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Communications on Pure & Applied Analysis

2011 , Volume 10 , Issue 2

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Bootstrapped Morawetz estimates and resonant decomposition for low regularity global solutions of cubic NLS on $R^2$
J. Colliander and Tristan Roy
2011, 10(2): 397-414 doi: 10.3934/cpaa.2011.10.397 +[Abstract](579) +[PDF](459.9KB)
Abstract:
We prove global well-posedness for the $L^2$-critical cubic defocusing nonlinear Schrödinger equation on $R^2$ with data $u_0 \in H^s(R^2)$ for $ s > \frac{1}{3}$. The proof combines a priori Morawetz estimates obtained in [4] and the improved almost conservation law obtained in [6]. There are two technical difficulties. The first one is to estimate the variation of the improved almost conservation law on intervals given in terms of Strichartz spaces rather than in terms of $X^{s,b}$ spaces. The second one is to control the error of the a priori Morawetz estimates on an arbitrary large time interval, which is performed by a bootstrap via a double layer in time decomposition.
A non-autonomous 3D Lagrangian averaged Navier-Stokes-$\alpha$ model with oscillating external force and its global attractor
T. Tachim Medjo
2011, 10(2): 415-433 doi: 10.3934/cpaa.2011.10.415 +[Abstract](618) +[PDF](462.2KB)
Abstract:
In this article, we consider a non-autonomous three-dimensional Lagrangian averaged Navier-Stokes-$\alpha$ model with a singulary oscillating external force depending on a small parameter $ \epsilon$. We prove the existence of the uniform global attractor $A^\epsilon$. Furthermore, using the method of [15] in the case of the two-dimensional Navier-Stokes systems, we study the convergence of $A^\epsilon $ as $\epsilon$ goes to zero.
The Boltzmann equation near Maxwellian in the whole space
Xinkuan Chai
2011, 10(2): 435-458 doi: 10.3934/cpaa.2011.10.435 +[Abstract](594) +[PDF](423.1KB)
Abstract:
A recent nonlinear energy method introduced in [19, 20] leads to another construction global solutions near Maxwellian for the Boltzmann equation over the whole space. Moreover, the optimal time decay, uniform stability and the optimal time stability of the solutions to the Boltzmann equation are all obtained via such a energy method.
Free boundary problem for compressible flows with density--dependent viscosity coefficients
Ping Chen, Daoyuan Fang and Ting Zhang
2011, 10(2): 459-478 doi: 10.3934/cpaa.2011.10.459 +[Abstract](552) +[PDF](443.8KB)
Abstract:
In this paper, we consider the free boundary problem of the spherically symmetric compressible isentropic Navier--Stokes equations in $R^n (n \geq 1)$, with density--dependent viscosity coefficients. Precisely, the viscosity coefficients $\mu$ and $\lambda$ are assumed to be proportional to $\rho^\theta$, $0 < \theta < 1$, where $\rho$ is the density. We obtain the global existence, uniqueness and continuous dependence on initial data of a weak solution, with a Lebesgue initial velocity $u_0\in L^{4 m}$, $4m>n$ and $\theta<\frac{4m-2}{4m+n}$. We weaken the regularity requirement of the initial velocity, and improve some known results of the one-dimensional system.
A system of the Hamilton--Jacobi and the continuity equations in the vanishing viscosity limit
Thomas Strömberg
2011, 10(2): 479-506 doi: 10.3934/cpaa.2011.10.479 +[Abstract](498) +[PDF](580.1KB)
Abstract:
We study the following system of the viscous Hamilton--Jacobi and the continuity equations in the limit as $\varepsilon \downarrow 0$:

$ S^\varepsilon_t+\frac{1}{2}|D S^\varepsilon|^2+V(x)-\varepsilon\Delta S^\varepsilon =0$ in $Q_T$, $S^\varepsilon(0,x)=S_0(x)$ in $R^n;$

$ \rho^\varepsilon_t+$ Div$(\rho^\varepsilon D S^\varepsilon)=0$ in $Q_T$, $\rho^\varepsilon(0,x)=\rho_0(x)$ in $R^n$.

Here $Q_T=(0,T]\times R^n$. The potential $V$ and the initial function $S_0$ are allowed to grow quadratically while $\rho_0$ is a Borel measure. The paper justifies and describes the vanishing viscosity transition to the corresponding inviscid system. The notion of weak solution employed for the inviscid system is that of a viscosity--measure solution $(S,\rho)$.

