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Communications on Pure & Applied Analysis

2012 , Volume 11 , Issue 5

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Lie groups related to Hörmander operators and Kolmogorov-Fokker-Planck equations
Andrea Bonfiglioli and  Ermanno Lanconelli
2012, 11(5): 1587-1614 doi: 10.3934/cpaa.2012.11.1587 +[Abstract](32) +[PDF](602.5KB)
If $\mathcal{L}=\sum_{j=1}^m X_j^2+X_0$ is a Hörmander partial differential operator in $\mathbb{R}^N$, we give sufficient conditions on the $X_j$'s for the existence of a Lie group structure $\mathbb{G}=(\mathbb{R}^N,*)$, not necessarily nilpotent, such that $\mathcal{L}$ is left invariant on $\mathbb{G}$. We also investigate the existence of a global fundamental solution $\Gamma$ for $\mathcal{L}$, providing results ensuring a suitable left invariance property of $\Gamma$. Examples are given for operators $\mathcal{L}$ to which our results apply: some are new, some appear in recent literature, usually quoted as Kolmogorov-Fokker-Planck type operators.
Positive solutions of a fourth-order boundary value problem involving derivatives of all orders
Zhilin Yang and  Jingxian Sun
2012, 11(5): 1615-1628 doi: 10.3934/cpaa.2012.11.1615 +[Abstract](29) +[PDF](400.5KB)
This paper is mainly concerned with the existence, multiplicity and uniqueness of positive solutions for the fourth-order boundary value problem \begin{eqnarray*} u^{(4)}=f(t,u,u^\prime,-u^{\prime\prime},-u^{\prime\prime\prime}),\\ u(0)=u^\prime(1)=u^{\prime\prime}(0)=u^{\prime\prime\prime}(1)=0, \end{eqnarray*} where $f\in C([0,1]\times\mathbb R_+^4,\mathbb R_+)(\mathbb R_+:=[0,\infty))$. Based on a priori estimates achieved by utilizing some integral identities and inequalities, we use fixed point index theory to prove the existence, multiplicity and uniqueness of positive solutions for the above problem. Finally, as a byproduct, our main results are applied to establish the existence, multiplicity and uniqueness of symmetric positive solutions for the fourth order Lidstone problem.
Improved Caffarelli-Kohn-Nirenberg and trace inequalities for radial functions
Pablo L. De Nápoli , Irene Drelichman and  Ricardo G. Durán
2012, 11(5): 1629-1642 doi: 10.3934/cpaa.2012.11.1629 +[Abstract](33) +[PDF](374.6KB)
We show that Caffarelli-Kohn-Nirenberg first order interpolation inequalities as well as weighted trace inequalities in $\mathbb{R}^n \times \mathbb{R}_+$ admit a better range of power weights if we restrict ourselves to the space of radially symmetric functions.
Regularizing rate estimates for mild solutions of the incompressible Magneto-hydrodynamic system
Qiao Liu and  Shangbin Cui
2012, 11(5): 1643-1660 doi: 10.3934/cpaa.2012.11.1643 +[Abstract](28) +[PDF](476.7KB)
We establish some regularizing rate estimates for mild solutions of the magneto-hydrodynamic system (MHD). These estimations ensure that there exist positive constants $K_1$ and $K_2$ such that for any $\beta\in\mathbb{Z}^{n}_{+}$ and any $t\in (0,T^\ast)$, where $T^\ast$ is the life-span of the solution, we have $\| (\partial_{x}^{\beta}u(t),\partial_{x}^{\beta}b(t))\|_{q}\leq K_{1}(K_{2}|\beta|)^{|\beta|}t^{-\frac{|\beta|}{2} -\frac{n}{2}(\frac{1}{n}-\frac{1}{q})}$. Spatial analyticity of the solution and temporal decay of global solutions are direct consequences of such estimates.
A class of large amplitude oscillating solutions for three dimensional Euler equations
Christophe Cheverry and  Mekki Houbad
2012, 11(5): 1661-1697 doi: 10.3934/cpaa.2012.11.1661 +[Abstract](21) +[PDF](725.1KB)
In this article, we construct large amplitude oscillating waves, noted $ (u ^ {\varepsilon}) _ {\varepsilon} $ where $ \varepsilon \in] 0,1] $ is a parameter going to zero, which are devised to be local solutions on some open domain of the time-space $ R_+ \times R^3$ of both the three dimensional Burger equations (without source term), the compressible Euler equations (with some constant pressure) and the incompressible Euler equations (without pressure). The functions $ u^\varepsilon (t,x) $ are characterized by the fact that the corresponding Jacobian matrices $ D_x u^\varepsilon (t,x) $ are nilpotent of rank one or two. Our purpose is to describe the interesting geometrical features of the expressions $ u^\varepsilon (t,x) $ which can be obtained by this way.
