\tilde v}^\gamma$, $b(\tilde u)=1+{\tilde u}^\eta$ with $\gamma, \eta >1$ a model example. We give conditions for the stability of constant steady state solutions and we prove that under suitable conditions Turing patterns arise considering $\delta$ as a bifurcation parameter.
2014, 34(2): 635-645
doi: 10.3934/dcds.2014.34.635
+[Abstract](1634)
+[PDF](346.3KB)
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Let $f$ be a continuous map on a compact metric space. In this paper, under the hypothesis that $f$ satisfies the specification property, we prove that the set consisting of those points for which the Birkhoff ergodic average does not exist is either residual or empty.
Let $f$ be a continuous map on a compact metric space. In this paper, under the hypothesis that $f$ satisfies the specification property, we prove that the set consisting of those points for which the Birkhoff ergodic average does not exist is either residual or empty.
2014, 34(2): 647-662
doi: 10.3934/dcds.2014.34.647
+[Abstract](2081)
+[PDF](421.0KB)
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In this paper we study the notion of topological entropy by unit length for the dynamical system given by the complex Ginzburg-Landau equation on the line (CGL). This equation has a global attractor $\mathcal{A}$ that attracts all the trajectories. We first prove the existence of the topological entropy by unit length for the topological dynamical system $(\mathcal{A},S)$ in a Hilbert space framework, where $S(t)$ is the semi-flow defined by CGL. Next we show that this topological entropy by unit length is bounded by the product of the upper fractal dimension per unit length (see [10]) with the expansion rate. Finally, we prove that this quantity is invariant for all $H^k$ metrics ($k\geq 0$).
In this paper we study the notion of topological entropy by unit length for the dynamical system given by the complex Ginzburg-Landau equation on the line (CGL). This equation has a global attractor $\mathcal{A}$ that attracts all the trajectories. We first prove the existence of the topological entropy by unit length for the topological dynamical system $(\mathcal{A},S)$ in a Hilbert space framework, where $S(t)$ is the semi-flow defined by CGL. Next we show that this topological entropy by unit length is bounded by the product of the upper fractal dimension per unit length (see [10]) with the expansion rate. Finally, we prove that this quantity is invariant for all $H^k$ metrics ($k\geq 0$).
2014, 34(2): 663-676
doi: 10.3934/dcds.2014.34.663
+[Abstract](1352)
+[PDF](369.0KB)
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We call a real number normal if for any block of digits the asymptotic frequency of this block in the $N$-adic expansion equals the expected one. In the present paper we consider non-normal numbers and, in particular, essentially and extremely non-normal numbers. We call a real number essentially non-normal if for each single digit there exists no asymptotic frequency of its occurrence. Furthermore we call a real number extremely non-normal if all possible probability vectors are accumulation points of the sequence of frequency vectors. Our aim now is to extend and generalize these results to Markov partitions.
We call a real number normal if for any block of digits the asymptotic frequency of this block in the $N$-adic expansion equals the expected one. In the present paper we consider non-normal numbers and, in particular, essentially and extremely non-normal numbers. We call a real number essentially non-normal if for each single digit there exists no asymptotic frequency of its occurrence. Furthermore we call a real number extremely non-normal if all possible probability vectors are accumulation points of the sequence of frequency vectors. Our aim now is to extend and generalize these results to Markov partitions.
2014, 34(2): 677-688
doi: 10.3934/dcds.2014.34.677
+[Abstract](1543)
+[PDF](355.5KB)
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This paper deals with normal forms about contact points (`turning points') of nilpotent type that one frequently encounters in the study of planar slow-fast systems. In case the contact point of an analytic slow-fast vector field is of order two, we prove that the slow-fast vector field can locally be written as a slow-fast Liénard equation up to exponentially small error. The proof is based on the use of Gevrey asymptotics. Furthermore, for slow-fast jump points, we eliminate the exponentially small remainder.