Bifurcations of some elliptic problems with a singular nonlinearity via Morse index
Zongming Guo, Zhongyuan Liu, Juncheng Wei and Feng Zhou
2011, 10(2): 507-525 doi: 10.3934/cpaa.2011.10.507 +[Abstract](711) +[PDF](451.1KB)
Abstract:
We study the boundary value problem

$\Delta u=\lambda |x|^\alpha f(u)$ in $\Omega, u=1$ on $\partial \Omega\qquad$ (1)

where $\lambda>0$, $\alpha \geq 0$, $\Omega$ is a bounded smooth domain in $R^N$ ($N \geq 2$) containing $0$ and $f$ is a $C^1$ function satisfying $\lim_{s \to 0^+} s^p f(s)=1$. We show that for each $\alpha \geq 0$, there is a critical power $p_c (\alpha)>0$, which is decreasing in $\alpha$, such that the branch of positive solutions possesses infinitely many bifurcation points provided $p > p_c (\alpha)$ or $p > p_c (0)$, and this relies on the shape of the domain $\Omega$. We get some important estimates of the Morse index of the regular and singular solutions. Moreover, we also study the radial solution branch of the related problems in the unit ball. We find that the branch possesses infinitely many turning points provided that $p>p_c (\alpha)$ and the Morse index of any radial solution (regular or singular) in this branch is finite provided that $0 < p \leq p_c (\alpha)$. This implies that the structure of the radial solution branch of (1) changes for $0 < p \leq p_c (\alpha)$ and $p > p_c (\alpha)$.

Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent
Xiaomei Sun and Wenyi Chen
2011, 10(2): 527-540 doi: 10.3934/cpaa.2011.10.527 +[Abstract](710) +[PDF](402.0KB)
Abstract:
In this paper, we consider the following semilinear elliptic equations with critical Hardy-Sobolev exponent:

$ -\Delta u+\lambda\frac{u}{|x-a|^2}-\gamma\frac{u}{|x|^2} =\frac{Q(x)}{|x|^s}|u|^{2^*(s)-2}u+g(x,u), u>0$ in $\Omega,$

$ \frac{\partial u}{\partial\nu}+\alpha(x)u=0 $ on $\partial\Omega. $

By variational method, the existence of positive solution is obtained.

A singular limit in a nonlinear problem arising in electromagnetism
Frank Jochmann
2011, 10(2): 541-559 doi: 10.3934/cpaa.2011.10.541 +[Abstract](602) +[PDF](433.7KB)
Abstract:
This paper deals with a generally nonlinear mixed-type initial-boundary value problem for the description of the electromagnetic field in a conducting medium that is surrounded by an insulating medium with a high dielectric permittivity. The main goals are the existence, uniqueness and the asymptotic behavior of the solutions to this system.
The optimal weighted $W^{2, p}$ estimates of elliptic equation with non-compatible conditions
Yi Cao, Dong Li and Lihe Wang
2011, 10(2): 561-570 doi: 10.3934/cpaa.2011.10.561 +[Abstract](537) +[PDF](382.1KB)
Abstract:
In this paper we study uniformly elliptic equations with non-compatible conditions, where $\Omega$ is a bounded Lipchitz domain, and the right-hand side term and the boundary value of the elliptic equations belong to $L^p (p \geq 2)$ space. Then the optimal weighted $W^{2, p}$ estimates will be given by Whitney decomposition and $L^p$ estimates of non-tangential maximal function associated to solutions of the elliptic equations.
Linking solutions for N-laplace elliptic equations with Hardy-Sobolev operator and indefinite weights
Guoqing Zhang, Jia-yu Shao and Sanyang Liu
2011, 10(2): 571-581 doi: 10.3934/cpaa.2011.10.571 +[Abstract](624) +[PDF](366.1KB)
Abstract:
In this paper, we investigate a class of N-Laplace elliptic equations with Hardy-Sobolev operator and indefinite weights

$ -\Delta_N u-\mu \frac{1}{(|x|\log(\frac{R}{|x|}))^N}|u|^{N-2}u= \lambda V(x)|u|^{N-2} u + f(x,u), u\in W_0^{1, N}(\Omega), $

where $\Omega$ be a bounded domain containing $0$ in $R^N$, $N \geq 2, 0 < \mu < (\frac{N-1}{N})^N$, and the weight function $V(x)$ may change sign and has nontrivial positive part. Using Moser-Trudinger inequality and nonstandard linking structure introduced by Degiovanni and Lancelotti [6], we prove the existence of a nontrivial solution for any $\lambda\in R$.

