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Communications on Pure & Applied Analysis

2010 , Volume 9 , Issue 2

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Low regularity global well-posedness for the nonlinear Schrödinger equation on closed manifolds
Takafumi Akahori
2010, 9(2): 261-280 doi: 10.3934/cpaa.2010.9.261 +[Abstract](42) +[PDF](266.1KB)
Abstract:
We consider the defocusing cubic nonlinear Schrödinger equation on two dimensional closed Riemannian manifolds. We prove global well-posedness below the energy class on manifolds satisfying some condition. The main ingredient for the proof is an application of the I-method introduced by Colliander, Keel, Staffilani, Takaoka and Tao.
Existence and concentration of solitary waves for a class of quasilinear Schrödinger equations
Daniele Cassani , João Marcos do Ó and  Abbas Moameni
2010, 9(2): 281-306 doi: 10.3934/cpaa.2010.9.281 +[Abstract](43) +[PDF](312.3KB)
Abstract:
In this paper we prove the existence and qualitative properties of positive bound state solutions for a class of quasilinear Schrödinger equations in dimension $N\ge 3$: we investigate the case of unbounded potentials which can not be covered in a standard function space setting. This leads us to use a nonstandard penalization technique, in a borderline Orlicz space framework, which enable us to build up a one parameter family of classical solutions which have finite energy and exhibit, as the parameter goes to zero, a concentrating behavior around some point which is localized.
Evolution by mean curvature flow in sub-Riemannian geometries: A stochastic approach
Nicolas Dirr , Federica Dragoni and  Max von Renesse
2010, 9(2): 307-326 doi: 10.3934/cpaa.2010.9.307 +[Abstract](45) +[PDF](265.7KB)
Abstract:
We study evolution by horizontal mean curvature flow in sub- Riemannian geometries by using stochastic approach to prove the existence of a generalized evolution in these spaces. In particular we show that the value function of suitable family of stochastic control problems solves in the viscosity sense the level set equation for the evolution by horizontal mean curvature flow.
Locally Lipschitz perturbations of bisemigroups
Mohamed Sami ElBialy
2010, 9(2): 327-349 doi: 10.3934/cpaa.2010.9.327 +[Abstract](34) +[PDF](290.3KB)
Abstract:
In this work we study the ill posed semilinear system $\dot{x}= Lx + f(\xi,t), \dot{y}= Ry + g(\xi,t)$, $\xi=(x,y)$, in Banach spaces where $L$ and $R$ are the infinitesimal generators of two $C_o$ semigroups $\{\mathcal{L}(t), t\geq 0\}$ and $\{\mathcal{R}(-t), t\geq 0\}$ respectively. The nonlinearity $h=(f,g)$ is continuous in $t$ and locally Lipschitz continuous in $\xi$ locally uniformly in $t$.
$\cdot $ We show the existence and uniqueness of what we call local dichotomous mild solutions (DMS) that take the form

$x(t) = e^{(t-t_1)L}x_1 + \int_{t_1}^{t} e^{(t-s)L} f(\xi(s), s)ds$

$ y(t)= e^{-(t_2-t)R} y_2 - \int_{t}^{t_2} e^{-(s-t)R} g(\xi(s), s) ds$

$ t_1 < t_2, \qquad t_1\leq t \leq t_2$

for any sufficiently small time interval $[t_1, t_2]$ and any given $\xi :=(x_1, y_2)$ in a sufficiently small neighbourhood.
$\cdot $ We show that in the uniform $C^0$-norm DMSs vary continuously with $[t_1, t_2]$ and Lipschitz-continuously with $\xi $.
$\cdot $ We study the regularity of DMSs under various hypotheses.
$\cdot $ A simple example that leads to a bisemigroup is a semilinear elliptic system that arises when considering solitary waves in an infinite cylinder:

$u_{x x}+\Delta u = f(u), \quad u|_{\Gamma} = 0, \quad\Gamma= \mathbb{R}\times \partial\Omega, \quad (x, y, u)\in \mathbb{R}\times \Omega\times\mathbb{R}^m

where $\Omega$ is a bounded region in $ \mathbb{R}^n$ with $C^2$ boundary and $\Delta$ is the Laplacian in the variable $y\in \Omega$.

