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Communications on Pure & Applied Analysis

2017 , Volume 16 , Issue 1

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Global weak solution to 3D compressible flows with density-dependent viscosity and free boundary
Mei Wang, Zilai Li and Zhenhua Guo
2017, 16(1): 1-24 doi: 10.3934/cpaa.2017001 +[Abstract](129) +[HTML](20) +[PDF](472.41KB)

In this paper, we obtain the global weak solution to the 3D spherically symmetric compressible isentropic Navier-Stokes equations with arbitrarily large, vacuum data and free boundary when the shear viscosity $\mu$ is a positive constant and the bulk viscosity $\lambda(\rho)=\rho^\beta$ with $\beta>0$. The analysis of the upper and lower bound of the density is based on some well-chosen functionals. In addition, the free boundary can be shown to expand outward at an algebraic rate in time.

Invariant tori of a nonlinear Schrödinger equation with quasi-periodically unbounded perturbations
Jie Liu and Jianguo Si
2017, 16(1): 25-68 doi: 10.3934/cpaa.2017002 +[Abstract](123) +[HTML](7) +[PDF](690.24KB)

This paper is concerned with the derivative nonlinear Schrödinger equation with quasi-periodic forcing under periodic boundary conditions

Assume that the frequency vector $\beta$ is co-linear with a fixed Diophantine vector $\bar{\beta}\in \mathbb{R}.{m}$, that is, $\beta=\lambda \bar{\beta}$, $\lambda \in [1/2, 3/2]$. We show that above equation possesses a Cantorian branch of invariant $n$--tori and exists many smooth quasi-periodic solutions with $(m+n)$ non-resonance frequencies $(\lambda\bar{\beta}, \omega_{\ast})$. The proof is based on a Kolmogorov--Arnold--Moser (KAM) iterative procedure for quasi-periodically unbounded vector fields and partial Birkhoff normal form.

An isomorphism theorem for parabolic problems in Hörmander spaces and its applications
Valerii Los, Vladimir A. Mikhailets and Aleksandr A. Murach
2017, 16(1): 69-98 doi: 10.3934/cpaa.2017003 +[Abstract](87) +[HTML](5) +[PDF](572.06KB)

We investigate a general parabolic initial-boundary value problem with zero Cauchy data in some anisotropic Hörmander inner product spaces. We prove that the operators corresponding to this problem are isomorphisms between appropriate Hörmander spaces. As an application of this result, we establish a theorem on the local increase in regularity of solutions to the problem. We also obtain new sufficient conditions under which the generalized derivatives, of a given order, of the solutions should be continuous.

Existence and symmetry result for fractional p-Laplacian in $\mathbb{R}^{n}$
2017, 16(1): 99-114 doi: 10.3934/cpaa.2017004 +[Abstract](272) +[HTML](15) +[PDF](412.83KB)

In this article we are interested in the following fractional $p$-Laplacian equation in $\mathbb{R}^n$

where $p\geq 2$, $0 < s < 1$, $n\geq 2$ and $f$ is $p$-superlinear. By using mountain pass theorem with Cerami condition we prove the existence of nontrivial solution. Furthermore, we show that this solution is radially simmetry.

A complete classification of ground-states for a coupled nonlinear Schrödinger system
Chuangye Liu and Zhi-Qiang Wang
2017, 16(1): 115-130 doi: 10.3934/cpaa.2017005 +[Abstract](170) +[HTML](4) +[PDF](413.1KB)

In this paper, we establish the existence of nontrivial ground-state solutions for a coupled nonlinear Schrödinger system

where $n=1, 2, 3, m\geq 2$ and $b_{ij}$ are positive constants satisfying $b_{ij}=b_{ji}.$ By nontrivial we mean a solution that has all components non-zero. Due to possible systems collapsing it is important to classify ground state solutions. For $m=3$, we get a complete picture that describes whether nontrivial ground-state solutions exist or not for all possible cases according to some algebraic conditions of the matrix $B = (b_{ij})$. In particular, there is a nontrivial ground-state solution provided that all coupling constants $b_{ij}, i\neq j$ are sufficiently large as opposed to cases in which any ground-state solution has at least a zero component when $b_{ij}, i\neq j$ are all sufficiently small. Moreover, we prove that any ground-state solution is synchronized when matrix $B=(b_{ij})$ is positive semi-definite.

Asymptotic behavior and uniqueness of traveling wave fronts in a delayed nonlocal dispersal competitive system
Kun Li, Jianhua Huang and Xiong Li
2017, 16(1): 131-150 doi: 10.3934/cpaa.2017006 +[Abstract](168) +[HTML](13) +[PDF](485.21KB)

This paper is concerned with the asymptotic behavior and uniqueness of traveling wave fronts connecting two half-positive equilibria in a delayed nonlocal dispersal competitive system. We first prove the existence results by applying abstract theories. And then, we show that the traveling wave fronts decay exponentially at both infinities. At last, the strict monotonicity and uniqueness of traveling wave fronts are obtained by using the sliding method in the absent of intraspecific competitive delays. Based on the uniqueness, the exact decay rate of the stronger competitor is established under certain conditions.

