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1078-0947
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Discrete & Continuous Dynamical Systems - A
December 2016 , Volume 36 , Issue 12
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2016, 36(12): 6623-6643
doi: 10.3934/dcds.2016087
+[Abstract](1951)
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Abstract:
We study the distribution of bulk superconductivity in presence of an applied magnetic field, supposed to be a step function, modeled by the Ginzburg-Landau theory. Our results are valid for the minimizers of the two-dimensional Ginzburg-Landau functional with a large Ginzburg-Landau parameter and with an applied magnetic field of intensity comparable with the Ginzburg-Landau parameter.
We study the distribution of bulk superconductivity in presence of an applied magnetic field, supposed to be a step function, modeled by the Ginzburg-Landau theory. Our results are valid for the minimizers of the two-dimensional Ginzburg-Landau functional with a large Ginzburg-Landau parameter and with an applied magnetic field of intensity comparable with the Ginzburg-Landau parameter.
2016, 36(12): 6645-6656
doi: 10.3934/dcds.2016088
+[Abstract](2438)
+[PDF](429.7KB)
Abstract:
The nonlinear recombination equation from population genetics has a long history and is notoriously difficult to solve, both in continuous and in discrete time. This is particularly so if one aims at full generality, thus also including degenerate parameter cases. Due to recent progress for the continuous time case via the identification of an underlying stochastic fragmentation process, it became clear that a direct general solution at the level of the corresponding ODE itself should also be possible. This paper shows how to do it, and how to extend the approach to the discrete-time case as well.
The nonlinear recombination equation from population genetics has a long history and is notoriously difficult to solve, both in continuous and in discrete time. This is particularly so if one aims at full generality, thus also including degenerate parameter cases. Due to recent progress for the continuous time case via the identification of an underlying stochastic fragmentation process, it became clear that a direct general solution at the level of the corresponding ODE itself should also be possible. This paper shows how to do it, and how to extend the approach to the discrete-time case as well.
2016, 36(12): 6657-6668
doi: 10.3934/dcds.2016089
+[Abstract](1989)
+[PDF](411.2KB)
Abstract:
We provide a general framework to study differentiability of SRB measures for one dimensional non-uniformly expanding maps. Our technique is based on inducing the non-uniformly expanding system to a uniformly expanding one, and on showing how the linear response formula of the non-uniformly expanding system is inherited from the linear response formula of the induced one. We apply this general technique to interval maps with a neutral fixed point (Pomeau-Manneville maps) to prove differentiability of the corresponding SRB measure. Our work covers systems that admit a finite SRB measure and it also covers systems that admit an infinite SRB measure. In particular, we obtain a linear response formula for both finite and infinite SRB measures. To the best of our knowledge, this is the first work that contains a linear response result for infinite measure preserving systems.
We provide a general framework to study differentiability of SRB measures for one dimensional non-uniformly expanding maps. Our technique is based on inducing the non-uniformly expanding system to a uniformly expanding one, and on showing how the linear response formula of the non-uniformly expanding system is inherited from the linear response formula of the induced one. We apply this general technique to interval maps with a neutral fixed point (Pomeau-Manneville maps) to prove differentiability of the corresponding SRB measure. Our work covers systems that admit a finite SRB measure and it also covers systems that admit an infinite SRB measure. In particular, we obtain a linear response formula for both finite and infinite SRB measures. To the best of our knowledge, this is the first work that contains a linear response result for infinite measure preserving systems.
2016, 36(12): 6669-6688
doi: 10.3934/dcds.2016090
+[Abstract](2483)
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Abstract:
We prove the stability of global equilibrium in a multi-species mixture, where the different species can have different masses, on the $3$-dimensional torus. We establish stability estimates in $L^\infty_{x,v}(w)$ where $w=w(v)$ is either polynomial or exponential, with explicit threshold. Along the way we extend recent estimates and stability results for the mono-species Boltzmann operator not only to the multi-species case but also to more general hard potential and Maxwellian kernels.
We prove the stability of global equilibrium in a multi-species mixture, where the different species can have different masses, on the $3$-dimensional torus. We establish stability estimates in $L^\infty_{x,v}(w)$ where $w=w(v)$ is either polynomial or exponential, with explicit threshold. Along the way we extend recent estimates and stability results for the mono-species Boltzmann operator not only to the multi-species case but also to more general hard potential and Maxwellian kernels.
2016, 36(12): 6689-6714
doi: 10.3934/dcds.2016091
+[Abstract](2297)
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Abstract:
We establish a global Calderón-Zygmund estimate for homogenization of a parabolic system in divergence form with discontinuous coefficients in a bounded nonsmooth domain under the assumptions that the coefficients have small BMO seminorms and the boundary of the domain is $\delta$-flat for some $\delta>0$ depending on the given data.
We establish a global Calderón-Zygmund estimate for homogenization of a parabolic system in divergence form with discontinuous coefficients in a bounded nonsmooth domain under the assumptions that the coefficients have small BMO seminorms and the boundary of the domain is $\delta$-flat for some $\delta>0$ depending on the given data.
