Showing papers in "Dynamics of Partial Differential Equations in 2008"
TL;DR: In this paper, the authors studied the existence, uniqueness, blow up and regularity properties of solutions of the Burgers equation with fractional dissipation, and proved the existence of the finite time blow up for the power of Laplacian α < 1/2, and global existence as well as analyticity of solution for α ≥ 1 2.
Abstract: The paper is a comprehensive study of the existence, uniqueness, blow up and regularity properties of solutions of the Burgers equation with fractional dissipation. We prove existence of the finite time blow up for the power of Laplacian α < 1/2, and global existence as well as analyticity of solution for α ≥ 1/2. We also prove the existence of solutions with very rough initial data u0 ∈ Lp, 1 < p < ∞. Many of the results can be extended to a more general class of equations, including the surface quasi-geostrophic equation.
179 citations
TL;DR: In this article, a model of non-isothermal phase transition taking place in a conned container is considered and the order parameter is governed by a Cahn-Hilliard type equation which is coupled with a nonlinear heat equation for the temperature.
Abstract: We consider a model of non-isothermal phase transition taking place in a conned container. The order parameter is governed by a Cahn- Hilliard type equation which is coupled with a nonlinear heat equation for the temperature . The former is subject to a nonlinear dynamic boundary condition recently proposed by some physicists to account for interactions of the material with the walls. The latter is endowed with a boundary condi- tion which can be a standard one (Dirichlet, Neumann or Robin). We thus formulate a class of initial and boundary value problems whose local exis- tence and uniqueness is proven by means of a Faedo-Galerkin approximation scheme. The local solution becomes global owing to suitable a priori estimates. Then we analyze the asymptotic behavior of the solutions within the theory of innite-dimension al dynamical systems. In particular, we demonstrate the existence of a nite dimensional global attractor as well as of an exponential attractor.
41 citations
TL;DR: In this paper, the authors give a short discussion of the nature of the listed solutions and show that the majority of them have been obtained by means of different symmetry methods, such as reduction with respect to Lie and non-Lie symmetries, separation of variables, equivalence transformations, etc.
Abstract: (1) where f = f(x), g = g(x), h = h(x), A = A(u) and B = B(u) are arbitrary smooth functions of their variables, f(x)g(x)A(u) 6. Our aim is not to give a physical interpretation of the solution of diffusion equations (that is too huge and cannot be reached in the scope of a short paper), but to list the already known exact solutions of equations from the class under consideration. However, in some cases we give a short discussion of the nature of the listed solutions. The majority of the listed solutions have been obtained by means of different symmetry methods, such as reduction with respect to Lie and non-Lie symmetries, separation of variables, equivalence transformations, etc. Let us note that the constant coefficient diffusion equations ( = g = 1, B = 0) are well
37 citations
TL;DR: In this article, the authors consider 2D localized rotating patterns which solve a parabolic system of PDEs on the spatial domain R 2 and prove nonlinear stability with asymptotic phase with respect to the norm in the Sobolev space H 2.
Abstract: We consider 2D localized rotating patterns which solve a parabolic system of PDEs on the spatial domain R2. Under suitable assumptions, we prove nonlinear stability with asymptotic phase with respect to the norm in the Sobolev space H2. The stability result is obtained by a combination of energy and resolvent estimates, after the dynamics is decomposed into an evolution within a three–dimensional group orbit and a transversal evolution towards the group orbit. The stability theorem is applied to the quintic–cubic Ginzburg–Landau equation and illustrated by numerical computations.
30 citations
TL;DR: In this article, the authors provided a rigorous mathematical derivation of the convergence in the long-wave transonic limit of the minimizing travelling waves for the two-dimensional Gross-Pitaevskii equation towards ground states for the Kadomtsev-Petviashvili equation (KP I).
Abstract: We provide a rigorous mathematical derivation of the convergence in the long-wave transonic limit of the minimizing travelling waves for the two-dimensional Gross-Pitaevskii equation towards ground states for the Kadomtsev-Petviashvili equation (KP I).
30 citations
TL;DR: In this paper, a system of Gross-Pitaevskii equations in R 2 mod- elling a mixture of two Bose-Einstein condensates with repulsive interaction is considered.
Abstract: We consider a system of Gross-Pitaevskii equations in R 2 mod- elling a mixture of two Bose-Einstein condensates with repulsive interaction. We aim to study the qualitative behaviour of ground and excited state so- lutions. We allow two different harmonic and off-centered trapping poten- tials and study the spatial patterns of the solutions within the Thomas- Fermi approximation as well as phase segregation phenomena within the large- interaction regime.
