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Communications on Pure & Applied Analysis

2014 , Volume 13 , Issue 4

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Infinite-energy solutions for the Navier-Stokes equations in a strip revisited
Peter Anthony and Sergey Zelik
2014, 13(4): 1361-1393 doi: 10.3934/cpaa.2014.13.1361 +[Abstract](801) +[PDF](566.3KB)
Abstract:
The paper deals with the Navier-Stokes equations in a strip in the class of spatially non-decaying (in nite-energy) solutions belonging to the properly chosen uniformly local Sobolev spaces. The global well-posedness and dissipativity of the Navier-Stokes equations in a strip in such spaces has been rst established in [22]. However, the proof given there contains a rather essential error and the aim of the present paper is to correct this error and to show that the main results of [22] remain true.
Global existence of solutions for the thermoelastic Bresse system
Yuming Qin, Xinguang Yang and Zhiyong Ma
2014, 13(4): 1395-1406 doi: 10.3934/cpaa.2014.13.1395 +[Abstract](737) +[PDF](367.4KB)
Abstract:
In this paper, using the semigroup approach, we obtain the global existence of solutions for linear (nonlinear) homogeneous (nonhomogeneous) thermoelastic Bresse System.
Green's functions for parabolic systems of second order in time-varying domains
Hongjie Dong and Seick Kim
2014, 13(4): 1407-1433 doi: 10.3934/cpaa.2014.13.1407 +[Abstract](638) +[PDF](539.2KB)
Abstract:
We construct Green's functions for divergence form, second order parabolic systems in non-smooth time-varying domains whose boundaries are locally represented as graph of functions that are Lipschitz continuous in the spatial variables and $1/2$-Hölder continuous in the time variable, under the assumption that weak solutions of the system satisfy an interior Hölder continuity estimate. We also derive global pointwise estimates for Green's function in such time-varying domains under the assumption that weak solutions of the system vanishing on a portion of the boundary satisfy a certain local boundedness estimate and a local Hölder continuity estimate. In particular, our results apply to complex perturbations of a single real equation.
Boundedness of second order Riesz transforms associated to Schrödinger operators on Musielak-Orlicz-Hardy spaces
Jun Cao, Der-Chen Chang, Dachun Yang and Sibei Yang
2014, 13(4): 1435-1463 doi: 10.3934/cpaa.2014.13.1435 +[Abstract](747) +[PDF](603.1KB)
Abstract:
Let $L:=-\Delta+V$ be a Schrödinger operator with the nonnegative potential $V$ belonging to the reverse Hölder class $RH_{q_0}(R^n)$ for some $q_0\in[n,\infty)$ with $n\geq 3$, and $\varphi: R^n\times[0,\infty)\to[0,\infty)$ a function such that $\varphi(x,\cdot)$ is an Orlicz function, $\varphi(\cdot,t)\in A_{\infty}(R^n)$ (the class of uniformly Muckenhoupt weights) and its uniformly critical lower type index $i(\varphi)\in (\frac{n}{n+1},1]$. In this article, the authors prove that the second order Riesz transform $\nabla^2L^{-1}$ associated with $L$ is bounded from the Musielak-Orlicz-Hardy space associated with $L$, $H_{\varphi,L}(R^n)$, to the Musielak-Orlicz-Hardy space $H_{\varphi}(R^n)$, via establishing an atomic characterization of $H_{\varphi,L}(R^n)$. As an application, the authors prove that the operator $VL^{-1}$ is bounded on the Musielak-Orlicz-Hardy space $H_{\varphi,L}(R^n)$, which further gives the maximal inequality associated with $L$ in $H_{\varphi,L}(R^n)$. All these results are new even when $\varphi(x,t):=t^p$, with $p\in(\frac{n}{n+1},1]$, for all $x\in R^n$ and $t\in[0,\infty)$.
Existence and nonexistence of local/global solutions for a nonhomogeneous heat equation
Xie Li and Zhaoyin Xiang
2014, 13(4): 1465-1480 doi: 10.3934/cpaa.2014.13.1465 +[Abstract](530) +[PDF](431.2KB)
Abstract:
In this paper, we study the existence of local/global solutions to the Cauchy problem \begin{eqnarray} \rho(x)u_t=\Delta u+q(x)u^p, (x,t)\in R^N \times (0,T),\\ u(x,0)=u_{0}(x)\ge 0, x \in R^N \end{eqnarray} with $p > 0$ and $N\ge 3$. We describe the sharp decay conditions on $\rho, q$ and the data $u_0$ at infinity that guarantee the local/global existence of nonnegative solutions.