2016, 36(12): 6715-6736
doi: 10.3934/dcds.2016092
+[Abstract](2558)
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Abstract:
In this paper we study normal forms of planar differential systems with a non-degenerate equilibrium on a single switching line, i.e., the equilibrium is a non-degenerate equilibrium of both the upper system and the lower one. In the sense of $C^0$ conjugation we find all normal forms for linear switching systems and use them together with switching near-identity transformations to normalize second order terms, showing the reduction of normal forms. We prove that only one of those 19 types of linear normal form decides if the system is monodromic. With the monodromic linear normal form, we compute the second order monodromic normal form, which gives a condition under which exactly one limit cycle arises from a Hopf bifurcation.
In this paper we study normal forms of planar differential systems with a non-degenerate equilibrium on a single switching line, i.e., the equilibrium is a non-degenerate equilibrium of both the upper system and the lower one. In the sense of $C^0$ conjugation we find all normal forms for linear switching systems and use them together with switching near-identity transformations to normalize second order terms, showing the reduction of normal forms. We prove that only one of those 19 types of linear normal form decides if the system is monodromic. With the monodromic linear normal form, we compute the second order monodromic normal form, which gives a condition under which exactly one limit cycle arises from a Hopf bifurcation.
2016, 36(12): 6737-6765
doi: 10.3934/dcds.2016093
+[Abstract](2000)
+[PDF](490.1KB)
Abstract:
In this paper we study the eigenvalue problems for a nonlocal operator of order $s$ that is analogous to the local pseudo $p-$Laplacian. We show that there is a sequence of eigenvalues $\lambda_n \to \infty$ and that the first one is positive, simple, isolated and has a positive and bounded associated eigenfunction. For the first eigenvalue we also analyze the limits as $p\to \infty$ (obtaining a limit nonlocal eigenvalue problem analogous to the pseudo infinity Laplacian) and as $s\to 1^-$ (obtaining the first eigenvalue for a local operator of $p-$Laplacian type). To perform this study we have to introduce anisotropic fractional Sobolev spaces and prove some of their properties.
In this paper we study the eigenvalue problems for a nonlocal operator of order $s$ that is analogous to the local pseudo $p-$Laplacian. We show that there is a sequence of eigenvalues $\lambda_n \to \infty$ and that the first one is positive, simple, isolated and has a positive and bounded associated eigenfunction. For the first eigenvalue we also analyze the limits as $p\to \infty$ (obtaining a limit nonlocal eigenvalue problem analogous to the pseudo infinity Laplacian) and as $s\to 1^-$ (obtaining the first eigenvalue for a local operator of $p-$Laplacian type). To perform this study we have to introduce anisotropic fractional Sobolev spaces and prove some of their properties.
2016, 36(12): 6767-6780
doi: 10.3934/dcds.2016094
+[Abstract](3126)
+[PDF](443.9KB)
Abstract:
In this paper, we classify the positive solutions to the following Lane-Emden type integral system with negative exponents \begin{equation*} \begin{cases} u(x)&= \displaystyle \int_{\mathbb{R}^{n}}|x-y|^{\tau} u^{-p}(y)v^{-q}(y) \, dy, ~x\in \mathbb{R}^{n}, \\ v(x)&= \displaystyle \int_{\mathbb{R}^{n}}|x-y|^{\tau}u^{-r}(y)v^{-s}(y) \, dy,~ x\in \mathbb{R}^{n}, \end{cases} \end{equation*}where $n \geq 1$ is an integer and $ \tau, p,q,r,s>0.$ Particularly, using an integral form of the method of moving spheres, we classify the positive solutions to the integral system whenever $$p+q=r+s=1 + 2n/\tau.$$ We also establish the non-existence of positive solutions under the condition $$\max\{p+q,r+s\} \leq 1 + 2n/\tau \,\text{ and }\, p + q + r + s < 2(1 + 2n/\tau).$$
In this paper, we classify the positive solutions to the following Lane-Emden type integral system with negative exponents \begin{equation*} \begin{cases} u(x)&= \displaystyle \int_{\mathbb{R}^{n}}|x-y|^{\tau} u^{-p}(y)v^{-q}(y) \, dy, ~x\in \mathbb{R}^{n}, \\ v(x)&= \displaystyle \int_{\mathbb{R}^{n}}|x-y|^{\tau}u^{-r}(y)v^{-s}(y) \, dy,~ x\in \mathbb{R}^{n}, \end{cases} \end{equation*}where $n \geq 1$ is an integer and $ \tau, p,q,r,s>0.$ Particularly, using an integral form of the method of moving spheres, we classify the positive solutions to the integral system whenever $$p+q=r+s=1 + 2n/\tau.$$ We also establish the non-existence of positive solutions under the condition $$\max\{p+q,r+s\} \leq 1 + 2n/\tau \,\text{ and }\, p + q + r + s < 2(1 + 2n/\tau).$$
2016, 36(12): 6781-6797
doi: 10.3934/dcds.2016095
+[Abstract](1970)
+[PDF](524.1KB)
Abstract:
We prove existence of certain compressible subsonic inviscid flows with vortex sheets separating from a solid boundary. To leading order, any perturbation of the upstream boundary causes positive drag. We also prove that if a region of irrotational inviscid flow bounded by a vortex sheet and slip condition wall is enclosed in an angle less than $180^\circ$, then the velocity is zero.
