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1078-0947
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Discrete & Continuous Dynamical Systems - A
July 2009 , Volume 25 , Issue 2
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2009, 25(2): 399-429
doi: 10.3934/dcds.2009.25.399
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Abstract:
The classical phase-field model has been introduced as a model for non-isothermal phase separation processes in materials. In order to overcome some of the model's shortcomings, a variety of extensions have recently been proposed which include nonlocal interactions as well as stochastic noise terms. In this paper, we study these extensions from a functional-analytic and numerical point of view. More precisely, we present a functional-analytic framework for establishing existence, uniqueness, and qualitative dynamical results, and then propose a spectral method for solving stochastic nonlocal phase-field models. In particular, we establish convergence properties of our method and study the effects of noise regularity and of the nonlocal interaction term on these convergence properties. Finally, numerical studies relating to the associated phase separation process are presented.
The classical phase-field model has been introduced as a model for non-isothermal phase separation processes in materials. In order to overcome some of the model's shortcomings, a variety of extensions have recently been proposed which include nonlocal interactions as well as stochastic noise terms. In this paper, we study these extensions from a functional-analytic and numerical point of view. More precisely, we present a functional-analytic framework for establishing existence, uniqueness, and qualitative dynamical results, and then propose a spectral method for solving stochastic nonlocal phase-field models. In particular, we establish convergence properties of our method and study the effects of noise regularity and of the nonlocal interaction term on these convergence properties. Finally, numerical studies relating to the associated phase separation process are presented.
2009, 25(2): 431-456
doi: 10.3934/dcds.2009.25.431
+[Abstract](1594)
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Abstract:
In this paper we first conduct a study of the spectrum of the negative $p$-Laplacian with Neumann boundary conditions. More precisely we investigate the first nonzero eigenvalue. We produce alternative variational characterizations, we examine its dependence on $p\in( 1,\infty) $ and on the weight function $m\in L^{\infty}(Z) _{+}$ and we prove that the isolation of the principal eigenvalue $\lambda_{0}=0,$ is uniform for all $p$ in a bounded closed interval. All these results are then used to prove an index formula (jumping theorem) for the $d_{( S)_{+}}-$degree map at the first nonzero eigenvalue. Finally the index formula is used to prove a multiplicity result for problems with a multivalued crossing nonlinearity.
In this paper we first conduct a study of the spectrum of the negative $p$-Laplacian with Neumann boundary conditions. More precisely we investigate the first nonzero eigenvalue. We produce alternative variational characterizations, we examine its dependence on $p\in( 1,\infty) $ and on the weight function $m\in L^{\infty}(Z) _{+}$ and we prove that the isolation of the principal eigenvalue $\lambda_{0}=0,$ is uniform for all $p$ in a bounded closed interval. All these results are then used to prove an index formula (jumping theorem) for the $d_{( S)_{+}}-$degree map at the first nonzero eigenvalue. Finally the index formula is used to prove a multiplicity result for problems with a multivalued crossing nonlinearity.
2009, 25(2): 457-466
doi: 10.3934/dcds.2009.25.457
+[Abstract](1700)
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Abstract:
At the first time, Razumikhin technique is applied for differential equations with piecewise constant argument of generalized type [1, 2]. Sufficient conditions are established for stability, uniform stability and uniform asymptotic stability of the trivial solution of such equations. We also provide appropriate examples to illustrate our results.
At the first time, Razumikhin technique is applied for differential equations with piecewise constant argument of generalized type [1, 2]. Sufficient conditions are established for stability, uniform stability and uniform asymptotic stability of the trivial solution of such equations. We also provide appropriate examples to illustrate our results.
2009, 25(2): 467-479
doi: 10.3934/dcds.2009.25.467
+[Abstract](1666)
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Abstract:
Let l be a compact convex subset of a Hausdorff topological vector space $(\mathcal{E},\tau)$ and $\sigma$ another Hausdorff vector topology in $\mathcal{E}$. We establish an approximate fixed point result for sequentially continuous maps f: (l,$\sigma$)$\to$ (l,$\tau$). As application, we obtain the weak-approximate fixed point property for demicontinuous self-mapping weakly compact convex sets in general Banach spaces and use this to prove new results in asymptotic fixed point theory. These results are also applied to study the existence of limiting-weak solutions for differential equations in reflexive Banach spaces.