A comparison between random and stochastic modeling for a SIR model
Tomás Caraballo and Renato Colucci
2017, 16(1): 151-162 doi: 10.3934/cpaa.2017007 +[Abstract](238) +[HTML](19) +[PDF](357.82KB)

In this article, a random and a stochastic version of a SIR nonautonomous model previously introduced in [19] is considered. In particular, the existence of a random attractor is proved for the random model and the persistence of the disease is analyzed as well. In the stochastic case, we consider some environmental effect on the model, in fact, we assume that one of the coefficients of the system is affected by some stochastic perturbation, and analyze the asymptotic behavior of the solutions. The paper is concluded with a comparison between the two different modeling strategies.

Resonant problems for fractional Laplacian
Yutong Chen and Jiabao Su
2017, 16(1): 163-188 doi: 10.3934/cpaa.2017008 +[Abstract](148) +[HTML](1) +[PDF](493.3KB)

In this paper we consider the following fractional Laplacian equation

where $ s\in (0, 1)$ is fixed, $\Omega$ is an open bounded set of $\mathbb{R}.N$, $N > 2s$, with smooth boundary, $(-\Delta).s$ is the fractional Laplace operator. By Morse theory we obtain the existence of nontrivial weak solutions when the problem is resonant at both infinity and zero.

Continuity of cost functional and optimal feedback controls for the stochastic Navier Stokes equation in 2D
Kerem Uǧurlu
2017, 16(1): 189-208 doi: 10.3934/cpaa.2017009 +[Abstract](160) +[HTML](2) +[PDF](466.67KB)

We show the continuity of a specific cost functional $J(\phi) =\mathbb{E} \sup_{ t \in [0, T]}(\varphi(\mathcal{L}[t, u_\phi(t), \phi(t)]))$ of the SNSE in 2D on an open bounded nonperiodic domain $\mathcal{O}$ with respect to a special set of feedback controls $\{\phi_n\}_{n \geq 0}$, where $\varphi(x) =\log(1 + x)^{1-\epsilon}$ with $0 < \epsilon < 1$.

Zero dissipation limit to rarefaction wave with vacuum for a one-dimensional compressible non-Newtonian fluid
Li Fang and Zhenhua Guo
2017, 16(1): 209-242 doi: 10.3934/cpaa.2017010 +[Abstract](144) +[HTML](1) +[PDF](525.99KB)

In this paper, we study the zero dissipation limit toward rarefaction waves for solutions to a one-dimensional compressible non-Newtonian fluid for general initial data, whose far fields are connected by a rarefaction wave to the corresponding Euler equations with one end state being vacuum. Given a rarefaction wave with one-side vacuum state to the compressible Euler equations, we construct a sequence of solutions to the one-dimensional compressible non-Newtonian fluid which converge to the above rarefaction wave with vacuum as the viscosity coefficient $\epsilon$ tends to zero. Moreover, the uniform convergence rate is obtained, based on one fact that the viscosity constant can control the degeneracies caused by the vacuum in rarefaction waves and another fact that the energy estimates are obtained under some a priori assumption.

Trudinger-Moser type inequality and existence of solution for perturbed non-local elliptic operators with exponential nonlinearity
Anouar Bahrouni
2017, 16(1): 243-252 doi: 10.3934/cpaa.2017011 +[Abstract](156) +[HTML](1) +[PDF](400.8KB)

In this paper we consider the following perturbed nonlocal problem with exponential nonlinearity

\begin{document}$\begin{cases}-\mathcal{L}_{K}u+ \left|u\right|^{p-2}u+h(u)= f \ \ \ \ \ \mbox{in} \ \ \Omega,\\u=0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{in}\ \ \mathbb{R}^{N}\setminus \Omega,\end{cases}$\end{document}

where $s\in (0, 1)$, $N=ps$, $p\geq 2$ and $f\in L.{\infty}(\mathbb{R}^{N})$. First, we generalize a suitable Trudinger-Moser inequality to a fractional functional space. Then, using the Ekeland's variational principle, we prove the existence of a solution of problem (1).