Global strong solution to the two-dimensional density-dependent magnetohydrodynamic equations with vaccum
Jishan Fan, Fucai Li and Gen Nakamura
2014, 13(4): 1481-1490 doi: 10.3934/cpaa.2014.13.1481 +[Abstract](629) +[PDF](349.2KB)
Abstract:
In this paper we establish the global existence of strong solution to the density-dependent incompressible magnetohydrodynamic equations with vaccum in a bounded domain in $R^2$. Furthermore, the limit as the heat conductivity coefficient tends to zero is also obtained.
Multiple solutions for a class of nonlinear Neumann eigenvalue problems
Leszek Gasiński and Nikolaos S. Papageorgiou
2014, 13(4): 1491-1512 doi: 10.3934/cpaa.2014.13.1491 +[Abstract](663) +[PDF](494.3KB)
Abstract:
We consider a parametric nonlinear equation driven by the Neumann $p$-Laplacian. Using variational methods we show that when the parameter $\lambda > \widehat{\lambda}_1$ (where $\widehat{\lambda}_1$ is the first nonzero eigenvalue of the negative Neumann $p$-Laplacian), then the problem has at least three nontrivial smooth solutions, two of constant sign (one positive and one negative) and the third nodal. In the semilinear case (i.e., $p=2$), using Morse theory and flow invariance argument, we show that the problem has three nodal solutions.
Retraction
Editorial Office
2014, 13(4): 1513-1513 doi: 10.3934/cpaa.2014.13.1513 +[Abstract](381) +[PDF](86.3KB)
Abstract:
This article was accidentally posted online but only to be discovered that the same article had been published (see [1]) in the previous issue of the same journal. Thus this publication is retracted. The Editorial Office offers apologies for the confusion and inconvenience it might have caused.
Semi discrete weakly damped nonlinear Klein-Gordon Schrödinger system
Olivier Goubet and Marilena N. Poulou
2014, 13(4): 1525-1539 doi: 10.3934/cpaa.2014.13.1525 +[Abstract](686) +[PDF](399.4KB)
Abstract:
We consider a semi-discrete in time relaxation scheme to discretize a damped forced nonlinear Klein-Gordon Schrödinger system. This provides us with a discrete infinite-dimensional dynamical system. We prove the existence of a finite dimensional global attractor for this dynamical system.
General decay estimates for a Cauchy viscoelastic wave problem
Belkacem Said-Houari and Salim A. Messaoudi
2014, 13(4): 1541-1551 doi: 10.3934/cpaa.2014.13.1541 +[Abstract](754) +[PDF](373.5KB)
Abstract:
In this paper, we consider the Cauchy problem of a viscoelatic wave equation and by using the energy method in the Fourier space, we show general decay estimates of the solution. This result improves and generalizes some other results in the literature.
Global existence of strong solutions to incompressible MHD
Huajun Gong and Jinkai Li
2014, 13(4): 1553-1561 doi: 10.3934/cpaa.2014.13.1553 +[Abstract](621) +[PDF](336.6KB)
Abstract:
We establish the global existence and uniqueness of strong solutions to the initial boundary value problem for the incompressible MHD equations in bounded smooth domains of $\mathbb R^3$ under some suitable smallness conditions. The initial density is allowed to have vacuum, in particular, it can vanish in a set of positive Lebessgue measure. More precisely, under the assumption that the production of the quantities $\|\sqrt\rho_0u_0\|_{L^2(\Omega)}^2+\|H_0\|_{L^2(\Omega)}^2$ and $\|\nabla u_0\|_{L^2(\Omega)}^2+\|\nabla H_0\|_{L^2(\Omega)}^2$ is suitably small, with the smallness depending only on the bound of the initial density and the domain, we prove that there is a unique strong solution to the Dirichlet problem of the incompressible MHD system.
Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data
Hiroyuki Hirayama
2014, 13(4): 1563-1591 doi: 10.3934/cpaa.2014.13.1563 +[Abstract](671) +[PDF](564.0KB)
Abstract:
In the present paper, we consider the Cauchy problem of a system of quadratic derivative nonlinear Schrödinger equations which was introduced by M. Colin and T. Colin (2004) as a model of laser-plasma interaction. The local existence of the solution of the system in the Sobolev space $H^s$ for $s > d/2+3$ is proved by M. Colin and T. Colin. We prove the well-posedness of the system with low regularity initial data. For some cases, we also prove the well-posedness and the scattering at the scaling critical regularity by using $U^2$ space and $V^2$ space which are applied to prove the well-posedness and the scattering for KP-II equation at the scaling critical regularity by Hadac, Herr and Koch (2009).