We prove existence of certain compressible subsonic inviscid flows with vortex sheets separating from a solid boundary. To leading order, any perturbation of the upstream boundary causes positive drag. We also prove that if a region of irrotational inviscid flow bounded by a vortex sheet and slip condition wall is enclosed in an angle less than $180^\circ$, then the velocity is zero.
2016, 36(12): 6799-6833
doi: 10.3934/dcds.2016096
+[Abstract](2332)
+[PDF](608.2KB)
Abstract:
In this work, we show that a family of non-linear mean-field equations on discrete spaces can be viewed as a gradient flow of a natural free energy functional with respect to a certain metric structure we make explicit. We also prove that this gradient flow structure arises as the limit of the gradient flow structures of a natural sequence of $N$-particle dynamics, as $N$ goes to infinity.
In this work, we show that a family of non-linear mean-field equations on discrete spaces can be viewed as a gradient flow of a natural free energy functional with respect to a certain metric structure we make explicit. We also prove that this gradient flow structure arises as the limit of the gradient flow structures of a natural sequence of $N$-particle dynamics, as $N$ goes to infinity.
2016, 36(12): 6835-6853
doi: 10.3934/dcds.2016097
+[Abstract](2402)
+[PDF](442.2KB)
Abstract:
We establish an improved form of the classical logarithmic Sobolev inequality for the Gaussian measure restricted to probability densities which satisfy a Poincaré inequality. The result implies a lower bound on the deficit in terms of the quadratic Kantorovich-Wasserstein distance. We similarly investigate the deficit in the Talagrand quadratic transportation cost inequality this time by means of an ${ L}^1$-Kantorovich-Wasserstein distance, optimal for product measures, and deduce a lower bound on the deficit in the logarithmic Sobolev inequality in terms of this metric. Applications are given in the context of the Bakry-Émery theory and the coherent state transform. The proofs combine tools from semigroup and heat kernel theory and optimal mass transportation.
We establish an improved form of the classical logarithmic Sobolev inequality for the Gaussian measure restricted to probability densities which satisfy a Poincaré inequality. The result implies a lower bound on the deficit in terms of the quadratic Kantorovich-Wasserstein distance. We similarly investigate the deficit in the Talagrand quadratic transportation cost inequality this time by means of an ${ L}^1$-Kantorovich-Wasserstein distance, optimal for product measures, and deduce a lower bound on the deficit in the logarithmic Sobolev inequality in terms of this metric. Applications are given in the context of the Bakry-Émery theory and the coherent state transform. The proofs combine tools from semigroup and heat kernel theory and optimal mass transportation.
2016, 36(12): 6855-6871
doi: 10.3934/dcds.2016098
+[Abstract](2011)
+[PDF](449.6KB)
Abstract:
Consider the delay differential equation \begin{equation*} \dot{x}(t)=a \Bigg(\sum_{i=1}^n b_i\big[x(t-s_i)- x(t-r_i)\big]\Bigg)-g(x(t)), \end{equation*} where $a>0$, $b_i>0$ and $0\leq s_i < r_i$ $(i\in \{1,\dots,n\})$ are parameters, $g\colon \mathbb{R} \to \mathbb{R}$ is an odd $C^1$ function with $g'(0)=0$, the map $(0,\infty)\ni \xi \mapsto g(\xi)/\xi\in\mathbb{R}$ is strictly increasing and $\sup_{\xi>0} g(\xi)/\xi>2a$. This equation can be interpreted as a price model, where $x(t)$ represents the price of an asset (e.g. price of share or commodity, currency exchange rate etc.) at time $t$. The first term on the right-hand side represents the positive response for the recent tendencies of the price and $-g(x(t))$ is responsible for the instantaneous negative feedback to the deviation from the equilibrium price.
We study the local and global stability of the unique, non-hyperbolic equilibrium point. The main result gives a sufficient condition for global asymptotic stability of the equilibrium. The region of attractivity is also estimated in case of local asymptotic stability.
Consider the delay differential equation \begin{equation*} \dot{x}(t)=a \Bigg(\sum_{i=1}^n b_i\big[x(t-s_i)- x(t-r_i)\big]\Bigg)-g(x(t)), \end{equation*} where $a>0$, $b_i>0$ and $0\leq s_i < r_i$ $(i\in \{1,\dots,n\})$ are parameters, $g\colon \mathbb{R} \to \mathbb{R}$ is an odd $C^1$ function with $g'(0)=0$, the map $(0,\infty)\ni \xi \mapsto g(\xi)/\xi\in\mathbb{R}$ is strictly increasing and $\sup_{\xi>0} g(\xi)/\xi>2a$. This equation can be interpreted as a price model, where $x(t)$ represents the price of an asset (e.g. price of share or commodity, currency exchange rate etc.) at time $t$. The first term on the right-hand side represents the positive response for the recent tendencies of the price and $-g(x(t))$ is responsible for the instantaneous negative feedback to the deviation from the equilibrium price.
We study the local and global stability of the unique, non-hyperbolic equilibrium point. The main result gives a sufficient condition for global asymptotic stability of the equilibrium. The region of attractivity is also estimated in case of local asymptotic stability.
