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Communications on Pure & Applied Analysis

2009 , Volume 8 , Issue 3

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Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations
P.E. Kloeden , Pedro Marín-Rubio and  José Real
2009, 8(3): 785-802 doi: 10.3934/cpaa.2009.8.785 +[Abstract](31) +[PDF](233.1KB)
A new proof of existence of solutions for the three dimensional system of globally modified Navier-Stokes equations introduced in [3] by Caraballo, Kloeden and Real is obtained using a smoother Galerkin scheme. This is then used to investigate the relationship between invariant measures and statistical solutions of this system in the case of temporally independent forcing term. Indeed, we are able to prove that a stationary statistical solution is also an invariant probability measure under suitable assumptions.
Exponential attractors for second order lattice dynamical systems
Ahmed Y. Abdallah
2009, 8(3): 803-813 doi: 10.3934/cpaa.2009.8.803 +[Abstract](34) +[PDF](189.4KB)
In [3], we introduced for the first time the study of exponential attractors for lattice dynamical systems, where a first order system has been investigated. Here we shall examine the existence of an exponential attractor for the solution semigroup of a second order lattice dynamical system acting on a closed bounded positively invariant set in the Hilbert space $l^2\times l^2$.
Global well-posedness and non-linear stability of periodic traveling waves for a Schrödinger-Benjamin-Ono system
Jaime Angulo , Carlos Matheus and  Didier Pilod
2009, 8(3): 815-844 doi: 10.3934/cpaa.2009.8.815 +[Abstract](46) +[PDF](392.9KB)
The objective of this paper is two-fold: firstly, we develop a local and global (in time) well-posedness theory for a system describing the motion of two fluids with different densities under capillary-gravity waves in a deep water flow (namely, a Schrödinger-Benjamin-Ono system) for low-regularity initial data in both periodic and continuous cases; secondly, a family of new periodic traveling waves for the Schrödinger-Benjamin-Ono system is given: by fixing a minimal period we obtain, via the implicit function theorem, a smooth branch of periodic solutions bifurcating a Jacobian elliptic function called dnoidal, and, moreover, we prove that all these periodic traveling waves are nonlinearly stable by perturbations with the same wavelength.
Classical limit for linear and nonlinear quantum Fokker-Planck systems
Roberta Bosi
2009, 8(3): 845-870 doi: 10.3934/cpaa.2009.8.845 +[Abstract](40) +[PDF](364.0KB)
We study the classical limit of some linear and nonlinear Quantum Fokker-Planck systems. In the nonlinear case we consider an Hartree-type potential. By the use of the Wigner transform and compactness methods, we prove the convergence of the system to a linear and nonlinear Vlasov Fokker- Planck equation respectively. The physical case with a Poisson coupling in three dimensions is included.
Smooth control of nanowires by means of a magnetic field
Gilles Carbou , Stéphane Labbé and  Emmanuel Trélat
2009, 8(3): 871-879 doi: 10.3934/cpaa.2009.8.871 +[Abstract](30) +[PDF](155.2KB)
We address the problem of control of the magnetic moment in a ferromagnetic nanowire by means of a magnetic field. Based on theoretical results for the 1D Landau-Lifschitz equation, we show a robust controllability result.
On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions
Gianni Gilardi , A. Miranville and  Giulio Schimperna
2009, 8(3): 881-912 doi: 10.3934/cpaa.2009.8.881 +[Abstract](63) +[PDF](395.6KB)
The Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions is considered and well-posedness results are proved.
Sharp threshold of global existence for the generalized Davey-Stewartson system in $R^2$
Zaihui Gan , Boling Guo and  Jian Zhang
2009, 8(3): 913-922 doi: 10.3934/cpaa.2009.8.913 +[Abstract](32) +[PDF](158.7KB)
This paper is concerned with the generalized Davey-Stewartson system in $\mathbf R^2$ which appears as mathematical models for the evolution of shallow-water waves having one predominant direction of travel. We obtain a sharp threshold of blowing up and global existence to the Cauchy problem of the system by constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow. Furthermore, we answer the question: How small are the initial data, the global solutions to the Cauchy problem of the system exist.
Global existence for nonlinear parabolic equations with a damping term
Daniela Giachetti and  Maria Michaela Porzio
2009, 8(3): 923-953 doi: 10.3934/cpaa.2009.8.923 +[Abstract](46) +[PDF](374.9KB)
This paper deal with existence of global solutions of nonlinear parabolic equations, possibly with degenerate or singular principal part, when a source term with a very general growth and a damping term are present.
Generalized solutions for the abstract singular Cauchy problem
Hernan R. Henriquez
2009, 8(3): 955-976 doi: 10.3934/cpaa.2009.8.955 +[Abstract](29) +[PDF](256.5KB)
In this work we study existence of solutions in convoluted sense for the abstract singular Cauchy problem. We relate the existence of convoluted solutions with the existence of a generalized singular evolution operator, and we establish a Hille-Yosida type theorem to characterize the existence of a local generalized singular evolution operator.
Magnetic vortices for a Ginzburg-Landau type energy with discontinuous constraint. II
Hassen Aydi and  Ayman Kachmar
2009, 8(3): 977-998 doi: 10.3934/cpaa.2009.8.977 +[Abstract](26) +[PDF](296.1KB)
We study vortex nucleation for minimizers of a Ginzburg-Landau energy with discontinuous constraint. For applied magnetic fields comparable with the first critical field of vortex nucleation, we determine the limiting vorticities.
Precise range of the existence of positive solutions of a nonlinear, indefinite in sign Neumann problem
Vladimir Lubyshev
2009, 8(3): 999-1018 doi: 10.3934/cpaa.2009.8.999 +[Abstract](32) +[PDF](270.9KB)
We study positive solutions of an elliptic problem with indefinite in sign nonlinear Neumann boundary condition that depends on a real parameter, $\lambda$. We find precise range, $I$, of those $\lambda$'s for which our problem possesses a positive solution, prove that $\lambda^$∗ = sup $I$ is a bifurcation point, and exhibit explicit max-min procedure for computing $\lambda^$∗. We also obtain some properties of the set of solutions.
A min-max principle for non-differentiable functions with a weak compactness condition
Roberto Livrea and  Salvatore A. Marano
2009, 8(3): 1019-1029 doi: 10.3934/cpaa.2009.8.1019 +[Abstract](46) +[PDF](183.5KB)
A general critical point result established by Ghoussoub is extended to the case of locally Lipschitz continuous functions satisfying a weak Palais-Smale hypothesis, which includes the so-called non-smooth Cerami condition. Some special cases are then pointed out.
Pairs of positive solutions for $p$--Laplacian equations with combined nonlinearities
Sophia Th. Kyritsi and  Nikolaos S. Papageorgiou
2009, 8(3): 1031-1051 doi: 10.3934/cpaa.2009.8.1031 +[Abstract](43) +[PDF](256.6KB)
We consider a nonlinear Dirichlet problem driven by the $p$--Laplacian differential operator, with a nonlinearity concave near the origin and a nonlinear perturbation of it. We look for multiple positive solutions. We consider two distinct cases. One when the perturbation is $p$--linear and resonant with respect to $\lambda_1>0$ (the principal eigenvalue of $(-\Delta_p,W^{1,p}_0(Z))$) at infinity and the other when the perturbation is $p$--superlinear at infinity. In both cases we obtain two positive smooth solutions. The approach is variational, coupled with the method of upper--lower solutions and with suitable truncation techniques.
Polynomial solutions of linear partial differential equations
Eugenia N. Petropoulou and  Panayiotis D. Siafarikas
2009, 8(3): 1053-1065 doi: 10.3934/cpaa.2009.8.1053 +[Abstract](52) +[PDF](183.0KB)
In this paper it is proved that the condition

