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Discrete & Continuous Dynamical Systems - A

November 2017 , Volume 37 , Issue 11

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Eulerian dynamics with a commutator forcing Ⅱ: Flocking
Roman Shvydkoy and Eitan Tadmor
2017, 37(11): 5503-5520 doi: 10.3934/dcds.2017239 +[Abstract](602) +[HTML](14) +[PDF](483.4KB)
Abstract:

We continue our study of one-dimensional class of Euler equations, introduced in [11], driven by a forcing with a commutator structure of the form \begin{document} $[{\mathcal L}_φ, u](ρ)$ \end{document}, where \begin{document} $u$ \end{document} is the velocity field and \begin{document} ${\mathcal L}_φ$ \end{document} belongs to a rather general class of convolution operators depending on interaction kernels \begin{document} $φ$ \end{document}.

In this paper we quantify the large-time behavior of such systems in terms of fast flocking, for two prototypical sub-classes of kernels: bounded positive \begin{document} $φ$ \end{document}'s, and singular \begin{document} $φ(r) = r^{-(1+α)}$ \end{document} of order \begin{document} $α∈ [1, 2)$ \end{document} associated with the action of the fractional Laplacian \begin{document} ${\mathcal L}_φ=-(-\partial_{xx})^{α/2}$ \end{document}. Specifically, we prove fast velocity alignment as the velocity \begin{document} $u(·, t)$ \end{document} approaches a constant state, \begin{document} $u \to \bar{u}$ \end{document}, with exponentially decaying slope and curvature bounds \begin{document} $|{u_x}( \cdot ,t){|_\infty } + |{u_{xx}}( \cdot ,t){|_\infty }\lesssim{e^{ - \delta t}}$ \end{document}. The alignment is accompanied by exponentially fast flocking of the density towards a fixed traveling state \begin{document} $ρ(·, t) -{ρ_{∞}}(x -\bar{u} t) \to 0$ \end{document}.

The 3D liquid crystal system with Cannone type initial data and large vertical velocity
Renhui Wan
2017, 37(11): 5521-5539 doi: 10.3934/dcds.2017240 +[Abstract](383) +[HTML](14) +[PDF](414.3KB)
Abstract:

The hydrodynamic theory of the nematic liquid crystals was established by Ericksen [4] and Leslie [8]. In this paper, based on a new technique, we obtain global well-posedness to a simplified model introduced by Lin [9] in the critical Besov space with Cannone type initial data and large vertical velocity, which improves the main result in [15]. In addition, the small condition on \begin{document} $u_0$ \end{document} is independent of another small condition on \begin{document} $d_0-\bar{d}_0$ \end{document}, which is quite different from the previous works [15,16].

A discrete Bakry-Emery method and its application to the porous-medium equation
Ansgar Jüngel and Stefan Schuchnigg
2017, 37(11): 5541-5560 doi: 10.3934/dcds.2017241 +[Abstract](371) +[HTML](20) +[PDF](765.0KB)
Abstract:

The exponential decay of the relative entropy associated to a fully discrete porous-medium equation in one space dimension is shown by means of a discrete Bakry-Emery approach. The first ingredient of the proof is an abstract discrete Bakry-Emery method, which states conditions on a sequence under which the exponential decay of the discrete entropy follows. The second ingredient is a new nonlinear summation-by-parts formula which is inspired by systematic integration by parts developed by Matthes and the first author. Numerical simulations illustrate the exponential decay of the entropy for various time and space step sizes.

Infinitely many positive solutions of fractional nonlinear Schrödinger equations with non-symmetric potentials
Weiwei Ao, Juncheng Wei and Wen Yang
2017, 37(11): 5561-5601 doi: 10.3934/dcds.2017242 +[Abstract](683) +[HTML](15) +[PDF](614.8KB)
Abstract:

We consider the fractional nonlinear Schrödinger equation

where \begin{document}$V(x)$\end{document} is a uniformly positive potential and $p>1.$ Assuming that

and \begin{document}$p,m,σ,s$\end{document} satisfy certain conditions, we prove the existence of infinitely many positive solutions for \begin{document}$N=2$\end{document}. For \begin{document}$s=1$\end{document}, this corresponds to the multiplicity result given by Del Pino, Wei, and Yao [24] for the classical nonlinear Schrödinger equation.