Coexistence and extinction in the Volterra-Lotka competition model with nonlocal dispersal
Georg Hetzer , Tung Nguyen and  Wenxian Shen
2012, 11(5): 1699-1722 doi: 10.3934/cpaa.2012.11.1699 +[Abstract](61) +[PDF](455.2KB)
Coexistence and extinction for two species Volterra-Lotka competition systems with nonlocal dispersal are investigated in this paper. Sufficient conditions in terms of diffusion, reproduction, self-limitation, and competition rates are established for existence, uniqueness, and stability of coexistence states as well as for the extinction of one species. The focus is on environments with hostile surroundings. In this case, our results correspond to those for random dispersal under Dirichlet boundary conditions. Similar results hold for environments with non-flux boundary and for periodic environments, which correspond to those for random dispersal under Neumann boundary conditions and periodic boundary conditions, respectively.
Generalized and weighted Strichartz estimates
Jin-Cheng Jiang , Chengbo Wang and  Xin Yu
2012, 11(5): 1723-1752 doi: 10.3934/cpaa.2012.11.1723 +[Abstract](43) +[PDF](632.2KB)
In this paper, we explore the relations between different kinds of Strichartz estimates and give new estimates in Euclidean space $\mathbb{R}^n$. In particular, we prove the generalized and weighted Strichartz estimates for a large class of dispersive operators including the Schrödinger and wave equation. As a sample application of these new estimates, we are able to prove the Strauss conjecture with low regularity for dimension $2$ and $3$.
Instability of coupled systems with delay
Reinhard Racke
2012, 11(5): 1753-1773 doi: 10.3934/cpaa.2012.11.1753 +[Abstract](40) +[PDF](502.5KB)
We consider linear initial-boundary value problems that are a coupling like second-order thermoelasticity, or the thermoelastic plate equation or its generalization (the $\alpha$-$\beta$-system introduced in [1, 26]). Now, there is a delay term given in part of the coupled system, and we demonstrate that the expected inherent damping will not prevent the system from not being stable; indeed, the systems will shown to be ill-posed: a sequence of bounded initial data may lead to exploding solutions (at any fixed time).
Stability of stationary waves for full Euler-Poisson system in multi-dimensional space
Ming Mei and  Yong Wang
2012, 11(5): 1775-1807 doi: 10.3934/cpaa.2012.11.1775 +[Abstract](33) +[PDF](545.4KB)
This paper is concerned with the nonisentropic unipolar hydrodynamic model of semiconductors in the form of multi-dimensional full Euler-Poisson system. By heuristically analyzing the exact gaps between the original solutions and the stationary waves at far fields, we ingeniously construct some correction functions to delete these gaps, and then prove the $L^\infty$-stability of stationary waves with an exponential decay rate in 1-D case. Furthermore, based on the 1-D convergence result, we show the stability of planar stationary waves with also some exponential decay rate in $m$-D case.
On the Lagrangian averaged Euler equations: local well-posedness and blow-up criterion
Xinwei Yu and  Zhichun Zhai
2012, 11(5): 1809-1823 doi: 10.3934/cpaa.2012.11.1809 +[Abstract](47) +[PDF](402.9KB)
In this article we study local and global well-posedness of the Lagrangian Averaged Euler equations. We show local well-posedness in Triebel-Lizorkin spaces and further prove a Beale-Kato-Majda type necessary and sufficient condition for global existence involving the stream function. We also establish new sufficient conditions for global existence in terms of mixed Lebesgue norms of the generalized Clebsch variables.
Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case
Zhiming Guo , Zhi-Chun Yang and  Xingfu Zou
2012, 11(5): 1825-1838 doi: 10.3934/cpaa.2012.11.1825 +[Abstract](40) +[PDF](355.7KB)
This paper deals with a class of non-local second order differential equations subject to the homogeneous Dirichlet boundary condition. The main concern is positive steady state of the boundary value problem, especially when the equation does not enjoy the monotonicity. Nonexistence, existence and uniqueness of positive steady state for the problem are addressed. In particular, developed is a technique that combines the method of super-sub solutions and the estimation of integral kernels, which enables us to obtain some sufficient conditions for the existence and uniqueness of a positive steady state. Two examples are given to illustrate the obtained results.