25 citations
22 citations
TL;DR: In this article, the authors studied regularity and partial regularity for a class of general quasi-linear elliptic equations and systems, which are of the main coecien ts satisfying the VMO conditions in x uniformly with respect to u, and of lower order items satisfying con- trollable growth.
Abstract: In this paper we study regularity and partial regularity for the weak solution of a class of general quasi-linear elliptic equations and systems, which are of the quasi-linear main coecien ts satisfying the VMO conditions in x uniformly with respect to u, and of the lower order items satisfying con- trollable growth.
16 citations
TL;DR: In this paper, lower bounds on the power of breather solutions ψn(t) = e −iΩtφn, Ω > 0 of a Discrete Nonlinear Schrodinger Equation with cubic or higher order nonlinearity and site-dependent anharmonic parameter, supplemented with Dirichlet boundary conditions are derived.
Abstract: We derive lower bounds on the power of breather solutions ψn(t) = e −iΩtφn, Ω > 0 of a Discrete Nonlinear Schrodinger Equation with cubic or higher order nonlinearity and site-dependent anharmonic parameter, supplemented with Dirichlet boundary conditions. For the case of a defocusing DNLS, one of the lower bounds depends not only on the dimension of the lattice, the lattice spacing, and the frequency of the periodic solution, but also on the excitation threshold of time periodic and spatially localized solutions of the focusing DNLS, proved by M. Weinstein in Nonlinearity 12, 673–691, 1999. Our simple proof via a direct variational method, makes use of the interpolation inequality proved by Weinstein, and its optimal constant related to the excitation threshold. We also provide existence results (via the mountain pass theorem) and lower bounds on the power of breather solutions for DNLS lattices with sign-changing anharmonic parameter. Numerical studies considering the classical defocusing DNLS, the case of a single nonlinear impurity, as well as a random DNLS lattice are performed, to test the efficiency of the lower bounds.
9 citations
TL;DR: In this paper, the existence of chaotic dynamics in the flow generated by an infinite system of strongly coupled ordinary differential equations with a finite dimensional hyperbolic part and an infinite dimensional center part was shown.
Abstract: This work is concerned with showing the existence of chaotic dynamics in the flow generated by an infinite system of strongly coupled ordinary differential equations with a finite dimensional hyperbolic part and an infinite dimensional center part. This theory can be applied to partial differential equations by using a Galerkin expansion which is illustrated by the problem of oscillations of a buckled elastic beam.
8 citations
TL;DR: In this paper, an asymptotic two-dimensional model for nonlinearly viscous fluids is rigorously derived as a limit of corresponding three-dimensional models using the analysis with respect to the thickness of the respective three dimensional domains.
Abstract: In this paper, an asymptotic two dimensional model for compress- ible nonlinearly viscous fluids is rigorously derived as a limit of corresponding three dimensional models using asymptotic analysis with respect to the thick- ness of the respective three dimensional domains.
TL;DR: In this article, the authors studied the behavior of a species which inhabits two independent habitat patches and found that frequent transfers happen between two patches with an exponentially decaying nonlinear transfer rate.
Abstract: Abstract. In this paper, we study the dynamical behavior of a species which inhabits two independent habitat patches. Due to the long range foraging behavior, frequent transfers happen between two patches with an exponentially decaying nonlinear transfer rate. Periodic oscillation is observed as a Hopf bifurcation occurs at some critical values of the delay τ . By applying the center manifold theorem, the Poincaré normal form and the approximate periodic solution near the critical delay values are obtained. The complete synchronization of variations of the population size of species in two patches is analyzed and numerical simulations under various parametric conditions are illustrated. The moment stability of the solution of the stochastic delay equation is also considered by applying the Itô integral.
TL;DR: In this paper, an interface problem derived by a reaction-diusion equation in two-and three-dimensional systems with radial symmetry is considered, and the ex-istence of Hopf bifurcation as a parameter varies.
Abstract: We consider an interface problem derived by a reaction-diusion equation in two- and three- dimensional system with radial symmetry. Ex- istence of Hopf bifurcation as a parameter varies will be studied in two- and three- dimensional spaces.
TL;DR: In this article, a wave map with smooth compactly supported initial data satisfying the smallness condition parallel to a parallel wave map parallel to an initial wave map over a fixed number of vertices is presented.
Abstract: Let u : R3+1 -> H-2 be a Wave Map with smooth compactly supported initial data satisfying the smallness condition parallel to u[0]parallel to ((H) over dot3/2 x (H) over dot1/2) 0. In particular, the Wave Map exists globally in time and is smooth. Then denoting u(0, x) = u(infinity) epsilon H for vertical bar x vertical bar large enough, we have