The index bundle and multiparameter bifurcation for discrete dynamical systems
Robert Skiba and Nils Waterstraat
2017, 37(11): 5603-5629 doi: 10.3934/dcds.2017243 +[Abstract](465) +[HTML](17) +[PDF](531.3KB)
Abstract:

We develop a K-theoretic approach to multiparameter bifurcation theory of homoclinic solutions of discrete non-autonomous dynamical systems from a branch of stationary solutions. As a byproduct we obtain a family index theorem for asymptotically hyperbolic linear dynamical systems which is of independent interest. In the special case of a single parameter, our bifurcation theorem weakens the assumptions in previous work by Pejsachowicz and the first author.

The initial-boundary value problems for a class of sixth order nonlinear wave equation
Runzhang Xu, Mingyou Zhang, Shaohua Chen, Yanbing Yang and Jihong Shen
2017, 37(11): 5631-5649 doi: 10.3934/dcds.2017244 +[Abstract](827) +[HTML](27) +[PDF](467.0KB)
Abstract:

This paper considers the initial boundary value problem of solutions for a class of sixth order 1-D nonlinear wave equations. We discuss the probabilities of the existence and nonexistence of global solutions and give some sufficient conditions for the global and non-global existence of solutions at three different initial energy levels, i.e., sub-critical level, critical level and sup-critical level.

Sign-changing tower of bubbles for a sinh-Poisson equation with asymmetric exponents
Angela Pistoia and Tonia Ricciardi
2017, 37(11): 5651-5692 doi: 10.3934/dcds.2017245 +[Abstract](476) +[HTML](21) +[PDF](651.1KB)
Abstract:

Motivated by the statistical mechanics description of stationary 2D-turbulence, for a sinh-Poisson type equation with asymmetric nonlinearity, we construct a concentrating solution sequence in the form of a tower of singular Liouville bubbles, each of which has a different degeneracy exponent. The asymmetry parameter \begin{document} $γ∈(0, 1]$ \end{document} corresponds to the ratio between the intensity of the negatively rotating vortices and the intensity of the positively rotating vortices. Our solutions correspond to a superposition of highly concentrated vortex configurations of alternating orientation; they extend in a nontrivial way some known results for \begin{document} $\gamma=1$ \end{document}. Thus, by analyzing the case \begin{document} $\gamma≠1$ \end{document} we emphasize specific properties of the physically relevant parameter \begin{document} $\gamma$ \end{document} in the vortex concentration phenomena.

2-manifolds and inverse limits of set-valued functions on intervals
Sina Greenwood and Rolf Suabedissen
2017, 37(11): 5693-5706 doi: 10.3934/dcds.2017246 +[Abstract](430) +[HTML](32) +[PDF](420.8KB)
Abstract:

Suppose for each \begin{document} $n\in\mathbb{N}$ \end{document}, \begin{document} $f_n \colon [0,1] \to 2^{[0,1]}$ \end{document} is a function whose graph \begin{document} $\Gamma(f_n) = \left\lbrace (x,y) \in [0,1]^2 \colon y \in f_n(x)\right\rbrace$ \end{document} is closed in \begin{document} $[0,1]^2$ \end{document} (here \begin{document} $2^{[0,1]}$ \end{document} is the space of non-empty closed subsets of \begin{document} $[0,1]$ \end{document}). We show that the generalized inverse limit \begin{document} $\varprojlim (f_n) = \left\lbrace (x_n) \in [0,1]^\mathbb{N} \colon \forall n \in \mathbb{N},\ x_n \in f_n(x_{n+1})\right\rbrace$ \end{document} of such a sequence of functions cannot be an arbitrary continuum, answering a long-standing open problem in the study of generalized inverse limits. In particular we show that if such an inverse limit is a 2-manifold then it is a torus and hence it is impossible to obtain a sphere.

On the Dirichlet problem for fully nonlinear elliptic equations on annuli of metric cones
Chunhui Qiu and Rirong Yuan
2017, 37(11): 5707-5730 doi: 10.3934/dcds.2017247 +[Abstract](687) +[HTML](16) +[PDF](551.3KB)
Abstract:

In this paper, we study a class of fully nonlinear elliptic equations on annuli of metric cones constructed from closed Sasakian manifolds and derive the a priori estimates assuming the existence of subsolutions. Moreover, such a priori estimates can be applied to certain degenerate equations. A condition for the solvability of Dirichlet problem for non-degenerate fully nonlinear elliptic equations is discovered. Furthermore, we also discuss degenerate equations.