Collocation methods for differential equations with piecewise linear delays
Hui Liang and  Hermann Brunner
2012, 11(5): 1839-1857 doi: 10.3934/cpaa.2012.11.1839 +[Abstract](30) +[PDF](420.1KB)
After analyzing the regularity of solutions to delay differential equations (DDEs) with piecewise continuous (linear) non-vanishing delays, we describe collocation schemes using continuous piecewise polynomials for their numerical solution. We show that for carefully designed meshes these collocation solutions exhibit optimal orders of global and local superconvergence analogous to the ones for DDEs with constant delays. Numerical experiments illustrate the theoretical superconvergence results.
On a singular Hamiltonian elliptic systems involving critical growth in dimension two
Manassés de Souza
2012, 11(5): 1859-1874 doi: 10.3934/cpaa.2012.11.1859 +[Abstract](31) +[PDF](417.6KB)
In this paper we study the existence of nontrivial solutions for the strongly indefinite elliptic system \begin{eqnarray*} -\Delta u + b(x) u = \frac{g(v)}{|x|^\alpha}, v > 0 in R^2, \\ -\Delta v + b(x) v = \frac{f(u)}{|x|^\beta}, u > 0 in R^2, \end{eqnarray*} where $\alpha, \beta \in [0,2)$, $b: \mathbb{R}^2\rightarrow \mathbb{R}$ is a continuous positive potential bounded away from zero and which can be ``large" at the infinity and the functions $f: \mathbb{R}\rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ behaves like $\exp(\gamma s^2)$ when $|s|\rightarrow+\infty$ for some $\gamma >0$.
Elliptic equations having a singular quadratic gradient term and a changing sign datum
Daniela Giachetti , Francesco Petitta and  Sergio Segura de León
2012, 11(5): 1875-1895 doi: 10.3934/cpaa.2012.11.1875 +[Abstract](42) +[PDF](461.5KB)
In this paper we study a singular elliptic problem whose model is \begin{eqnarray*} - \Delta u= \frac{|\nabla u|^2}{|u|^\theta}+f(x), in \Omega\\ u = 0, on \partial \Omega; \end{eqnarray*} where $\theta\in (0,1)$ and $f \in L^m (\Omega)$, with $m\geq \frac{N}{2}$. We do not assume any sign condition on the lower order term, nor assume the datum $f$ has a constant sign. We carefully define the meaning of solution to this problem giving sense to the gradient term where $u=0$, and prove the existence of such a solution. We also discuss related questions as the existence of solutions when the datum $f$ is less regular or the boundedness of the solutions when the datum $f \in L^m (\Omega)$ with $m> \frac{N}{2}$.
Local maximum principle for $L^p$-viscosity solutions of fully nonlinear elliptic PDEs with unbounded coefficients
Shigeaki Koike and  Andrzej Świech
2012, 11(5): 1897-1910 doi: 10.3934/cpaa.2012.11.1897 +[Abstract](43) +[PDF](410.8KB)
We establish local maximum principle for $L^p$-viscosity solutions of fully nonlinear elliptic partial differential equations with unbounded ingredients.
On the characteristic curvature operator
Vittorio Martino
2012, 11(5): 1911-1922 doi: 10.3934/cpaa.2012.11.1911 +[Abstract](34) +[PDF](352.3KB)
We introduce the Characteristic Curvature as the curvature of the trajectories of the Hamiltonian vector field with respect to the normal direction to the isoenergetic surfaces; by using the Second Fundamental Form we relate it to the Classical and Levi Mean Curvature. Then we prove existence and uniqueness of viscosity solutions for the related Dirichlet problem and we show the Lipschitz regularity of the solutions under suitable hypotheses. At the end we show that neither Strong Comparison Principle nor Hopf Lemma hold for the Characteristic Curvature Operator.
Global well-posedness and scattering for Skyrme wave maps
Dan-Andrei Geba , Kenji Nakanishi and  Sarada G. Rajeev
2012, 11(5): 1923-1933 doi: 10.3934/cpaa.2012.11.1923 +[Abstract](32) +[PDF](383.7KB)
We study equivariant solutions for two models ([13]-[15], [1]) arising in high energy physics, which are generalizations of the wave maps theory (i.e., the classical nonlinear $\sigma$ model) in 3 + 1 dimensions. We prove global existence and scattering for small initial data in critical Sobolev-Besov spaces.