2016, 36(12): 6873-6898
doi: 10.3934/dcds.2016099
+[Abstract](2501)
+[PDF](507.5KB)
Abstract:
We consider the following problem: \begin{equation*} \left\{\begin{array}{ll} (-\Delta)^s u=K(y)u^{p-1} \hbox { in } \ \mathbb{R}^N, \\ u>0, \ y \in \mathbb{R}^N, \end{array}\right. (P) \end{equation*} where $s\in(0,\frac{1}{2})$ for $N=3$, $s\in(0,1)$ for $N\geq4$ and $p=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent. Under the condition that the function $K(y)$ has a local maximum point, we prove the existence of infinitely many non-radial solutions for the problem $(P)$.
We consider the following problem: \begin{equation*} \left\{\begin{array}{ll} (-\Delta)^s u=K(y)u^{p-1} \hbox { in } \ \mathbb{R}^N, \\ u>0, \ y \in \mathbb{R}^N, \end{array}\right. (P) \end{equation*} where $s\in(0,\frac{1}{2})$ for $N=3$, $s\in(0,1)$ for $N\geq4$ and $p=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent. Under the condition that the function $K(y)$ has a local maximum point, we prove the existence of infinitely many non-radial solutions for the problem $(P)$.
2016, 36(12): 6899-6919
doi: 10.3934/dcds.2016100
+[Abstract](2580)
+[PDF](483.3KB)
Abstract:
In this paper, we consider the viscoelastic equation with damping and source terms and acoustic boundary conditions. We prove a global existence of solutions and uniform decay rates of the energy without imposing any restrictive growth assumption on the damping term and weakening of the usual assumptions on the relaxation function. This paper is to improve the result of [4] by applying the method developed in [17].
In this paper, we consider the viscoelastic equation with damping and source terms and acoustic boundary conditions. We prove a global existence of solutions and uniform decay rates of the energy without imposing any restrictive growth assumption on the damping term and weakening of the usual assumptions on the relaxation function. This paper is to improve the result of [4] by applying the method developed in [17].
Global well-posedness for inhomogeneous Navier-Stokes equations with logarithmical hyper-dissipation
2016, 36(12): 6921-6941
doi: 10.3934/dcds.2016101
+[Abstract](2897)
+[PDF](470.0KB)
Abstract:
This paper is devoted to studying the global well-posedness for 3D inhomogeneous logarithmical hyper-dissipative Navier-Stokes equations with dissipative terms $D^2u$. Here we consider the supercritical case, namely, the symbol of the Fourier multiplier $D$ takes the form $h(\xi)=|\xi|^{\frac{5}{4}}/g(\xi)$, where $g(\xi)=\log^{\frac{1}{4}}(2+|\xi|^2)$. This generalizes the work of Tao [17] to the inhomogeneous system, and can also be viewed as a generalization of Fang and Zi [12], in which they considered the critical case $h(\xi)=|\xi|^{\frac{5}{4}}$.
This paper is devoted to studying the global well-posedness for 3D inhomogeneous logarithmical hyper-dissipative Navier-Stokes equations with dissipative terms $D^2u$. Here we consider the supercritical case, namely, the symbol of the Fourier multiplier $D$ takes the form $h(\xi)=|\xi|^{\frac{5}{4}}/g(\xi)$, where $g(\xi)=\log^{\frac{1}{4}}(2+|\xi|^2)$. This generalizes the work of Tao [17] to the inhomogeneous system, and can also be viewed as a generalization of Fang and Zi [12], in which they considered the critical case $h(\xi)=|\xi|^{\frac{5}{4}}$.
2016, 36(12): 6943-6974
doi: 10.3934/dcds.2016102
+[Abstract](2594)
+[PDF](578.3KB)
Abstract:
We consider the Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity $(i\partial _t + \Delta ) u= \pm \partial (\overline{u}^m)$ on $\mathbb{R} ^d$, $d \ge 1$, with random initial data, where $\partial$ is a first order derivative with respect to the spatial variable, for example a linear combination of $\frac{\partial}{\partial x_1} , \, \dots , \, \frac{\partial}{\partial x_d}$ or $|\nabla |= \mathcal{F}^{-1}[|\xi | \mathcal{F}]$. We prove that almost sure local in time well-posedness, small data global in time well-posedness and scattering hold in $H^s(\mathbb{R} ^d)$ with $s> \max \left( \frac{d-1}{d} s_c , \frac{s_c}{2}, s_c - \frac{d}{2(d+1)} \right)$ for $d+m \ge 5$, where $s$ is below the scaling critical regularity $s_c := \frac{d}{2}-\frac{1}{m-1}$.
We consider the Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity $(i\partial _t + \Delta ) u= \pm \partial (\overline{u}^m)$ on $\mathbb{R} ^d$, $d \ge 1$, with random initial data, where $\partial$ is a first order derivative with respect to the spatial variable, for example a linear combination of $\frac{\partial}{\partial x_1} , \, \dots , \, \frac{\partial}{\partial x_d}$ or $|\nabla |= \mathcal{F}^{-1}[|\xi | \mathcal{F}]$. We prove that almost sure local in time well-posedness, small data global in time well-posedness and scattering hold in $H^s(\mathbb{R} ^d)$ with $s> \max \left( \frac{d-1}{d} s_c , \frac{s_c}{2}, s_c - \frac{d}{2(d+1)} \right)$ for $d+m \ge 5$, where $s$ is below the scaling critical regularity $s_c := \frac{d}{2}-\frac{1}{m-1}$.