A Liouville theorem for indefinite fractional diffusion equations and its application to existence of solutions
Begoña Barrios, Leandro Del Pezzo, Jorge García-Melián and Alexander Quaas
2017, 37(11): 5731-5746 doi: 10.3934/dcds.2017248 +[Abstract](541) +[HTML](16) +[PDF](450.4KB)
Abstract:

In this work we obtain a Liouville theorem for positive, bounded solutions of the equation

where \begin{document} $(-\Delta)^s$ \end{document} stands for the fractional Laplacian with \begin{document} $s∈ (0, 1)$ \end{document}, and the functions \begin{document} $h$ \end{document} and \begin{document} $f$ \end{document} are nondecreasing. The main feature is that the function \begin{document} $h$ \end{document} changes sign in \begin{document} $\mathbb R$ \end{document}, therefore the problem is sometimes termed as indefinite. As an application we obtain a priori bounds for positive solutions of some boundary value problems, which give existence of such solutions by means of bifurcation methods.

Rectifiability of a class of invariant measures with one non-vanishing Lyapunov exponent
Gabriel Fuhrmann and Jing Wang
2017, 37(11): 5747-5761 doi: 10.3934/dcds.2017249 +[Abstract](502) +[HTML](20) +[PDF](464.4KB)
Abstract:

We study order-preserving \begin{document} $\mathcal{C}^1$ \end{document}-circle diffeomorphisms driven by irrational rotations with a Diophantine rotation number. We show that there is a non-empty open set of one-parameter families of such diffeomorphisms where the ergodic measures of nearly all family members are one-rectifiable, that is, absolutely continuous with respect to the restriction of the one-dimensional Hausdorff measure to a countable union of Lipschitz graphs.

On the uniqueness of an ergodic measure of full dimension for non-conformal repellers
Nuno Luzia
2017, 37(11): 5763-5780 doi: 10.3934/dcds.2017250 +[Abstract](499) +[HTML](19) +[PDF](470.0KB)
Abstract:

We give a subclass \begin{document} $\mathcal{L}$ \end{document} of Non-linear Lalley-Gatzouras carpets and an open set \begin{document} $\mathcal{U}$ \end{document} in \begin{document} $\mathcal{L}$ \end{document} such that any carpet in \begin{document} $\mathcal{U}$ \end{document} has a unique ergodic measure of full dimension. In particular, any Lalley-Gatzouras carpet which is close to a non-trivial general Sierpinski carpet has a unique ergodic measure of full dimension.

Non-degenerate locally connected models for plane continua and Julia sets
Alexander Blokh, Lex Oversteegen and Vladlen Timorin
2017, 37(11): 5781-5795 doi: 10.3934/dcds.2017251 +[Abstract](416) +[HTML](18) +[PDF](383.0KB)
Abstract:

Every plane continuum admits a finest locally connected model. The latter is a locally connected continuum onto which the original continuum projects in a monotone fashion. It may so happen that the finest locally connected model is a singleton. For example, this happens if the original continuum is indecomposable. In this paper, we provide sufficient conditions for the existence of a non-degenerate model depending on the existence of subcontinua with certain properties. Applications to complex polynomial dynamics are discussed.

Mixed elliptic problems involving the $p-$Laplacian with nonhomogeneous boundary conditions
Gabriele Bonanno and Giuseppina D'Aguì
2017, 37(11): 5797-5817 doi: 10.3934/dcds.2017252 +[Abstract](588) +[HTML](21) +[PDF](488.4KB)
Abstract:

In this paper, mixed elliptic problems involving the p-Laplacian and with nonhomogeneous boundary conditions are investigated. At first, the existence of one non-trivial solution, under a suitable behaviour on the nonlinearity and without requiring neither conditions at zero nor conditions at infinity, is established. Then, by adding a condition at infinity on the nonlinearity, also a second non-trivial solution is guaranteed. Some special cases are pointed out as, in particular, the existence of one non-trivial solution when the datum is \begin{document} $(p-1)-$ \end{document}sublinear at zero and the existence of two non-trivial solutions when the nonlinear term is again \begin{document} $(p-1)-$ \end{document}sublinear at zero and, in addition, more than \begin{document} $(p-1)-$ \end{document}superlinear at infinity. As a consequence, the existence of two non-trivial solutions for concave-convex nonlinearities is emphasized. Finally, the case of a simple \begin{document} $(p-1)-$ \end{document}superlinearity at infinity is considered and it is also observed that the same results hold when the nonlinear behaviour, described before for the datum, is instead assumed by the nonhomogeneous Neumann boundary conditions. Concrete examples of applications are also given. The approach is based on variational methods and critical point theory. Precisely, a non-zero local minimum theorem and a two non-zero critical points theorem are applied.

Energy transfer model for the derivative nonlinear Schrödinger equations on the torus
Hideo Takaoka
2017, 37(11): 5819-5841 doi: 10.3934/dcds.2017253 +[Abstract](455) +[HTML](34) +[PDF](523.3KB)
Abstract:

We consider the nonlinear derivative Schrödinger equation with a quintic nonlinearity, on the one dimensional torus. We exhibit that the nonlinear dynamic properties of the particular solution consisting of four frequency modes initially excited, whose frequencies include the resonant clusters and phase matched resonant interactions of nonlinearities. The proof is based on the analysis of resonant dynamics via a finite dimensional ordinary differential system.

Estimating the fractal dimension of sets determined by nonergodic parameters
Joseph Squillace
2017, 37(11): 5843-5859 doi: 10.3934/dcds.2017254 +[Abstract](547) +[HTML](19) +[PDF](439.2KB)
Abstract:

Given fixed and irrational \begin{document} $0<α, θ<1$ \end{document}, consider the billiard table \begin{document} $B_{α}$ \end{document} formed by a \begin{document} $\frac{1}{2}×1$ \end{document} rectangle with a horizontal barrier of length \begin{document} $α$ \end{document} emanating from the midpoint of a vertical side and a billiard flow with trajectory angle \begin{document} $θ$ \end{document}. In 1969, Veech introduced two subsets \begin{document} $K_{0}(θ)$ \end{document} and \begin{document} $K_{1}(θ)$ \end{document} of \begin{document} $\mathbb{R}/\mathbb{Z}$ \end{document} that are defined in terms of the continued fraction representation of \begin{document} $θ∈\mathbb{R}/\mathbb{Z}$ \end{document}, and Veech showed that these sets have Hausdorff dimension \begin{document} $0$ \end{document} when \begin{document} $θ$ \end{document} is rational. Moreover, the set \begin{document} $K_{1}(θ)$ \end{document} describes the set of all \begin{document} $α$ \end{document} such that the billiard flow on \begin{document} $B_{α}$ \end{document} in direction \begin{document} $θ$ \end{document} is nonergodic. We show that the Hausdorff dimension of the sets \begin{document} $K_{0}(θ)$ \end{document} and \begin{document} $K_{1}(θ)$ \end{document} can attain any value in \begin{document} $[0, 1]$ \end{document} by considering the continued fraction expansion of \begin{document} $θ$ \end{document}. This result resolves an analogue of work completed by Cheung, Hubert, and Pascal in which they consider, for fixed \begin{document} $α$ \end{document}, the set of \begin{document} $θ$ \end{document} such that the flow on \begin{document} $B_{α}$ \end{document} in direction \begin{document} $θ$ \end{document} is nonergodic.

Hitting times distribution and extreme value laws for semi-flows
Maria José Pacifico and Fan Yang
2017, 37(11): 5861-5881 doi: 10.3934/dcds.2017255 +[Abstract](372) +[HTML](20) +[PDF](450.7KB)
Abstract:

For flows whose return map on a cross section has sufficient mixing property, we show that the hitting time distribution of the flow to balls is exponential in limit. We also establish a link between the extreme value distribution of the flow and its hitting time distribution, generalizing a previous work by Freitas et al in the discrete time case. Finally we show that for maps that can be modeled by Young's tower with polynomial tail, the extreme value laws hold.