$\lambda=a_1 (n-2)(n-1)+\gamma_1 (m-2)(m-1)+\beta_1 (n-1)(m-1)+\delta_1 (n-1)+\epsilon_1 (m-1),$

where $n=1,2,...,N$, $m=1,2,...,M$ is a necessary and sufficient condition for the linear partial differential equation

$(a_1x^2+a_2x+a_3)u_{x x}+(\beta_1xy+\beta_2x+\beta_3y+\beta_4)u_{x y} $

$+(\gamma_1y^2+\gamma_2y+\gamma_3)u_{y y}+(\delta_1x+\delta_2)u_x+(\epsilon_1y+\epsilon_2)u_y=\lambda u, $

where $a_i$, $\beta_j$, $\gamma_i$, $\delta_s$, $\epsilon_s$, $i=1,2,3$, $j=1,2,3,4$, $s=1,2$ are real or complex constants, to have polynomial solutions of the form

$u(x,y)=\sum_{n=1}^N\sum_{m=1}^Mu_{n m}x^{n-1}y^{m-1}.$

The proof of this result is obtained using a functional analytic method which reduces the problem of polynomial solutions of such partial differential equations to an eigenvalue problem of a specific linear operator in an abstract Hilbert space. The main result of this paper generalizes previously obtained results by other researchers.

On global regularity of incompressible Navier-Stokes equations in $\mathbf R^3$
Keyan Wang
2009, 8(3): 1067-1072 doi: 10.3934/cpaa.2009.8.1067 +[Abstract](32) +[PDF](133.5KB)
In this paper we prove the global regularity of classical solutions to the incompressible Navier-Stokes equations in $\mathbf R^3$ for a family of large initial data with finite energy.
Well-posedness and stability of classical solutions to the multidimensional full hydrodynamic model for semiconductors
Jiang Xu
2009, 8(3): 1073-1092 doi: 10.3934/cpaa.2009.8.1073 +[Abstract](36) +[PDF](273.3KB)
This paper is concerned with the global well-posedness and stability of classical solutions to the Cauchy problem for the multidimensional full hydrodynamic model in semiconductors on the framework of Besov space. By using the high- and low- frequency decomposition method, we obtain the exponential decay of classical solutions (close to equilibrium). Moreover, it is also shown that the vorticity decays to zero exponentially in the 2D and 3D space. The work weakens the regularity requirement of the initial data and improves some known results in Sobolev space.
A quasilinear thermoviscoelastic system for shape memory alloys with temperature dependent specific heat
Shuji Yoshikawa , Irena Pawłow and  Wojciech M. Zajączkowski
2009, 8(3): 1093-1115 doi: 10.3934/cpaa.2009.8.1093 +[Abstract](35) +[PDF](288.9KB)
This article extends the previous author's paper [28] on the existence of solutions to a quasilinear thermoviscoelasticity system arising in shape memory alloys. In the present setup we admit a modified energy equation with temperature growing specific heat. Due to such a modification we can solve the problem with stronger thermomechanical nonlinearity which was left open in [28].
$L^2$-concentration phenomenon for Zakharov system below energy norm II
Sijia Zhong and  Daoyuan Fang
2009, 8(3): 1117-1132 doi: 10.3934/cpaa.2009.8.1117 +[Abstract](33) +[PDF](222.7KB)
In this paper, we will prove a $L^2$-concentration result of Zakharov system in space dimension two, with initial data $(u_0,n_0,n_1)\in H^s\times L^2\times H^{-1}$ ($\frac {1 2}{1 3} < s < 1$), when blow up of the solution happens, by resonant decomposition and I-method, which is an improvement of [13].

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