On regularity criteria for the 3D magneto-micropolar fluid equations in the critical Morrey-Campanato space
Jinbo Geng, Xiaochun Chen and Sadek Gala
2011, 10(2): 583-592 doi: 10.3934/cpaa.2011.10.583 +[Abstract](624) +[PDF](369.6KB)
Abstract:
In this paper, some improved regularity criteria for the 3D magneto-micropolar fluid equations are established in critical Morrey-Campanato spaces. It is proved that if the velocity field satisfies

$u\in L^{\frac{2}{1-r}}(0,T; M_{2,\frac{3}{r}}(R^3)) $ with $r\in (0, 1)$ or $u\in C(0, T; M_{2,3}(R^3))$

or the gradient field of velocity satisfies

$ \nabla u\in L^{\frac{2}{2-r}}(0, T; M_{2,\frac{3}{ r}}(R^3))$ with $r\in (0,1], $

then the solution remains smooth on $[0,T] $.

Breaking of resonance for elliptic problems with strong degeneration at infinity
Francesco Della Pietra and Ireneo Peral
2011, 10(2): 593-612 doi: 10.3934/cpaa.2011.10.593 +[Abstract](580) +[PDF](433.7KB)
Abstract:
In this paper we study the problem

-div$(\frac{Du}{(1+u)^\theta})+|Du|^q =\lambda g(x)u +f$ in $\Omega,$

$u=0$ on $\partial \Omega, $

$u\geq 0$ in $\Omega,$

where $\Omega$ is a bounded open set of $R^n$, $1 < q \leq 2$, $\theta\geq 0$, $f\in L^1(\Omega)$, and $f>0$. The main feature is to show that even for large values of $\theta$ there is solution for all $\lambda>0$.
The problem could be seen as a reaction-diffusion model which produces a saturation effect, that is, the diffusion goes to zero when $u$ go to infinity.