2016, 36(12): 6975-7000
doi: 10.3934/dcds.2016103
+[Abstract](3000)
+[PDF](498.2KB)
Abstract:
In this paper, we study the Cauchy problem for an integrable multi-component ($2N$-component) peakon system which is involved in an arbitrary polynomial function. Based on a generalized Ovsyannikov type theorem, we first prove the existence and uniqueness of solutions for the system in the Gevrey-Sobolev spaces with the lower bound of the lifespan. Then we show the continuity of the data-to-solution map for the system. Furthermore, by introducing a family of continuous diffeomorphisms of a line and utilizing the fine structure of the system, we demonstrate the system exhibits unique continuation.
In this paper, we study the Cauchy problem for an integrable multi-component ($2N$-component) peakon system which is involved in an arbitrary polynomial function. Based on a generalized Ovsyannikov type theorem, we first prove the existence and uniqueness of solutions for the system in the Gevrey-Sobolev spaces with the lower bound of the lifespan. Then we show the continuity of the data-to-solution map for the system. Furthermore, by introducing a family of continuous diffeomorphisms of a line and utilizing the fine structure of the system, we demonstrate the system exhibits unique continuation.
2016, 36(12): 7001-7020
doi: 10.3934/dcds.2016104
+[Abstract](2311)
+[PDF](428.0KB)
Abstract:
A new method is presented to prove finiteness of the fractal and Hausdorff dimensions of the global attractor for the strong solutions to the 3D Primitive Equations (PEs) with viscosity, which is applicable to more general situations than the recent result of [8] in the sense that it removes all extra technical conditions imposed by previous analyses. More specifically, the dimensions of the global attractor are proved finite for heat source $Q\in L^2$, exactly the same condition for well-posedness of global strong solutions and existence of the global attractor of these solutions; while the best previous result obtained recently in [8] still requires the extra condition that $∂_zQ\in L^2$ for finiteness of the dimensions of the global attractor. The key new idea is that Ladyzhenskaya's squeezing property of the semigroup for the strong solutions can be established without higher solution regularity of Primitive Equations. This has the general interest for dissipative evolution equations. For this reason, the new method especially has the advantange of dealing with more complicated boundary conditions which present essential difficulties for previous methods. In particular, the case of 3D viscous PEs with `` physical boundary conditions'' can be treated by the new method in the same way as presented in this article, however, it seems rather difficult for previous methods.
A new method is presented to prove finiteness of the fractal and Hausdorff dimensions of the global attractor for the strong solutions to the 3D Primitive Equations (PEs) with viscosity, which is applicable to more general situations than the recent result of [8] in the sense that it removes all extra technical conditions imposed by previous analyses. More specifically, the dimensions of the global attractor are proved finite for heat source $Q\in L^2$, exactly the same condition for well-posedness of global strong solutions and existence of the global attractor of these solutions; while the best previous result obtained recently in [8] still requires the extra condition that $∂_zQ\in L^2$ for finiteness of the dimensions of the global attractor. The key new idea is that Ladyzhenskaya's squeezing property of the semigroup for the strong solutions can be established without higher solution regularity of Primitive Equations. This has the general interest for dissipative evolution equations. For this reason, the new method especially has the advantange of dealing with more complicated boundary conditions which present essential difficulties for previous methods. In particular, the case of 3D viscous PEs with `` physical boundary conditions'' can be treated by the new method in the same way as presented in this article, however, it seems rather difficult for previous methods.
2016, 36(12): 7021-7028
doi: 10.3934/dcds.2016105
+[Abstract](2084)
+[PDF](397.4KB)
Abstract:
We will show that, in the the geometric Lorenz flow, the set of initial states which give rise to orbits with historic behavior is residual in a trapping region.
We will show that, in the the geometric Lorenz flow, the set of initial states which give rise to orbits with historic behavior is residual in a trapping region.
2016, 36(12): 7029-7056
doi: 10.3934/dcds.2016106
+[Abstract](1870)
+[PDF](290.5KB)
Abstract:
We consider MacKay's renormalization operator for pairs of area-preserving maps, near the fixed point obtained in [1]. Of particular interest is the restriction $\mathfrak{R}_{0}$ of this operator to pairs that commute and have a zero Calabi invariant. We prove that a suitable extension of $\mathfrak{R}_{0}^{3}$ is hyperbolic at the fixed point, with a single expanding direction. The pairs in this direction are presumably commuting, but we currently have no proof for this. Our analysis yields rigorous bounds on various ``universal'' quantities, including the expanding eigenvalue.