This paper deals with normal forms about contact points (`turning points') of nilpotent type that one frequently encounters in the study of planar slow-fast systems. In case the contact point of an analytic slow-fast vector field is of order two, we prove that the slow-fast vector field can locally be written as a slow-fast Liénard equation up to exponentially small error. The proof is based on the use of Gevrey asymptotics. Furthermore, for slow-fast jump points, we eliminate the exponentially small remainder.
2014, 34(2): 689-707
doi: 10.3934/dcds.2014.34.689
+[Abstract](1549)
+[PDF](277.8KB)
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In this paper, we consider the non-autonomous Benjamin-Ono equation $$u_t+\mathscr{H}u_{xx}- uu_x- (F(\omega t,x,u))_x=0$$ under periodic boundary conditions. Using an abstract infinite dimensional KAM theorem dealing with unbounded perturbation vector-field and partial Birkhoff normal form, we will prove that there exists a Cantorian branch of KAM tori and thus many time quasi-periodic solutions for the above equation.
In this paper, we consider the non-autonomous Benjamin-Ono equation $$u_t+\mathscr{H}u_{xx}- uu_x- (F(\omega t,x,u))_x=0$$ under periodic boundary conditions. Using an abstract infinite dimensional KAM theorem dealing with unbounded perturbation vector-field and partial Birkhoff normal form, we will prove that there exists a Cantorian branch of KAM tori and thus many time quasi-periodic solutions for the above equation.
2014, 34(2): 709-732
doi: 10.3934/dcds.2014.34.709
+[Abstract](1378)
+[PDF](295.0KB)
Abstract:
In this paper we study the family of $\alpha$-Farey-Minkowski functions $\theta_\alpha$, for an arbitrary countable partition $\alpha$ of the unit interval with atoms which accumulate only at the origin, which are the conjugating homeomorphisms between each of the $\alpha$-Farey systems and the tent map. We first show that each function $\theta_\alpha$ is singular with respect to the Lebesgue measure and then demonstrate that the unit interval can be written as the disjoint union of the following three sets: $\Theta_0 : = \{x \in [0,1] : \theta_\alpha'(x)=0\}, \Theta_{\infty} : = \{ x \in [0,1] : \theta_\alpha'(x)=\infty \} $ and $\Theta_\sim : = \{ x \in [0,1] : \theta_\alpha'(x)\ does\ not\ exist \} $. The main result is that \[ \dim_{\mathrm{H}}(\Theta_\infty)=\dim_{\mathrm{H}}(\Theta_\sim)=\sigma_\alpha(\log2)<\dim_{\mathrm{H}}(\Theta_0)=1, \] where $\sigma_\alpha(\log2)$ denotes the Hausdorff dimension of the level set $\{x\in [0,1]:\Lambda(F_\alpha, x)=\log2\}$ and $\Lambda(F_\alpha, x)$ is the Lyapunov exponent of the map $F_\alpha$ at the point $x$. The proof of the theorem employs the multifractal formalism for $\alpha$-Farey systems.
In this paper we study the family of $\alpha$-Farey-Minkowski functions $\theta_\alpha$, for an arbitrary countable partition $\alpha$ of the unit interval with atoms which accumulate only at the origin, which are the conjugating homeomorphisms between each of the $\alpha$-Farey systems and the tent map. We first show that each function $\theta_\alpha$ is singular with respect to the Lebesgue measure and then demonstrate that the unit interval can be written as the disjoint union of the following three sets: $\Theta_0 : = \{x \in [0,1] : \theta_\alpha'(x)=0\}, \Theta_{\infty} : = \{ x \in [0,1] : \theta_\alpha'(x)=\infty \} $ and $\Theta_\sim : = \{ x \in [0,1] : \theta_\alpha'(x)\ does\ not\ exist \} $. The main result is that \[ \dim_{\mathrm{H}}(\Theta_\infty)=\dim_{\mathrm{H}}(\Theta_\sim)=\sigma_\alpha(\log2)<\dim_{\mathrm{H}}(\Theta_0)=1, \] where $\sigma_\alpha(\log2)$ denotes the Hausdorff dimension of the level set $\{x\in [0,1]:\Lambda(F_\alpha, x)=\log2\}$ and $\Lambda(F_\alpha, x)$ is the Lyapunov exponent of the map $F_\alpha$ at the point $x$. The proof of the theorem employs the multifractal formalism for $\alpha$-Farey systems.