New dissipated energy for the unstable thin film equation
Marina Chugunova and Roman M. Taranets
2011, 10(2): 613-624 doi: 10.3934/cpaa.2011.10.613 +[Abstract](612) +[PDF](231.7KB)
Abstract:
The fluid thin film equation $h_t = - (h^n h_{x x x})_x - a_1 (h^m h_x)_x$ is known to conserve mass $\int h dx$, and in the case of $a_1 \leq 0$, to dissipate entropy $\int h^{3/2 - n} dx$ (see [8]) and the $L^2$-norm of the gradient $\int h_x^2 dx$ (see [3]). For the special case of $a_1 = 0$ a new dissipated quantity $\int h^{\alpha} h_x^2 dx $ was recently discovered for positive classical solutions by Laugesen (see [15]). We extend it in two ways. First, we prove that Laugesen's functional dissipates strong nonnegative generalized solutions. Second, we prove the full $\alpha$-energy $\int (\frac{1}{2} h^\alpha h_x^2 - $ $ \frac {a_1 h^{\alpha + m - n + 2}}{(\alpha + m - n + 1)(\alpha + m - n + 2)} ) dx $ dissipation for strong nonnegative generalized solutions in the case of the unstable porous media perturbation $a_1> 0$ and the critical exponent $m = n+2$.
On the collapsing sandpile problem
S. Dumont and Noureddine Igbida
2011, 10(2): 625-638 doi: 10.3934/cpaa.2011.10.625 +[Abstract](639) +[PDF](782.0KB)
Abstract:
We are interested in the modeling of collapsing sandpiles. We use the collapsing model introduced by Evans, Feldman and Gariepy in [13], to provide a description of the phenomena in terms of a composition of projections onto interlocked convex sets around the set of stable sandpiles.
Uniform attractor for non-autonomous nonlinear Schrödinger equation
Olivier Goubet and Wided Kechiche
2011, 10(2): 639-651 doi: 10.3934/cpaa.2011.10.639 +[Abstract](622) +[PDF](401.0KB)
Abstract:
We consider a weakly coupled system of nonlinear Schrödinger equations which models a Bose Einstein condensate with an impurity. The first equation is dissipative, while the second one is conservative. We consider this dynamical system into the framework of non-autonomous dynamical systems, the solution to the conservative equation being the symbol of the semi-process. We prove that the first equation possesses a uniform attractor, which attracts the solutions for the weak topology of the underlying energy space. We then study the limit of this attractor when the coupling parameter converges towards $0$.
Global wellposedness for a transport equation with super-critial dissipation
Xumin Gu
2011, 10(2): 653-665 doi: 10.3934/cpaa.2011.10.653 +[Abstract](553) +[PDF](339.9KB)
Abstract:
We study a one-dimensional transport equation with non-local velocity and supercritial dissipation. Using the methods of modulus of continuity introduced in [1] and fractional Laplacian representaiton introduced in [2], we prove its global well-posedness for small periodic initial data in Holder spaces.
Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks
Mokhtar Kirane, Belkacem Said-Houari and Mohamed Naim Anwar
2011, 10(2): 667-686 doi: 10.3934/cpaa.2011.10.667 +[Abstract](753) +[PDF](395.6KB)
Abstract:
We study the exponential stability of the Timoshenko beam system by interior time-dependent delay term feedbacks. The beam is clamped at the two hand points subject to two internal feedbacks: one with a time-varying delay and the other without delay. Using the variable norm technique of Kato, it is proved that the system is well-posed whenever an hypothesis between the weight of the delay term in the feedback, the weight of the term without delay and the wave speeds. By introducing an appropriate Lyapunov functional the exponential stability of the system is proved. Under the imposed constrain on the weights of the feedbacks and the wave speeds, the exponential decay of the energy is established via a suitable Lyapunov functional.
Kernel sections and (almost) periodic solutions of a non-autonomous parabolic PDE with a discrete state-dependent delay
Xiang Li and Zhixiang Li
2011, 10(2): 687-700 doi: 10.3934/cpaa.2011.10.687 +[Abstract](590) +[PDF](358.3KB)
Abstract:
In this paper, we consider the long time behavior of a non-autonomous parabolic PDE with a discrete state-dependent delay. We study the existence of compact kernel sections and unique complete trajectory of the corresponding problem. Furthermore, we obtain the (almost) periodic solution which attracts all solutions provided the time dependent terms are (almost) periodic with respect to time $t$.
An eigenvalue problem possessing a continuous family of eigenvalues plus an isolated eigenvalue
Mihai Mihăilescu
2011, 10(2): 701-708 doi: 10.3934/cpaa.2011.10.701 +[Abstract](548) +[PDF](296.8KB)
Abstract:
In this paper we analyze an eigenvalue problem, involving a homogeneous Neumann boundary condition, in a smooth bounded domain. We show that the problem possesses, on the one hand, a continuous family of eigenvalues and, on the other hand, exactly one more eigenvalue which is isolated in the set of eigenvalues of the problem.
Existence results for the Klein-Gordon-Maxwell equations in higher dimensions with critical exponents
Paulo Cesar Carrião, Patrícia L. Cunha and Olímpio Hiroshi Miyagaki
2011, 10(2): 709-718 doi: 10.3934/cpaa.2011.10.709 +[Abstract](663) +[PDF](353.6KB)
Abstract:
In this paper we study the existence of radially symmetric solitary waves in $R^N$ for the nonlinear Klein-Gordon equations coupled with the Maxwell's equations when the nonlinearity exhibits critical growth. The main feature of this kind of problem is the lack of compactness arising in connection with the use of variational methods.
On existence and nonexistence of the positive solutions of non-newtonian filtration equation
Emil Novruzov
2011, 10(2): 719-730 doi: 10.3934/cpaa.2011.10.719 +[Abstract](631) +[PDF](186.1KB)
Abstract:
The subject of this investigation is existence and nonexistence of positive solutions of the following nonhomogeneous equation

$ \rho (|x|) \frac{\partial u}{\partial t}- \sum_{i=1}^N D_i(u^{m-1}|D_i u|^{\lambda -1}D_i u)+g(u)+lu=f(x)$ (1)

or, after the change $v=u^{\sigma}$, $\sigma =\frac{m+\lambda -1}{\lambda }, $ of equation

$\rho (|x|) \frac{\partial v^{\frac{1}{ \sigma }}}{\partial t}-\sigma ^{-\lambda }\sum_{i=1} ^N D_i(|D_i v|^{\lambda -1}D_i v)+g(v^{\frac{1}{\sigma }}) +lv^{\frac{1}{ \sigma }}=f(x),$ (1')

in unbounded domain $R_+\times R^N,$ where the term $g(s)$ is supposed to satisfy just a lower polynomial growth condition and $g'(s) > -l_1$. The existence of the solution in $ L^{1+1/\sigma}(0, T; L^{1+1/\sigma}(R^N))\cap L^{\lambda +1}(0, T; W^{1,\lambda +1}(R^N))$ is proved. Also, under some condition on $g(s)$ and $u_0$ is shown a nonexistence of the solution.