We consider MacKay's renormalization operator for pairs of area-preserving maps, near the fixed point obtained in [1]. Of particular interest is the restriction $\mathfrak{R}_{0}$ of this operator to pairs that commute and have a zero Calabi invariant. We prove that a suitable extension of $\mathfrak{R}_{0}^{3}$ is hyperbolic at the fixed point, with a single expanding direction. The pairs in this direction are presumably commuting, but we currently have no proof for this. Our analysis yields rigorous bounds on various ``universal'' quantities, including the expanding eigenvalue.
2016, 36(12): 7057-7061
doi: 10.3934/dcds.2016107
+[Abstract](2019)
+[PDF](255.6KB)
Abstract:
We show that a spatially periodic solution to the irrotational two-dimensional gravity water wave problem, with the property that the horizontal velocity component at the flat bed is symmetric, while the acceleration at the flat bed is anti-symmetric with respect to a common axis of symmetry, necessarily constitutes a traveling wave. The proof makes use complex variables and structural properties of the governing equations for nonlinear water waves.
We show that a spatially periodic solution to the irrotational two-dimensional gravity water wave problem, with the property that the horizontal velocity component at the flat bed is symmetric, while the acceleration at the flat bed is anti-symmetric with respect to a common axis of symmetry, necessarily constitutes a traveling wave. The proof makes use complex variables and structural properties of the governing equations for nonlinear water waves.
2016, 36(12): 7063-7079
doi: 10.3934/dcds.2016108
+[Abstract](2252)
+[PDF](460.3KB)
Abstract:
In this paper, we study the dynamics of a non-autonomous semilinear degenerate parabolic equation $u_{t}-div(\sigma(x)\nabla u)+ f(u)=g(x, t)$ defined on a bounded domain $\Omega\subset \mathbb{R}^N$ with smooth boundary. We first establish a Nash-Moser-Alikakos type a priori estimate for the difference of solutions near the initial time; Then we prove that the solution process $U(t,\tau)$ is continuous from $L^{2}(\Omega)$ to $\mathscr{D}_{0}^{1}(\Omega, \sigma)$ w.r.t. initial data; And finally show that the known $(L^{2}(\Omega), L^{2}(\Omega))$ pullback $\mathscr{D}_{\lambda}$-attractor indeed can attract in $\mathscr{D}_{0}^{1}(\Omega, \sigma)$-norm. Any differentiability on the forcing term is not required.
In this paper, we study the dynamics of a non-autonomous semilinear degenerate parabolic equation $u_{t}-div(\sigma(x)\nabla u)+ f(u)=g(x, t)$ defined on a bounded domain $\Omega\subset \mathbb{R}^N$ with smooth boundary. We first establish a Nash-Moser-Alikakos type a priori estimate for the difference of solutions near the initial time; Then we prove that the solution process $U(t,\tau)$ is continuous from $L^{2}(\Omega)$ to $\mathscr{D}_{0}^{1}(\Omega, \sigma)$ w.r.t. initial data; And finally show that the known $(L^{2}(\Omega), L^{2}(\Omega))$ pullback $\mathscr{D}_{\lambda}$-attractor indeed can attract in $\mathscr{D}_{0}^{1}(\Omega, \sigma)$-norm. Any differentiability on the forcing term is not required.
2016, 36(12): 7081-7115
doi: 10.3934/dcds.2016109
+[Abstract](2195)
+[PDF](543.6KB)
Abstract:
In this paper, we study the nonlinear Schrödinger equation with non-symmetric electromagnetic fields $$ \Big(\frac{\nabla}{i}-A_{\epsilon}(x)\Big)^{2}u+V_{\epsilon}(x)u=f(u),~~~~~~u\in H^{1}(\mathbb{R}^{N},\mathbb{C}), $$ where $A_{\epsilon}(x)=(A_{\epsilon,1}(x),A_{\epsilon,2}(x),\cdots,A_{\epsilon,N}(x))$ is a magnetic field satisfying that $A_{\epsilon,j}(x)(j=1,\ldots,N)$ is a real $C^{1}$ bounded function on $\mathbb{R}^{N}$ and $V_{\epsilon}(x)$ is an electric potential. Both of them satisfy some decay conditions but without any symmetric conditions and $f(u)$ is a superlinear nonlinearity satisfying some non-degeneracy condition. Applying two times finite reduction methods and localized energy method, we prove that there exists some $\epsilon_{0 }> 0$ such that for $0 < \epsilon < \epsilon_{0 }$, the above problem has infinitely many complex-valued solutions.
In this paper, we study the nonlinear Schrödinger equation with non-symmetric electromagnetic fields $$ \Big(\frac{\nabla}{i}-A_{\epsilon}(x)\Big)^{2}u+V_{\epsilon}(x)u=f(u),~~~~~~u\in H^{1}(\mathbb{R}^{N},\mathbb{C}), $$ where $A_{\epsilon}(x)=(A_{\epsilon,1}(x),A_{\epsilon,2}(x),\cdots,A_{\epsilon,N}(x))$ is a magnetic field satisfying that $A_{\epsilon,j}(x)(j=1,\ldots,N)$ is a real $C^{1}$ bounded function on $\mathbb{R}^{N}$ and $V_{\epsilon}(x)$ is an electric potential. Both of them satisfy some decay conditions but without any symmetric conditions and $f(u)$ is a superlinear nonlinearity satisfying some non-degeneracy condition. Applying two times finite reduction methods and localized energy method, we prove that there exists some $\epsilon_{0 }> 0$ such that for $0 < \epsilon < \epsilon_{0 }$, the above problem has infinitely many complex-valued solutions.