Visco-Energetic solutions to one-dimensional rate-independent problems
Luca Minotti
2017, 37(11): 5883-5912 doi: 10.3934/dcds.2017256 +[Abstract](403) +[HTML](17) +[PDF](669.6KB)
Abstract:

Visco-Energetic solutions of rate-independent systems (recently introduced in [17]) are obtained by solving a modified time Incremental Minimization Scheme, where at each step the dissipation is reinforced by a viscous correction \begin{document} $δ$ \end{document}, typically a quadratic perturbation of the dissipation distance. Like Energetic and Balanced Viscosity solutions, they provide a variational characterization of rate-independent evolutions, with an accurate description of their jump behaviour.

In the present paper we study Visco-Energetic solutions in the scalar-valued case and we obtain a full characterization for a broad class of energy functionals. In particular, we prove that they exhibit a sort of intermediate behaviour between Energetic and Balanced Viscosity solutions, which can be finely tuned according to the choice of the viscous correction \begin{document} $δ$ \end{document}.

Well-posedness for vanishing viscosity solutions of scalar conservation laws on a network
Boris P. Andreianov, Giuseppe Maria Coclite and Carlotta Donadello
2017, 37(11): 5913-5942 doi: 10.3934/dcds.2017257 +[Abstract](527) +[HTML](23) +[PDF](584.3KB)
Abstract:

We provide a complete study of the model investigated in [Coclite, Garavello, SIAM J. Math. Anal., 2010]. We prove well-posedness of solutions obtained as vanishing viscosity limits for the Cauchy problem for scalar conservation laws \begin{document} $ ρ_{h, t} + f_h(ρ_h)_x = 0$ \end{document}, for \begin{document} $h∈ \{1, ..., m+n\}$ \end{document}, on a junction where \begin{document} $m$ \end{document} incoming and \begin{document} $n$ \end{document} outgoing edges meet. Our analysis and the definition of the admissible solution rely upon the complete description of the set of edge-wise constant solutions and its properties, which is of some interest on its own. The Riemann solver at the junction is characterized. In order to prove uniqueness, we introduce a family of Kruzhkov-type adapted entropies at the junction. Existence is justified both by the vanishing viscosity method and via the proof of convergence of a monotone well-balanced finite volume discretization. Beyond the classical vanishing viscosity framework, the numerical procedure and the uniqueness argument can be applied to general junction solvers enjoying the crucial order-preservation property.

Asymptotic large time behavior of singular solutions of the fast diffusion equation
Kin Ming Hui and Soojung Kim
2017, 37(11): 5943-5977 doi: 10.3934/dcds.2017258 +[Abstract](563) +[HTML](23) +[PDF](626.1KB)
Abstract:

We study the asymptotic large time behavior of singular solutions of the fast diffusion equation \begin{document} $u_t=Δ u^m$ \end{document} in \begin{document} $({\mathbb R}^n\setminus\{0\})×(0, ∞)$ \end{document} in the subcritical case \begin{document} $0<m<\frac{n-2}{n}$ \end{document}, \begin{document} $n≥3$ \end{document}. Firstly, we prove the existence of the singular solution \begin{document} $u$ \end{document} of the above equation that is trapped in between self-similar solutions of the form of \begin{document} $t^{-α} f_i(t^{-β}x)$ \end{document}, \begin{document} $i=1, 2$ \end{document}, with the initial value \begin{document} $u_0$ \end{document} satisfying \begin{document} $A_1|x|^{-γ}≤ u_0≤ A_2|x|^{-γ}$ \end{document} for some constants \begin{document} $A_2>A_1>0$ \end{document} and \begin{document} $\frac{2}{1-m}<γ<\frac{n-2}{m}$ \end{document}, where \begin{document} $β:=\frac{1}{2-γ(1-m)}$, $α:=\frac{2\beta-1}{1-m}, $ \end{document} and the self-similar profile \begin{document} $f_i$ \end{document} satisfies the elliptic equation

with $\lim_{|x|\to0}|x|^{\frac{ α}{ β}}f_i(x)=A_i$ and $\lim_{|x|\to∞}|x|^{\frac{n-2}{m}}{f_i}(x)= D_{A_i} $ for some constants $D_{A_i}>0$. When $\frac{2}{1-m} < γ < n$, under an integrability condition on the initial value $u_0$ of the singular solution $u$, we prove that the rescaled function

converges to some self-similar profile $f$ as $τ\to∞$.

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