Let l be a compact convex subset of a Hausdorff topological vector space $(\mathcal{E},\tau)$ and $\sigma$ another Hausdorff vector topology in $\mathcal{E}$. We establish an approximate fixed point result for sequentially continuous maps f: (l,$\sigma$)$\to$ (l,$\tau$). As application, we obtain the weak-approximate fixed point property for demicontinuous self-mapping weakly compact convex sets in general Banach spaces and use this to prove new results in asymptotic fixed point theory. These results are also applied to study the existence of limiting-weak solutions for differential equations in reflexive Banach spaces.
2009, 25(2): 481-493
doi: 10.3934/dcds.2009.25.481
+[Abstract](1715)
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Abstract:
We determine topological and algebraic conditions for a germ of holomorphic foliation $\mathcal{F}_X$ induced by a generic vector field $X$ on $(\mathbb{C}^{3},0)$ to have a holomorphic first integral, i.e., a germ of holomorphic map $F$ : $(\mathbb{C}^{3},0)\rightarrow(\mathbb{C}^{2},0)$ such that the leaves of $\mathcal{F}_X$ are contained in the level curves of $F$.
We determine topological and algebraic conditions for a germ of holomorphic foliation $\mathcal{F}_X$ induced by a generic vector field $X$ on $(\mathbb{C}^{3},0)$ to have a holomorphic first integral, i.e., a germ of holomorphic map $F$ : $(\mathbb{C}^{3},0)\rightarrow(\mathbb{C}^{2},0)$ such that the leaves of $\mathcal{F}_X$ are contained in the level curves of $F$.
2009, 25(2): 495-509
doi: 10.3934/dcds.2009.25.495
+[Abstract](1843)
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Abstract:
We present a version of the Poincaré-Bendixson Theorem on the Klein bottle $K^2$ for continuous vector fields. As a consequence, we obtain the fact that $K^2$ does not admit continuous vector fields having a $\omega$-recurrent injective trajectory.
We present a version of the Poincaré-Bendixson Theorem on the Klein bottle $K^2$ for continuous vector fields. As a consequence, we obtain the fact that $K^2$ does not admit continuous vector fields having a $\omega$-recurrent injective trajectory.
2009, 25(2): 511-535
doi: 10.3934/dcds.2009.25.511
+[Abstract](1594)
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Abstract:
We propose a program for finding the cyclicity of period annuli of quadratic systems with centers of genus one. As a first step, we classify all such systems and determine the essential one-parameter quadratic perturbations which produce the maximal number of limit cycles. We compute the associated Poincaré-Pontryagin-Melnikov functions whose zeros control the number of limit cycles. To illustrate our approach, we determine the cyclicity of the annuli of two particular reversible systems.
We propose a program for finding the cyclicity of period annuli of quadratic systems with centers of genus one. As a first step, we classify all such systems and determine the essential one-parameter quadratic perturbations which produce the maximal number of limit cycles. We compute the associated Poincaré-Pontryagin-Melnikov functions whose zeros control the number of limit cycles. To illustrate our approach, we determine the cyclicity of the annuli of two particular reversible systems.
2009, 25(2): 537-544
doi: 10.3934/dcds.2009.25.537
+[Abstract](1483)
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Abstract:
We study the geometrical and dynamical properties of a holomorphic vector field on a complex surface, assumed to be transverse to the boundary of a domain which is a non-smooth manifold with boundary and corners. We obtain hyperbolicity and prove a compact leaf result. For a pseudoconvex domain with boundary diffeomorphic to the boundary of a bidisc in $\mathbb C^2$ the foliation is pull-back of a liner hyperbolic foliation. If moreover the diffeomorphism is transversely holomorphic then we have linearization.
We study the geometrical and dynamical properties of a holomorphic vector field on a complex surface, assumed to be transverse to the boundary of a domain which is a non-smooth manifold with boundary and corners. We obtain hyperbolicity and prove a compact leaf result. For a pseudoconvex domain with boundary diffeomorphic to the boundary of a bidisc in $\mathbb C^2$ the foliation is pull-back of a liner hyperbolic foliation. If moreover the diffeomorphism is transversely holomorphic then we have linearization.