2014, 34(2): 733-748
doi: 10.3934/dcds.2014.34.733
+[Abstract](1417)
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Abstract:
We consider the defocusing $\dot{H}^{1/2}$-critical nonlinear Schrödinger equation in dimensions $d\geq 4.$ In the spirit of Kenig and Merle [10], we combine a concentration-compactness approach with the Lin--Strauss Morawetz inequality to prove that if a solution $u$ is bounded in $\dot{H}^{1/2}$ throughout its lifespan, then $u$ is global and scatters.
We consider the defocusing $\dot{H}^{1/2}$-critical nonlinear Schrödinger equation in dimensions $d\geq 4.$ In the spirit of Kenig and Merle [10], we combine a concentration-compactness approach with the Lin--Strauss Morawetz inequality to prove that if a solution $u$ is bounded in $\dot{H}^{1/2}$ throughout its lifespan, then $u$ is global and scatters.
2014, 34(2): 749-760
doi: 10.3934/dcds.2014.34.749
+[Abstract](1446)
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Abstract:
We present a survey of recent results concerning heat and telegraph equations, equipped with Goldstein-Wentzell boundary conditions (already known as general Wentzell boundary conditions). We focus on the generation of analytic semigroups and continuous dependence of the solutions of the associated Cauchy problems from the boundary conditions.
We present a survey of recent results concerning heat and telegraph equations, equipped with Goldstein-Wentzell boundary conditions (already known as general Wentzell boundary conditions). We focus on the generation of analytic semigroups and continuous dependence of the solutions of the associated Cauchy problems from the boundary conditions.
2014, 34(2): 761-787
doi: 10.3934/dcds.2014.34.761
+[Abstract](1498)
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Abstract:
Let $Ω\subset\mathbb{R}^N$ ($N\ge 2$) be a bounded domain with a boundary $∂Ω$ of class $C^2$ and let $\alpha,\beta$ be maximal monotone graphs in $\mathbb{R}^2$ satisfying $\alpha(0)\cap\beta(0)\ni 0$. Given $f\in L^1(Ω)$ and $g\in L^1(∂Ω)$, we characterize the existence and uniqueness of weak solutions to the semi-linear elliptic equation $-\Delta u+\alpha(u)\ni f$ in $Ω$ with the nonlinear general Wentzell boundary conditions $-\Delta_{\Gamma} u+\frac{\partial u}{\partial\nu}+\beta(u)\ni g$ on $∂Ω$. We also show the well-posedness of the associated parabolic problem on the Banach space $L^1(Ω)\times L^1(∂Ω)$.
Let $Ω\subset\mathbb{R}^N$ ($N\ge 2$) be a bounded domain with a boundary $∂Ω$ of class $C^2$ and let $\alpha,\beta$ be maximal monotone graphs in $\mathbb{R}^2$ satisfying $\alpha(0)\cap\beta(0)\ni 0$. Given $f\in L^1(Ω)$ and $g\in L^1(∂Ω)$, we characterize the existence and uniqueness of weak solutions to the semi-linear elliptic equation $-\Delta u+\alpha(u)\ni f$ in $Ω$ with the nonlinear general Wentzell boundary conditions $-\Delta_{\Gamma} u+\frac{\partial u}{\partial\nu}+\beta(u)\ni g$ on $∂Ω$. We also show the well-posedness of the associated parabolic problem on the Banach space $L^1(Ω)\times L^1(∂Ω)$.