Two sequences of solutions for indefinite superlinear-sublinear elliptic equations with nonlinear boundary conditions
Ryuji Kajikiya and Daisuke Naimen
2014, 13(4): 1593-1612 doi: 10.3934/cpaa.2014.13.1593 +[Abstract](604) +[PDF](430.6KB)
Abstract:
In this paper, we study semilinear elliptic equations with nonlinear Neumann boundary conditions. We prove the existence of a sequence of solutions converging to zero if the nonlinear term is locally sublinear and the existence of a sequence of solutions diverging to infinity if the nonlinear term is locally superlinear.
Global existence of strong solutions for $2$-dimensional Navier-Stokes equations on exterior domains with growing data at infinity
Michele Campiti, Giovanni P. Galdi and Matthias Hieber
2014, 13(4): 1613-1627 doi: 10.3934/cpaa.2014.13.1613 +[Abstract](641) +[PDF](410.0KB)
Abstract:
It is proved the existence of a unique, global strong solution to the two-dimensional Navier-Stokes initial-value problem in exterior domains in the case where the velocity field tends, at large spatial distance, to a prescribed velocity field that is allowed to grow linearly.
Note on evolutionary free piston problem for Stokes equations with slip boundary conditions
Boris Muha and Zvonimir Tutek
2014, 13(4): 1629-1639 doi: 10.3934/cpaa.2014.13.1629 +[Abstract](604) +[PDF](387.7KB)
Abstract:
In this paper we study a free boundary fluid-rigid body interaction problem, the free piston problem. We consider an evolutionary incompressible viscous fluid flow through a junction of two pipes. Inside the "vertical" pipe there is a heavy piston which can freely slide along the pipe. On the lateral boundary of the "vertical" pipe we prescribe Navier's slip boundary conditions. We prove the existence of a local in time weak solution. Furthermore, we show that the obtained solution is a strong solution.
The classification of constant weighted curvature curves in the plane with a log-linear density
Doan The Hieu and Tran Le Nam
2014, 13(4): 1641-1652 doi: 10.3934/cpaa.2014.13.1641 +[Abstract](461) +[PDF](3720.2KB)
Abstract:
In this paper, we classify the class of constant weighted curvature curves in the plane with a log-linear density, or in other words, classify all traveling curved fronts with a constant forcing term in $R^2.$ The classification gives some interesting phenomena and consequences including: the family of curves converge to a round point when the weighted curvature of curves (or equivalently the forcing term of traveling curved fronts) goes to infinity, a simple proof for a main result in [13] as well as some well-known facts concerning to the isoperimetric problem in the plane with density $e^y.$
Existence and uniqueness of a positive connection for the scalar viscous shallow water system in a bounded interval
Marta Strani
2014, 13(4): 1653-1667 doi: 10.3934/cpaa.2014.13.1653 +[Abstract](510) +[PDF](549.5KB)
Abstract:
We study the existence and the uniqueness of a {\bf positive connection}, that is a stationary solution connecting the boundary data, for the initial-boundary value problem for the viscous shallow water system \begin{eqnarray} \partial_t u + \partial_x v=0, \partial_t v+\partial_x( \frac{v^2}{u}+ P(u))= \varepsilon\partial_x ( u \partial_x(\frac{v}{u})) \end{eqnarray} in a bounded interval $(-l,l)$ of the real line. We firstly consider the general case where the term of pressure $P(u)$ satisfies \begin{eqnarray} P(0)=0, P(+\infty)=+\infty, P'(u) \quad and \quad P''(u)>0 \ \forall u >0, \end{eqnarray} and then we show properties of the steady state in the relevant case $P(u)=\kappa u^{\gamma}$, $\gamma>1$. The viscous Saint-Venant system, corresponding to $\gamma=2$, fits in the general framework.
Low regularity well-posedness for the 2D Maxwell-Klein-Gordon equation in the Coulomb gauge
Magdalena Czubak and Nina Pikula
2014, 13(4): 1669-1683 doi: 10.3934/cpaa.2014.13.1669 +[Abstract](563) +[PDF](447.3KB)
Abstract:
We consider the Maxwell-Klein-Gordon equation in 2D in the Coulomb gauge. We establish local well-posedness for $s=\frac 14+\epsilon$ for data for the spatial part of the gauge potentials and for $s=\frac 58+\epsilon$ for the solution $\phi$ of the gauged Klein-Gordon equation. The main tool for handling the wave equations is the product estimate established by D'Ancona, Foschi, and Selberg. Due to low regularity, we are unable to use the conventional approaches to handle the elliptic variable $A_0$, so we provide a new approach.

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