2009, 25(2): 545-565
doi: 10.3934/dcds.2009.25.545
+[Abstract](1766)
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Abstract:
The paper deals with Bean's critical state model for the description of the electromagnetic field in superconductors. A variational inequality for the quasi-stationary approximation on a part of the spatial domain is given. The main goals are the existence, uniqueness and asymptotic behavior as $t\rightarrow\infty$ of the solutions to that system.
The paper deals with Bean's critical state model for the description of the electromagnetic field in superconductors. A variational inequality for the quasi-stationary approximation on a part of the spatial domain is given. The main goals are the existence, uniqueness and asymptotic behavior as $t\rightarrow\infty$ of the solutions to that system.
2009, 25(2): 567-574
doi: 10.3934/dcds.2009.25.567
+[Abstract](1457)
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Abstract:
We investigate iterative systems consisting of Möbius transformations on the extended real line. We characterize systems with unique attractor and give some sufficient conditions for minimality.
We investigate iterative systems consisting of Möbius transformations on the extended real line. We characterize systems with unique attractor and give some sufficient conditions for minimality.
2009, 25(2): 575-583
doi: 10.3934/dcds.2009.25.575
+[Abstract](2829)
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Abstract:
In this paper we derive a criterion for the breakdown of classical solutions to the incompressible magnetohydrodynamic equations with zero viscosity and positive resistivity in $\mathbb{R}^3$. This result is analogous to the celebrated Beale-Kato-Majda's breakdown criterion for the inviscid Eluer equations of incompressible fluids. In $\mathbb{R}^2$ we establish global weak solutions to the magnetohydrodynamic equations with zero viscosity and positive resistivity for initial data in Sobolev space $H^1(\mathbb{R}^2)$.
In this paper we derive a criterion for the breakdown of classical solutions to the incompressible magnetohydrodynamic equations with zero viscosity and positive resistivity in $\mathbb{R}^3$. This result is analogous to the celebrated Beale-Kato-Majda's breakdown criterion for the inviscid Eluer equations of incompressible fluids. In $\mathbb{R}^2$ we establish global weak solutions to the magnetohydrodynamic equations with zero viscosity and positive resistivity for initial data in Sobolev space $H^1(\mathbb{R}^2)$.
2009, 25(2): 585-615
doi: 10.3934/dcds.2009.25.585
+[Abstract](2095)
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Abstract:
Rate-independent systems allow for solutions with jumps that need additional modeling. Here we suggest a formulation that arises as limit of viscous regularization of the solutions in the extended state space. Hence, our parametrized metric solutions of a rate-independent system are absolutely continuous mappings from a parameter interval into the extended state space. Jumps appear as generalized gradient flows during which the time is constant. The closely related notion of BV solutions is developed afterwards. Our approach is based on the abstract theory of gradient flows in metric spaces, and comparison with other notions of solutions is given.
Rate-independent systems allow for solutions with jumps that need additional modeling. Here we suggest a formulation that arises as limit of viscous regularization of the solutions in the extended state space. Hence, our parametrized metric solutions of a rate-independent system are absolutely continuous mappings from a parameter interval into the extended state space. Jumps appear as generalized gradient flows during which the time is constant. The closely related notion of BV solutions is developed afterwards. Our approach is based on the abstract theory of gradient flows in metric spaces, and comparison with other notions of solutions is given.
2009, 25(2): 617-626
doi: 10.3934/dcds.2009.25.617
+[Abstract](1958)
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Abstract:
We present a new and simpler proof that the nonlinear scattering operator $\S$ is analytic on energy space. We apply it in particular to a fourth-order nonlinear wave equation in Rn. In addition, we prove that $\S$ determines the scatterer uniquely and that for small powers there is no scattering.
We present a new and simpler proof that the nonlinear scattering operator $\S$ is analytic on energy space. We apply it in particular to a fourth-order nonlinear wave equation in Rn. In addition, we prove that $\S$ determines the scatterer uniquely and that for small powers there is no scattering.