Symmetries and blow-up phenomena for a Dirichlet problem with a large parameter
Pierpaolo Esposito and  Maristella Petralla
2012, 11(5): 1935-1957 doi: 10.3934/cpaa.2012.11.1935 +[Abstract](31) +[PDF](517.2KB)
\noindent For the Dirichlet problem $-\Delta u+\lambda V(x) u=u^p$ in $\Omega \subset \mathbb R^N$, $N\geq 3$, in the regime $\lambda \to +\infty$ we aim to give a description of the blow-up mechanism. For solutions with symmetries an uniform bound on the ``invariant" Morse index provides a localization of the blow-up orbits in terms of c.p.'s of a suitable modified potential. The main difficulty here is related to the presence of fixed points for the underlying group action.
Solvability and asymptotic analysis of a generalization of the Caginalp phase field system
Giacomo Canevari and  Pierluigi Colli
2012, 11(5): 1959-1982 doi: 10.3934/cpaa.2012.11.1959 +[Abstract](28) +[PDF](494.5KB)
We study a diffusion model of phase field type, which consists of a system of two partial differential equations involving as variables the thermal displacement, that is basically the time integration of temperature, and the order parameter. Our analysis covers the case of a non-smooth (maximal monotone) graph along with a smooth anti-monotone function in the phase equation. Thus, the system turns out a generalization of the well-known Caginalp phase field model for phase transitions when including a diffusive term for the thermal displacement in the balance equation. Systems of this kind have been extensively studied by Miranville and Quintanilla. We prove existence and uniqueness of a weak solution to the initial-boundary value problem, as well as various regularity results ensuring that the solution is strong and with bounded components. Then we investigate the asymptotic behaviour of the solutions as the coefficient of the diffusive term for the thermal displacement tends to zero and prove convergence to the Caginalp phase field system as well as error estimates for the difference of the solutions.
A pointwise gradient estimate for solutions of singular and degenerate pde's in possibly unbounded domains with nonnegative mean curvature
Diego Castellaneta , Alberto Farina and  Enrico Valdinoci
2012, 11(5): 1983-2003 doi: 10.3934/cpaa.2012.11.1983 +[Abstract](71) +[PDF](517.6KB)
We consider a singular or degenerate elliptic problem in a proper domain and we prove a gradient bound and some symmetry results.
Double resonance for Dirichlet problems with unbounded indefinite potential and combined nonlinearities
Shouchuan Hu and  Nikolaos S. Papageorgiou
2012, 11(5): 2005-2021 doi: 10.3934/cpaa.2012.11.2005 +[Abstract](36) +[PDF](408.4KB)
We study a semilinear parametric Dirichlet equation with an indefinite and unbounded potential. The reaction is the sum of a sublinear (concave) term and of an asymptotically linear resonant term. The resonance is with respect to any nonprincipal nonnegative eigenvalue of the differential operator. Using variational methods based on the critical point theory and Morse theory (critical groups), we show that when the parameter $\lambda>0$ is small, the problem has at least three nontrivial smooth solutions.
Evaluating cyclicity of cubic systems with algorithms of computational algebra
Viktor Levandovskyy , Gerhard Pfister and  Valery G. Romanovski
2012, 11(5): 2023-2035 doi: 10.3934/cpaa.2012.11.2023 +[Abstract](39) +[PDF](423.7KB)
We describe an algorithmic approach to studying limit cycle bifurcations in a neighborhood of an elementary center or focus of a polynomial system. Using it we obtain an upper bound for cyclicity of a family of cubic systems. Then using a theorem by Christopher [3] we study bifurcation of limit cycles from each component of the center variety. We obtain also the sharp bound for the cyclicity of a generic time-reversible cubic system.
Smooth attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities
Varga K. Kalantarov and  Sergey Zelik
2012, 11(5): 2037-2054 doi: 10.3934/cpaa.2012.11.2037 +[Abstract](33) +[PDF](456.4KB)
We prove the existence of regular dissipative solutions and global attractors for the 3D Brinkmann-Forchheimer equations with a nonlinearity of arbitrary polynomial growth rate. In order to obtain this result, we prove the maximal regularity estimate for the corresponding semi-linear stationary Stokes problem using some modification of the nonlinear localization technique. The applications of our results to the Brinkmann-Forchheimer equation with the Navier-Stokes inertial term are also considered.