2016, 36(12): 7117-7136
doi: 10.3934/dcds.2016110
+[Abstract](2603)
+[PDF](529.5KB)
Abstract:
We consider the dynamic elasticity equations with a locally distributed damping of Kelvin-Voigt type in a bounded domain. The damping is localized in a suitable open subset, of the domain under consideration, which satisfies the piecewise multipliers condition of Liu. Using multiplier techniques combined with the frequency domain method, we show that: i) the energy of this system decays polynomially when the damping coefficient is only bounded measurable, ii) the energy of this system decays exponentially when the damping coefficient as well as its gradient are bounded measurable, and the damping coefficient further satisfies a structural condition. These results generalize and improve, at the same time, on an earlier result of Liu and Rao involving the wave equation with Kelvin-Voigt damping; those authors proved the exponential decay of the energy provided that the damping region is a neighborhood of the whole boundary, and further restrictions are imposed on the damping coefficient.
We consider the dynamic elasticity equations with a locally distributed damping of Kelvin-Voigt type in a bounded domain. The damping is localized in a suitable open subset, of the domain under consideration, which satisfies the piecewise multipliers condition of Liu. Using multiplier techniques combined with the frequency domain method, we show that: i) the energy of this system decays polynomially when the damping coefficient is only bounded measurable, ii) the energy of this system decays exponentially when the damping coefficient as well as its gradient are bounded measurable, and the damping coefficient further satisfies a structural condition. These results generalize and improve, at the same time, on an earlier result of Liu and Rao involving the wave equation with Kelvin-Voigt damping; those authors proved the exponential decay of the energy provided that the damping region is a neighborhood of the whole boundary, and further restrictions are imposed on the damping coefficient.
2016, 36(12): 7137-7168
doi: 10.3934/dcds.2016111
+[Abstract](3015)
+[PDF](585.7KB)
Abstract:
In this paper, we concern with the following semilinear Kirchhoff type equation \begin{equation*} \begin{cases} -\left(\varepsilon^{2}a+b\varepsilon\int_{\mathbb{R}^{3}}|\nabla u|^{2}\right)\Delta u+V(x)u=K(x)f(u)+Q(x)|u|^{p-2}u, &x\in\mathbb{R}^{3},\\ u\in H^{1}(\mathbb{R}^{3}), u>0 &x\in\mathbb{R}^{3}, \end{cases} \end{equation*} where $\varepsilon>0$ is a small parameter, $a, b$ are positive constants, $V, K$ and $Q$ are positive bounded functions and $p\in(4,6]$, $f$ is a continuous superlinear and subcritical nonlinearity. On the one hand, for subcritical case, i.e., $p\in(4,6)$, we prove that there are three families of semiclassical positive solutions for $\varepsilon>0$ small, one is concentrating on the set of minima of $V$, the rest of two families of solutions are concentrating on the sets of maxima of $K$ and $Q$ respectively. On the other hand, we also prove the multiplicity and concentration of positive solutions for critical case($p=6$). The novelty is that we prove some new concentration phenomena for the positive solutions.
In this paper, we concern with the following semilinear Kirchhoff type equation \begin{equation*} \begin{cases} -\left(\varepsilon^{2}a+b\varepsilon\int_{\mathbb{R}^{3}}|\nabla u|^{2}\right)\Delta u+V(x)u=K(x)f(u)+Q(x)|u|^{p-2}u, &x\in\mathbb{R}^{3},\\ u\in H^{1}(\mathbb{R}^{3}), u>0 &x\in\mathbb{R}^{3}, \end{cases} \end{equation*} where $\varepsilon>0$ is a small parameter, $a, b$ are positive constants, $V, K$ and $Q$ are positive bounded functions and $p\in(4,6]$, $f$ is a continuous superlinear and subcritical nonlinearity. On the one hand, for subcritical case, i.e., $p\in(4,6)$, we prove that there are three families of semiclassical positive solutions for $\varepsilon>0$ small, one is concentrating on the set of minima of $V$, the rest of two families of solutions are concentrating on the sets of maxima of $K$ and $Q$ respectively. On the other hand, we also prove the multiplicity and concentration of positive solutions for critical case($p=6$). The novelty is that we prove some new concentration phenomena for the positive solutions.