2009, 25(2): 627-650
doi: 10.3934/dcds.2009.25.627
+[Abstract](1946)
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Abstract:
We derive the multifractal analysis of the conformal measure (or, equivalently, of the invariant measure) associated to a family of weights imposed upon a graph directed Markov system (GDMS) using balls as the filtration. Our analysis is done over a subset of the limit set, a subset which is often large. In particular, this subset is the entire limit set when the GDMS under scrutiny satisfies a boundary separation condition. Our analysis also applies to more general situations such as real and complex continued fractions.
We derive the multifractal analysis of the conformal measure (or, equivalently, of the invariant measure) associated to a family of weights imposed upon a graph directed Markov system (GDMS) using balls as the filtration. Our analysis is done over a subset of the limit set, a subset which is often large. In particular, this subset is the entire limit set when the GDMS under scrutiny satisfies a boundary separation condition. Our analysis also applies to more general situations such as real and complex continued fractions.
2009, 25(2): 651-667
doi: 10.3934/dcds.2009.25.651
+[Abstract](1679)
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Abstract:
The Riemann problem for the simplest scalar nonconvex CJ combustion model is considered. A set of entropy conditions are summarized, including pointwise entropy conditions and global entropy conditions. The later is according to the requirement of the structural stability. The unique self-similar entropy solution of the Riemann problem is constructed case by case. Transition from deflagration to detonation is shown, which dose not occur for the convex model.
The Riemann problem for the simplest scalar nonconvex CJ combustion model is considered. A set of entropy conditions are summarized, including pointwise entropy conditions and global entropy conditions. The later is according to the requirement of the structural stability. The unique self-similar entropy solution of the Riemann problem is constructed case by case. Transition from deflagration to detonation is shown, which dose not occur for the convex model.
2009, 25(2): 669-685
doi: 10.3934/dcds.2009.25.669
+[Abstract](1439)
+[PDF](218.2KB)
Abstract:
We prove an inequality for Hölder continuous differential forms on compact manifolds in which the integral of the form over the boundary of a sufficiently small, smoothly immersed disk is bounded by a certain multiplicative convex combination of the volume of the disk and the area of its boundary. This inequality has natural applications in dynamical systems, where Hölder continuity is ubiquitous. We give two such applications. In the first one, we prove a criterion for the existence of global cross sections to Anosov flows in terms of their expansion-contraction rates. The second application provides an analogous criterion for non-accessibility of partially hyperbolic diffeomorphisms.
We prove an inequality for Hölder continuous differential forms on compact manifolds in which the integral of the form over the boundary of a sufficiently small, smoothly immersed disk is bounded by a certain multiplicative convex combination of the volume of the disk and the area of its boundary. This inequality has natural applications in dynamical systems, where Hölder continuity is ubiquitous. We give two such applications. In the first one, we prove a criterion for the existence of global cross sections to Anosov flows in terms of their expansion-contraction rates. The second application provides an analogous criterion for non-accessibility of partially hyperbolic diffeomorphisms.
2009, 25(2): 687-699
doi: 10.3934/dcds.2009.25.687
+[Abstract](1421)
+[PDF](184.9KB)
Abstract:
A variational problem
A variational problem
Minimize $\quad \int_a^bL(x,u(x),u'(x),u''(x))dx,\quad u\in\Omega$
with a second-order Lagrangian is considered. In absence of any smoothness or convexity condition on $L$, we present an existence theorem by means of the integro-extremal technique. We also discuss the monotonicity property of the minimizers. An application to the extended Fisher-Kolmogorov model is included.
2009, 25(2): 701-718
doi: 10.3934/dcds.2009.25.701
+[Abstract](1588)
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Abstract:
In this paper we prove Gevrey smoothness of the persisting invariant tori for small perturbations of analytic linear reversible systems with Rüssmann's non-degeneracy condition by an improved KAM iteration method with parameters.
In this paper we prove Gevrey smoothness of the persisting invariant tori for small perturbations of analytic linear reversible systems with Rüssmann's non-degeneracy condition by an improved KAM iteration method with parameters.
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