On singular Navier-Stokes equations and irreversible phase transitions
José Luiz Boldrini , Luís H. de Miranda and  Gabriela Planas
2012, 11(5): 2055-2078 doi: 10.3934/cpaa.2012.11.2055 +[Abstract](34) +[PDF](857.0KB)
We analyze a singular system of partial differential equations corresponding to a model for the evolution of an irreversible solidification of certain pure materials by taking into account the effects of fluid flow in the molten regions. The model consists of a system of highly non-linear free-boundary parabolic equations and includes: a heat equation, a doubly nonlinear inclusion for the phase change and Navier-Stokes equations singularly perturbed by a Carman-Kozeny type term to take care of the flow in the mushy region and a Boussinesq term for the buoyancy forces due to thermal differences. Our approach to show existence of generalized solutions of this system involves time-discretization, a suitable regularization procedure and fixed point arguments.
An abstract existence theorem for parabolic systems
Hans Wilhelm Alt
2012, 11(5): 2079-2123 doi: 10.3934/cpaa.2012.11.2079 +[Abstract](52) +[PDF](495.6KB)
In this paper we prove an abstract existence theorem which can be applied to solve parabolic problems in a wide range of applications. It also applies to parabolic variational inequalities. The abstract theorem is based on a Gelfand triple $(V,H,V^*)$, where the standard realization for parabolic systems of second order is $(W^{1, 2}(\Omega),L^2(\Omega), W^{1,2}(\Omega)^*)$. But also realizations to other problems are possible, for example, to fourth order systems.
In all applications to boundary value problems the set $M\subset V$ is an affine subspace, whereas for variational inequalities the constraint $M$ is a closed convex set.
The proof is purely abstract and new.
The corresponding compactness theorem is based on [5].
The present paper is suitable for lectures, since it relays on the corresponding abstract elliptic theory.
Macrotransport in nonlinear, reactive, shear flows
Eric S. Wright
2012, 11(5): 2125-2146 doi: 10.3934/cpaa.2012.11.2125 +[Abstract](26) +[PDF](417.9KB)
In 1953, G.I. Taylor published his paper concerning the transport of a contaminant in a fluid flowing through a narrow tube. He demonstrated that the transverse variations in the fluid's velocity field and the transverse diffusion of the solute interact to yield an effective longitudinal mixing mechanism for the transverse average of the solute. This mechanism has been dubbed ``Taylor Dispersion.'' Since then, many related studies have surfaced. However, few of these addressed the effects of nonlinear chemical reactions upon a system of solutes undergoing Taylor Dispersion. In this paper, I present a mathematical model for the evolution of the transverse averages of reacting solutes in a fluid flowing down a pipe of arbitrary cross-section. The technique for deriving the model is a generalization of an approach by introduced by P.C. Fife. The key outcome is that while one still finds an effective mechanism for longitudinal mixing, there is also a effective mechanism for nonlinear advection.
Blow-up for the heat equation with a general memory boundary condition
Keng Deng and  Zhihua Dong
2012, 11(5): 2147-2156 doi: 10.3934/cpaa.2012.11.2147 +[Abstract](42) +[PDF](320.8KB)
In this paper, we study the long-time behavior of nonnegative solutions of the heat equation with a general memory boundary condition. We first present conditions on the memory term for finite time blow-up. We then establish global existence results through both analytical and numerical methods. Finally, we show that under certain conditions blow-up occurs only on the boundary.
Neuronal Fiber--tracking via optimal mass transportation
A. Daducci , A. Marigonda , G. Orlandi and  R. Posenato
2012, 11(5): 2157-2177 doi: 10.3934/cpaa.2012.11.2157 +[Abstract](24) +[PDF](1071.6KB)
Diffusion Magnetic Resonance Imaging (MRI) is used to (non-invasively) study neuronal fibers in the brain white matter. Reconstructing fiber paths from such data (tractography problem) is relevant in particular to study the connectivity between two given cerebral regions. Fiber-tracking models rely on how water molecules diffusion is represented in each MRI voxel. The Diffusion Spectrum Imaging (DSI) technique represents the diffusion as a probability density function (DDF) defined on a set of predefined directions inside each voxel. DSI is able to describe complex tissue configurations (compared e.g. with Diffusion Tensor Imaging), but ignores the actual density of fibers forming bundle trajectories among adjacent voxels, preventing any evaluation of the real physical dimension of these fiber bundles.
By considering the fiber paths between two given areas as geodesics of a suitable well-posed optimal control problem (related to optimal mass transportation) which takes into account the whole information given by the DDF, we are able to provide a quantitative criterion to estimate the connectivity between two given cerebral regions, and to recover the actual distribution of neuronal fibers between them.

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