2014, 34(2): 789-802
doi: 10.3934/dcds.2014.34.789
+[Abstract](2263)
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Abstract:
This paper deals with the global existence and boundedness of the solutions for the chemotaxis system with logistic source \begin{eqnarray*} \left\{ \begin{array}{llll} u_t=\nabla\cdot(\phi(u)\nabla u)-\nabla\cdot(\varphi(u)\nabla v)+f(u),\quad &x\in \Omega,\quad t>0,\\ v_t=\Delta v-v+u,\quad &x\in\Omega,\quad t>0,\\ \end{array} \right. \end{eqnarray*} under homogeneous Neumann boundary conditions in a convex smooth bounded domain $\Omega\subset \mathbb{R}^n (n\geq2),$ with non-negative initial data $u_0\in C^0(\overline{\Omega})$ and $v_0\in W^{1,\theta}{(\Omega)}$ (with some $\theta>n$). The nonlinearities $\phi$ and $\varphi$ are assumed to generalize the prototypes \begin{eqnarray*} \phi(u)=(u+1)^{-\alpha},\,\,\,\,\,\, \varphi(u)=u(u+1)^{\beta-1} \end{eqnarray*} with $\alpha\in \mathbb{R}$ and $\beta\in \mathbb{R}$. $f(u)$ is a smooth function generalizing the logistic function \begin{eqnarray*} f(u)=ru-bu^\gamma,\,\,\,\,\,\, u\geq0,\,\,\text{with}\,\, r\geq0,\,\,b>0\,\,\text{and}\,\,\gamma>1. \end{eqnarray*} It is proved that the corresponding initial-boundary value problem possesses a unique global classical solution that is uniformly bounded provided that some technical conditions are fulfilled.
This paper deals with the global existence and boundedness of the solutions for the chemotaxis system with logistic source \begin{eqnarray*} \left\{ \begin{array}{llll} u_t=\nabla\cdot(\phi(u)\nabla u)-\nabla\cdot(\varphi(u)\nabla v)+f(u),\quad &x\in \Omega,\quad t>0,\\ v_t=\Delta v-v+u,\quad &x\in\Omega,\quad t>0,\\ \end{array} \right. \end{eqnarray*} under homogeneous Neumann boundary conditions in a convex smooth bounded domain $\Omega\subset \mathbb{R}^n (n\geq2),$ with non-negative initial data $u_0\in C^0(\overline{\Omega})$ and $v_0\in W^{1,\theta}{(\Omega)}$ (with some $\theta>n$). The nonlinearities $\phi$ and $\varphi$ are assumed to generalize the prototypes \begin{eqnarray*} \phi(u)=(u+1)^{-\alpha},\,\,\,\,\,\, \varphi(u)=u(u+1)^{\beta-1} \end{eqnarray*} with $\alpha\in \mathbb{R}$ and $\beta\in \mathbb{R}$. $f(u)$ is a smooth function generalizing the logistic function \begin{eqnarray*} f(u)=ru-bu^\gamma,\,\,\,\,\,\, u\geq0,\,\,\text{with}\,\, r\geq0,\,\,b>0\,\,\text{and}\,\,\gamma>1. \end{eqnarray*} It is proved that the corresponding initial-boundary value problem possesses a unique global classical solution that is uniformly bounded provided that some technical conditions are fulfilled.
2014, 34(2): 803-820
doi: 10.3934/dcds.2014.34.803
+[Abstract](1898)
+[PDF](454.8KB)
Abstract:
In this paper, we study the generalized Novikov equation which describes the motion of shallow water waves. By using the Littlewood-Paley decomposition and nonhomogeneous Besov spaces, we prove that the Cauchy problem for the generalized Novikov equation is locally well-posed in Besov space $B_{p,r}^{s}$ with $1\leq p, r\leq +\infty$ and $s>{\rm max}\{1+\frac{1}{p},\frac{3}{2}\}$. We also show the persistence property of the strong solutions which implies that the solution decays at infinity in the spatial variable provided that the initial function does.
In this paper, we study the generalized Novikov equation which describes the motion of shallow water waves. By using the Littlewood-Paley decomposition and nonhomogeneous Besov spaces, we prove that the Cauchy problem for the generalized Novikov equation is locally well-posed in Besov space $B_{p,r}^{s}$ with $1\leq p, r\leq +\infty$ and $s>{\rm max}\{1+\frac{1}{p},\frac{3}{2}\}$. We also show the persistence property of the strong solutions which implies that the solution decays at infinity in the spatial variable provided that the initial function does.