Remarks on the orbital instability of standing waves for the wave-Schrödinger system in higher dimensions
Hiroaki Kikuchi
2010, 9(2): 351-364 doi: 10.3934/cpaa.2010.9.351 +[Abstract](37) +[PDF](207.0KB)
Abstract:
In this paper, we study the orbital instability of standing wave for the wave-Schrödinger system in 4 or 5 space dimensions. More precisely, we show that there exists a solution which is not uniformly bounded no matter how the initial data is close to the standing wave.
Non-relativistic global limits to the three dimensional relativistic euler equations with spherical symmetry
Xingwen Hao , Yachun Li and  Zejun Wang
2010, 9(2): 365-386 doi: 10.3934/cpaa.2010.9.365 +[Abstract](105) +[PDF](245.8KB)
Abstract:
As a fundamental and important step to understand the existence and behavior of solution to the multi-dimensional problem, we study in this paper the three dimensional relativistic Euler equations with spherical symmetry. We obtain the non-relativistic global limits of entropy solutions to the Cauchy problem of the spherically symmetric relativistic Euler equations.
Begehr-Hile operator and its applications to some differential equations
Hua Liu and  Zhaosheng Feng
2010, 9(2): 387-395 doi: 10.3934/cpaa.2010.9.387 +[Abstract](37) +[PDF](152.4KB)
Abstract:
In the present paper, we are concerned with the integral hierarchy operator defined by Begehr and Hile in 1997. We show that the Begehr--Hile operator $T_{m,n}$ can be interpreted as the iteration of $T$ and $\bar {T}$ under certain conditions. Applications are also illustrated to some differential equations and singular integral equations in the complex plane.
Fast rate of dead core for fast diffusion equation with strong absorption
Chunlai Mu , Jun Zhou and  Yuhuan Li
2010, 9(2): 397-411 doi: 10.3934/cpaa.2010.9.397 +[Abstract](32) +[PDF](212.9KB)
Abstract:
This paper deals with the dead core problem for the fast diffusion equation with strong absorption and positive boundary values. We prove that the dead core rate is faster than the one given by the corresponding ODE, which is contrary to the known results for the related extinction, quenching and blow up problems. Moreover, we find the dead core rate is quite unstable: the ODE rate can be recovered if the absorption term is replaced by $-a(t,x)u^p$ for a suitable bounded, uniformly positive function $a(t,x)$. As an application of the above results, some new and relatively simple examples of fast blow up are provided, and a phenomenon of strong sensitivity to gradient perturbations is exhibited. Furthermore, the blow up rate is found to depend on a constant in the perturbation term, and sharp estimates are also obtained for the profile of dead core and blow up.
Stability properties of periodic standing waves for the Klein-Gordon-Schrödinger system
Fábio Natali and  Ademir Pastor
2010, 9(2): 413-430 doi: 10.3934/cpaa.2010.9.413 +[Abstract](53) +[PDF](265.5KB)
Abstract:
We study the existence and orbital stability/instability of periodic standing wave solutions for the Klein-Gordon-Schrödinger system with Yukawa and cubic interactions. We prove the existence of periodic waves depending on the Jacobian elliptic functions. For one hand, the approach used to obtain the stability results is the classical Grillakis, Shatah and Strauss theory in the periodic context. On the other hand, to show the instability results we employ a general criterium introduced by Grillakis, which get orbital instability from linear instability.
The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system$^*$
Lihui Guo , Wancheng Sheng and  Tong Zhang
2010, 9(2): 431-458 doi: 10.3934/cpaa.2010.9.431 +[Abstract](133) +[PDF](1276.3KB)
Abstract:
The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system consists of interactions of four planar elementary waves. Different from polytropic gas, all of them are contact discontinuities due to the system is full linear degenerate, i.e., the three eigenvalues of the system are linear degenerate. They include compressive one ($S^\pm$), rarefactive one ($R^\pm$) and slip lines ($J^\pm$). We still call $S^\pm$ as shock and $R^\pm$ as rarefaction wave. In this paper, we study the problem systematically. According to different combination of four elementary waves, we deliver a complete classification to the problem. It contains 14 cases in all. The Riemann solutions are self-similar, and the flow is transonic in self-similar plane $(x/t,y/t)$. The boundaries of the interaction domains are obtained. Solutions in supersonic domains are constructed in no $J$ cases. While in the rest cases, the structure of solutions are conjectured except for the case $2J^++2J^-$. Especially, delta waves and simple waves appear in some cases. The Dirichlet boundary value problems in subsonic domains or the boundary value problems for transonic flow are formed case by case. The domains are convex for two cases, and non-convex for the rest cases. The boundaries of the domains are composed of sonic curves and/or slip lines.
Spherically symmetric Navier-Stokes equations with degenerate viscosity coefficients and vacuum
Wenjun Wang and  Lei Yao
2010, 9(2): 459-481 doi: 10.3934/cpaa.2010.9.459 +[Abstract](44) +[PDF](259.2KB)
Abstract:
This paper is concerned with the free boundary problem for the spherically symmetric compressible Navier-Stokes equations with degenerate viscosity coefficients and vacuum. A local (in time) existence result is established.
Bounds on Sobolev norms for the defocusing nonlinear Schrödinger equation on general flat tori
F. Catoire and  W. M. Wang
2010, 9(2): 483-491 doi: 10.3934/cpaa.2010.9.483 +[Abstract](32) +[PDF](147.2KB)
Abstract:
We prove Strichartz estimates on general flat $d$-torus for arbitrary $d$. Using these estimates, we prove local wellposedness for the cubic nonlinear Schrödinger equations in appropriate Sobolev spaces. In dimensions $2$ and $3$, we prove polynomial bounds on the possible growth of Sobolev norms of smooth solutions.
Stokes-Brinkman transmission problems on Lipschitz and $C^1$ domains in Riemannian manifolds
Mirela Kohr , Cornel Pintea and  Wolfgang L. Wendland
2010, 9(2): 493-537 doi: 10.3934/cpaa.2010.9.493 +[Abstract](30) +[PDF](530.0KB)
Abstract:
The purpose of this paper is to treat some transmission problems for the Stokes-Brinkman-coupled system on Lipschitz or $C^1$ domains in Riemannian manifolds, by using the method of boundary integral equations.
Asymptotic profiles of eigenfunctions for some 1-dimensional linearized eigenvalue problems
Tohru Wakasa and  Shoji Yotsutani
2010, 9(2): 539-561 doi: 10.3934/cpaa.2010.9.539 +[Abstract](31) +[PDF](434.2KB)
Abstract:
We are interested in the asymptotic profiles of all eigenfunctions for 1-dimensional linearized eigenvalue problems to nonlinear boundary value problems with a diffusion coefficient $\varepsilon$. For instance, it seems that they have simple and beautiful properties for sufficiently small $\varepsilon$ in the balanced bistable nonlinearity case. As the first step to give rigorous proofs for the above case, we study the case $f(u)=\sin u$ precisely. We show that two special eigenfunctions completely control the asymptotic profiles of other eigenfunctions.
Semi-positone nonlocal boundary value problems of arbitrary order
J. R. L. Webb and  Gennaro Infante
2010, 9(2): 563-581 doi: 10.3934/cpaa.2010.9.563 +[Abstract](40) +[PDF](267.9KB)
Abstract:
We give a unified approach to the study of existence of multiple positive solutions for semi-positone boundary value problems of arbitrary order. We cover local and nonlocal boundary conditions. Our nonlocal boundary conditions are quite general, they involve positive linear functionals on the space $C[0,1]$, given by Stieltjes integrals. With our general theory, we can, for the first time in semi-positone problems, allow any number of the boundary conditions to be nonlocal.

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