Higher order asymptotic for Burgers equation and Adhesion model
Engu Satynarayana, Manas R. Sahoo and Manasa M
2017, 16(1): 253-272 doi: 10.3934/cpaa.2017012 +[Abstract](137) +[HTML](6) +[PDF](456.38KB)

This paper is focused on the study of the large time asymptotic for solutions to the viscous Burgers equation and also to the adhesion model via heat equation. Using generalization of the truncated moment problem to a complex measure space, we construct asymptotic N-wave approximate solution to the heat equation subject to the initial data whose moments exist upto the order $2n+m$ and $i$-th order moment vanishes, for $i=0, 1, 2\dots m-1$. We provide a different proof for a theorem given by Duoandikoetxea and Zuazua [3], which plays a crucial role in error estimations. In addition to this we describe a simple way to construct an initial data in Schwartz class whose $m$ moments are equal to the $m$ moments of given initial data.

Quasineutral limit for the quantum Navier-Stokes-Poisson equations
Min Li, Xueke Pu and Shu Wang
2017, 16(1): 273-294 doi: 10.3934/cpaa.2017013 +[Abstract](132) +[HTML](2) +[PDF](446.14KB)

In this paper, we study the quasineutral limit and asymptotic behaviors for the quantum Navier-Stokes-Possion equation. We apply a formal expansion according to Debye length and derive the neutral incompressible Navier-Stokes equation. To establish this limit mathematically rigorously, we derive uniform (in Debye length) estimates for the remainders, for well-prepared initial data. It is demonstrated that the quantum effect do play important roles in the estimates and the norm introduced depends on the Planck constant $\hbar>0$.

The vanishing pressure limits of Riemann solutions to the Chaplygin gas equations with a source term
Lihui Guo, Tong Li and Gan Yin
2017, 16(1): 295-310 doi: 10.3934/cpaa.2017014 +[Abstract](135) +[HTML](83) +[PDF](528.23KB)

We study the vanishing pressure limits of Riemann solutions to the Chaplygin gas equations with a source term. The phenomena of concentration and cavitation to Chaplygin gas equations with a friction term are identified and analyzed as the pressure vanishes. Due to the influence of source term, the Riemann solutions are no longer self-similar. When the pressure vanishes, the Riemann solutions to the inhomogeneous Chaplygin gas equations converge to the Riemann solutions to the pressureless gas dynamics model with a friction term.

Struwe's decomposition for a polyharmonic operator on a compact Riemannian manifold with or without boundary
Saikat Mazumdar
2017, 16(1): 311-330 doi: 10.3934/cpaa.2017015 +[Abstract](136) +[HTML](2) +[PDF](461.8KB)

Given a high-order elliptic operator on a compact manifold with or without boundary, we perform the decomposition of Palais-Smale sequences for a nonlinear problem as a sum of bubbles. This is a generalization of the celebrated 1984 result of Struwe [19]. Unlike the case of second-order operators, bubbles close to the boundary might appear. Our result includes the case of a smooth bounded domain of $\mathbb{R}^{n}$.

Periodic solutions for nonlocal fractional equations
Vincenzo Ambrosio and Giovanni Molica Bisci
2017, 16(1): 331-344 doi: 10.3934/cpaa.2017016 +[Abstract](141) +[HTML](1) +[PDF](420.42KB)

The purpose of this paper is to study the existence of (weak) periodic solutions for nonlocal fractional equations with periodic boundary conditions. These equations have a variational structure and, by applying a critical point result coming out from a classical Pucci-Serrin theorem in addition to a local minimum result for differentiable functionals due to Ricceri, we are able to prove the existence of at least two periodic solutions for the treated problems. As far as we know, all these results are new.

Global well posedness for the ghost effect system
Bilal Al Taki
2017, 16(1): 345-368 doi: 10.3934/cpaa.2017017 +[Abstract](143) +[HTML](1) +[PDF](474.4KB)

The aim of this paper is to discuss the issue of global existence of weak solutions of the so called ghost effect system which has been derived recently in [C. D. LEVERMORE, W. SUN, K. TRIVISA, SIAM J. Math. Anal. 2012]. We extend the local existence of solutions proved in [C.D. LEVERMORE, W. SUN, K. TRIVISA, Indiana Univ. J., 2011] to a global existence result. The key tool in this paper is a new functional inequality inspired of what proposed in [A. JÜNGEL, D. MATTHES, SIAM J. Math. Anal., 2008]. Such an inequality being adapted in [D. BRESCH, A. VASSEUR, C. YU, 2016] to be useful for compressible Navier-Stokes equations with degenerate viscosities. Our strategy to prove the global existence of solution builds upon the framework developed in [D. BRESCH, V. GIOVANGILI, E. ZATORSKA, J. Math. Pures Appl., 2015] for low Mach number system.

Erratum: "On the nonlocal Cahn-Hilliard-Brinkman and Cahn-Hilliard-Hele-Shaw systems" [Comm. Pure Appl. Anal. 15 (2016), 299--317]
FRANCESCO DELLA PORTA and Maurizio Grasselli
2017, 16(1): 369-372 doi: 10.3934/cpaa.2017018 +[Abstract](121) +[HTML](4) +[PDF](277.9KB)

2016  Impact Factor: 0.801




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