The maximal number of interior peak solutions concentrating on hyperplanes for a singularly perturbed Neumann problem
Yang Wang
2011, 10(2): 731-744 doi: 10.3934/cpaa.2011.10.731 +[Abstract](466) +[PDF](429.6KB)
Abstract:
We consider the following singularly perturbed elliptic problem

$\varepsilon^2 \Delta u-u+f(u)=0, u>0 $ in $B_1$,

$\frac{\partial u}{\partial \nu}=0 $ on $\partial B_1,$

where $\Delta = \sum_{i=1}^N \frac{\partial^2}{\partial x_i^2}$ is the Laplace operator, $B_1$ is the unit ball centered at the origin in $R^N$ $(N\ge 3)$, $\nu$ denotes the unit outer normal to $\partial B_1$, $\varepsilon > 0$ is a constant, and $f$ is a superlinear, subcritical nonlinearity . We will show that when $\e$ is sufficiently small there exists a solution with K interior peaks located on a hyperplane, where $1\le K \varepsilon\frac{C}{(\varepsilon)^{N-1}}$ with $C$ a positive constant depending on $N$ and $f$ only. As a consequence, we obtain that there exists at least $[\frac{C}{(\varepsilon)^{N-1}}]$ number of solutions for $\varepsilon$ sufficiently small.

The heat kernel and Heisenberg inequalities related to the Kontorovich-Lebedev transform
Semyon Yakubovich
2011, 10(2): 745-760 doi: 10.3934/cpaa.2011.10.745 +[Abstract](585) +[PDF](410.5KB)
Abstract:
We introduce a notion of the heat kernel related to the familiar Kontorovich-Lebedev transform. We study differential and semigroup properties of this kernel and construct fundamental solutions of a generalized diffusion equation. An integral transformation with the heat kernel is considered. By using the Plancherel $L_2$-theory for the Kontorovich-Lebedev transform and norm estimates for its convolution we establish analogs of the classical Heisenberg inequality and uncertainty principle for this transformation. The proof is also based on the norm inequalities for the Mellin transform of the heat kernel.
On the accuracy of invariant numerical schemes
Marx Chhay and Aziz Hamdouni
2011, 10(2): 761-783 doi: 10.3934/cpaa.2011.10.761 +[Abstract](629) +[PDF](368.7KB)
Abstract:
In this paper we present a method of construction of invariant numerical schemes for partial differential equations. The resulting schemes preserve the Lie-symmetry group of the continuous equation and they are at least as accurate as the original scheme. The improvement of the numerical properties thanks to the Lie-symmetry preservation is illustrated on the example of the Burgers equation.
A conjecture on multiple solutions of a nonlinear elliptic boundary value problem: some numerical evidence
Lisa Hollman and P. J. McKenna
2011, 10(2): 785-802 doi: 10.3934/cpaa.2011.10.785 +[Abstract](587) +[PDF](1241.6KB)
Abstract:
We investigate a conjecture regarding the number of solutions of a second order elliptic boundary value problem with an asymmetric nonlinearity. This investigation makes use of several computer assisted techniques. First, we compute approximate solutions using Newton's Iteration for small $b$ and then use a continuation method to show that the number of solutions becomes large as $b$ increases.
Energy minimizers of a thin film equation with born repulsion force
Huiqiang Jiang
2011, 10(2): 803-815 doi: 10.3934/cpaa.2011.10.803 +[Abstract](700) +[PDF](335.4KB)
Abstract:
In this paper, we consider a singular elliptic equation modeling steady states of thin film equation with both Van der Waal force and Born repulsion force. We prove the existence of smooth positive energy minimizing solutions. We also investigate the regularity of local minimizers and the limiting behavior of energy minimizer as the Born repulsion force tends to zero.

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