2016, 36(12): 7169-7189
doi: 10.3934/dcds.2016112
+[Abstract](2566)
+[PDF](443.1KB)
Abstract:
In this paper, we study the elliptic equation with a multi-singular inverse square potential $$-\Delta u=\mu\sum_{i=1}^{k}\frac{u}{|x-a_i|^2}-u^p,\ \ x\in \mathbb{R}^N\backslash\{a_i:i\in K\},$$ where $N\geq 3$, $p>1$ and $\mu>(N-2)^2/4k$. In our discussions, the domain is the entire space, and the equation contains multiple singular points. We not only demonstrate the behavior of positive solutions near each singular point $a_i$, but also obtain the behavior of positive solutions as $|x|\rightarrow \infty$. Under suitable conditions, we show that the equation has a unique positive solution $w$, which satisfies $$\lim\limits_{|x|\rightarrow\infty}\frac{w(x)}{|x|^{-\frac{2}{p-1}}}=\left[k\mu+\frac{2}{p-1}\left(\frac{2}{p-1}+2-N\right)\right]^{1/(p-1)}$$ and $$\lim\limits_{|x-a_i|\rightarrow 0}\frac{w(x)}{|x-a_i|^{-\frac{2}{p-1}}}=\left[\mu+\frac{2}{p-1}\left(\frac{2}{p-1}+2-N\right)\right]^{1/(p-1)}.$$
In this paper, we study the elliptic equation with a multi-singular inverse square potential $$-\Delta u=\mu\sum_{i=1}^{k}\frac{u}{|x-a_i|^2}-u^p,\ \ x\in \mathbb{R}^N\backslash\{a_i:i\in K\},$$ where $N\geq 3$, $p>1$ and $\mu>(N-2)^2/4k$. In our discussions, the domain is the entire space, and the equation contains multiple singular points. We not only demonstrate the behavior of positive solutions near each singular point $a_i$, but also obtain the behavior of positive solutions as $|x|\rightarrow \infty$. Under suitable conditions, we show that the equation has a unique positive solution $w$, which satisfies $$\lim\limits_{|x|\rightarrow\infty}\frac{w(x)}{|x|^{-\frac{2}{p-1}}}=\left[k\mu+\frac{2}{p-1}\left(\frac{2}{p-1}+2-N\right)\right]^{1/(p-1)}$$ and $$\lim\limits_{|x-a_i|\rightarrow 0}\frac{w(x)}{|x-a_i|^{-\frac{2}{p-1}}}=\left[\mu+\frac{2}{p-1}\left(\frac{2}{p-1}+2-N\right)\right]^{1/(p-1)}.$$
2016, 36(12): 7191-7206
doi: 10.3934/dcds.2016113
+[Abstract](2851)
+[PDF](439.7KB)
Abstract:
By proving an amenable version of Katok's entropy formula and handling the quasi tiling techniques, we establish large deviations bounds for countable discrete amenable group actions. This generalizes the classical results of Lai-Sang Young [21].
By proving an amenable version of Katok's entropy formula and handling the quasi tiling techniques, we establish large deviations bounds for countable discrete amenable group actions. This generalizes the classical results of Lai-Sang Young [21].
2016, 36(12): 7207-7234
doi: 10.3934/dcds.2016114
+[Abstract](3151)
+[PDF](571.3KB)
Abstract:
In this paper we study a powered integral inequality involving a finite sum, which can be used to solve the inequalities with singular kernels. We present that the solution of the inequality is decided by a finite recursion, whose result is proved to be a continuous, bounded or asymptotic function. Meanwhile, in order to overcome an obstacle from powers of integrals, we modify the method of monotonization into the powered monotonization. Furthermore, relying on the result and our technique of concavification, we discuss a generalized stochastic integral inequality, and give an estimate of the mean square. In the end, as applications, we study uniform boundedness and continuous dependence of solutions for a class of stochastic differential equation in mean square.
In this paper we study a powered integral inequality involving a finite sum, which can be used to solve the inequalities with singular kernels. We present that the solution of the inequality is decided by a finite recursion, whose result is proved to be a continuous, bounded or asymptotic function. Meanwhile, in order to overcome an obstacle from powers of integrals, we modify the method of monotonization into the powered monotonization. Furthermore, relying on the result and our technique of concavification, we discuss a generalized stochastic integral inequality, and give an estimate of the mean square. In the end, as applications, we study uniform boundedness and continuous dependence of solutions for a class of stochastic differential equation in mean square.
2016, 36(12): 7235-7256
doi: 10.3934/dcds.2016115
+[Abstract](2784)
+[PDF](502.4KB)
Abstract:
Considered herein is the blow-up mechanism to the periodic modified Camassa-Holm equation with varying linear dispersion. We first consider the case when linear dispersion is absent and derive a finite-time blow-up result. The key feature is the ratio between solution and its gradient. Using the continuity of the solutions and the right transformation, we then obtain this blow-up criterion to the case with negative linear dispersion and determine that the finite time blow-up can still occur if the initial momentum density is bounded below by the magnitude of the linear dispersion and the initial datum has a local mild-oscillation region. Finally, we demonstrate that when the linear dispersion is non-negative, formation of singularity can be induced by an initial datum with a sufficiently steep profile.
Considered herein is the blow-up mechanism to the periodic modified Camassa-Holm equation with varying linear dispersion. We first consider the case when linear dispersion is absent and derive a finite-time blow-up result. The key feature is the ratio between solution and its gradient. Using the continuity of the solutions and the right transformation, we then obtain this blow-up criterion to the case with negative linear dispersion and determine that the finite time blow-up can still occur if the initial momentum density is bounded below by the magnitude of the linear dispersion and the initial datum has a local mild-oscillation region. Finally, we demonstrate that when the linear dispersion is non-negative, formation of singularity can be induced by an initial datum with a sufficiently steep profile.
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