2014, 34(2): 821-841
doi: 10.3934/dcds.2014.34.821
+[Abstract](1637)
+[PDF](310.2KB)
Abstract:
In this paper we investigate a nonlinear diffusion equation with the Neumann boundary condition, which was proposed by Nagylaki in [19] to describe the evolution of two types of genes in population genetics. For such a model, we obtain the existence of nontrivial solutions and the limiting profile of such solutions as the diffusion rate $d\rightarrow0$ or $d\rightarrow\infty$. Our results show that as $d\rightarrow0$, the location of nontrivial solutions relative to trivial solutions plays a very important role for the existence and shape of limiting profile. In particular, an example is given to illustrate that the limiting profile does not exist for some nontrivial solutions. Moreover, to better understand the dynamics of this model, we analyze the stability and bifurcation of solutions. These conclusions provide a different angle to understand that obtained in [17,21].
In this paper we investigate a nonlinear diffusion equation with the Neumann boundary condition, which was proposed by Nagylaki in [19] to describe the evolution of two types of genes in population genetics. For such a model, we obtain the existence of nontrivial solutions and the limiting profile of such solutions as the diffusion rate $d\rightarrow0$ or $d\rightarrow\infty$. Our results show that as $d\rightarrow0$, the location of nontrivial solutions relative to trivial solutions plays a very important role for the existence and shape of limiting profile. In particular, an example is given to illustrate that the limiting profile does not exist for some nontrivial solutions. Moreover, to better understand the dynamics of this model, we analyze the stability and bifurcation of solutions. These conclusions provide a different angle to understand that obtained in [17,21].
2014, 34(2): 843-867
doi: 10.3934/dcds.2014.34.843
+[Abstract](2476)
+[PDF](541.0KB)
Abstract:
This paper deals with the Cauchy problem for a weakly dissipative shallow water equation with high-order nonlinearities $y_{t}+u^{m+1}y_{x}+bu^{m}u_{x}y+\lambda y=0$, where $\lambda,b$ are constants and $m\in\mathbb{N}$, the notation $y:= (1-\partial_x^2) u$, which includes the famous $b$-equation and Novikov equations as special cases. The local well-posedness of solutions for the Cauchy problem in Besov space $B^s_{p,r} $ with $1\leq p,r \leq +\infty$ and $s>\max\{1+\frac{1}{p},\frac{3}{2}\}$ is obtained. Under some assumptions, the existence and uniqueness of the global solutions to the equation are shown, and conditions that lead to the development of singularities in finite time for the solutions are acquired, moreover, the propagation behaviors of compactly supported solutions are also established. Finally, the weak solution and analytic solution for the equation are considered.
This paper deals with the Cauchy problem for a weakly dissipative shallow water equation with high-order nonlinearities $y_{t}+u^{m+1}y_{x}+bu^{m}u_{x}y+\lambda y=0$, where $\lambda,b$ are constants and $m\in\mathbb{N}$, the notation $y:= (1-\partial_x^2) u$, which includes the famous $b$-equation and Novikov equations as special cases. The local well-posedness of solutions for the Cauchy problem in Besov space $B^s_{p,r} $ with $1\leq p,r \leq +\infty$ and $s>\max\{1+\frac{1}{p},\frac{3}{2}\}$ is obtained. Under some assumptions, the existence and uniqueness of the global solutions to the equation are shown, and conditions that lead to the development of singularities in finite time for the solutions are acquired, moreover, the propagation behaviors of compactly supported solutions are also established. Finally, the weak solution and analytic solution for the equation are considered.
2014, 34(2): 869-882
doi: 10.3934/dcds.2014.34.869
+[Abstract](1484)
+[PDF](384.9KB)
Abstract:
In this paper, $C^0$ random perturbations of a partially hyperbolic diffeomorphism are considered. It is shown that a partially hyperbolic diffeomorphism is quasi-stable under such perturbations.
In this paper, $C^0$ random perturbations of a partially hyperbolic diffeomorphism are considered. It is shown that a partially hyperbolic diffeomorphism is quasi-stable under such perturbations.
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