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Communications on Pure & Applied Analysis

2011 , Volume 10 , Issue 4

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On the structure of solutions of nonlinear hyperbolic systems of conservation laws
Gui-Qiang Chen and Monica Torres
2011, 10(4): 1011-1036 doi: 10.3934/cpaa.2011.10.1011 +[Abstract](391) +[PDF](541.6KB)
Abstract:
We are concerned with entropy solutions $u$ in $L^\infty$ of nonlinear hyperbolic systems of conservation laws. It is shown that, given any entropy function $\eta$ and any hyperplane $t=const.$, if $u$ satisfies a vanishing mean oscillation property on the half balls, then $\eta(u)$ has a trace $H^d$-almost everywhere on the hyperplane. For the general case, given any set $E$ of finite perimeter and its inner unit normal $\nu: \partial^*E \to S^d$ and assuming the vanishing mean oscillation property of $u$ on the half balls, we show that the weak trace of the vector field $(\eta(u), q(u))$, defined in Chen-Torres-Ziemer [9], satisfies a stronger property for any entropy pair $(\eta, q)$. We then introduce an approach to analyze the structure of bounded entropy solutions for the isentropic Euler equations.
Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents
Yimin Zhang, Youjun Wang and Yaotian Shen
2011, 10(4): 1037-1054 doi: 10.3934/cpaa.2011.10.1037 +[Abstract](493) +[PDF](403.3KB)
Abstract:
By using a change of variable, the quasilinear Schrödinger equation is reduced to semilinear elliptic equation. Then, Mountain Pass theorem without $(PS)_c$ condition in a suitable Orlicz space is employed to prove the existence of positive standing wave solutions for a class of quasilinear Schrödinger equations involving critical Sobolev-Hardy exponents.
Nonlinear Neumann equations driven by a nonhomogeneous differential operator
Shouchuan Hu and Nikolaos S. Papageorgiou
2011, 10(4): 1055-1078 doi: 10.3934/cpaa.2011.10.1055 +[Abstract](467) +[PDF](461.5KB)
Abstract:
We consider a nonlinear Neumann problem driven by a nonhomogeneous nonlinear differential operator and with a reaction which is $(p-1)$-superlinear without necessarily satisfying the Ambrosetti-Rabinowitz condition. A particular case of our differential operator is the $p$-Laplacian. By combining variational methods based on critical point theory with truncation techniques and Morse theory, we show that the problem has at least three nontrivial smooth solutions, two of which have constant sign (one positive and the other negative).
Nonlinear hyperbolic-elliptic systems in the bounded domain
N. V. Chemetov
2011, 10(4): 1079-1096 doi: 10.3934/cpaa.2011.10.1079 +[Abstract](346) +[PDF](453.0KB)
Abstract:
In the article we study a hyperbolic-elliptic system of PDE. The system can describe two different physical phenomena: 1st one is the motion of magnetic vortices in the II-type superconductor and 2nd one is the collective motion of cells. Motivated by real physics, we consider this system with boundary conditions, describing the flux of vortices (and cells, respectively) through the boundary of the domain. We prove the global solvability of this problem. To show the solvability result we use a "viscous" parabolic-elliptic system. Since the viscous solutions do not have a compactness property, we justify the limit transition on a vanishing viscosity, using a kinetic formulation of our problem. As the final result of all considerations we have solved a very important question related with a so-called "boundary layer problem", showing the strong convergence of the viscous solutions to the solution of our hyperbolic-elliptic system.
The Cauchy-Kovalevskaya extension theorem in discrete Clifford analysis
Hilde De Ridder, Hennie De Schepper and Frank Sommen
2011, 10(4): 1097-1109 doi: 10.3934/cpaa.2011.10.1097 +[Abstract](305) +[PDF](543.0KB)
Abstract:
Discrete Clifford analysis is a higher dimensional discrete function theory based on skew Weyl relations. It is centered around the study of Clifford algebra valued null solutions, called discrete monogenic functions, of a discrete Dirac operator, i.e. a first order, Clifford vector valued difference operator. In this paper, we establish a Cauchy-Kovalevskaya extension theorem for discrete monogenic functions defined on the standard $Z^m$ grid. Based on this extension principle, discrete Fueter polynomials, forming a basis of the space of discrete spherical monogenics, i.e. homogeneous discrete monogenic polynomials, are introduced. As an illustrative example we moreover explicitly construct the Cauchy-Kovalevskaya extension of the discrete delta function. These results are then generalized for a grid with variable mesh width $h$.
Regularity of solutions to an integral equation associated with Bessel potential
Xiaolong Han and Guozhen Lu
2011, 10(4): 1111-1119 doi: 10.3934/cpaa.2011.10.1111 +[Abstract](469) +[PDF](335.7KB)
Abstract:
In this paper, we study the regularity of the positive solutions to an integral equation associated with the Bessel potential. The kernel estimates for the Bessel potential plays an essential role in deriving such regularity results. First, we apply the regularity lifting by contracting operators to get the $L^\infty$ estimate. Then, we use the regularity lifting by combinations of contracting and shrinking operators, which was recently developed in [4] and [5], to prove the Lipschitz continuity estimate. Our regularity results here have been recently extended to positive solutions to an integral system associated with Bessel potential [9].
Remarks on some dispersive estimates
Yonggeun Cho, Tohru Ozawa and Suxia Xia
2011, 10(4): 1121-1128 doi: 10.3934/cpaa.2011.10.1121 +[Abstract](711) +[PDF](370.3KB)
Abstract:
In this paper we consider the initial value problem for $i\partial_t u + \omega(|\nabla|) u = 0$. Under suitable smoothness and growth conditions on $\omega$, we derive dispersive estimates which is the generalization of time decay and Strichartz estimates. We unify and also simplify dispersive estimates by utilizing the Bessel function. Another main ingredient of this paper is to revisit oscillatory integrals of [2].
A continuum of extinction rates for the fast diffusion equation
Marek Fila, Juan-Luis Vázquez and Michael Winkler
2011, 10(4): 1129-1147 doi: 10.3934/cpaa.2011.10.1129 +[Abstract](422) +[PDF](456.7KB)
Abstract:
We find a continuum of extinction rates for solutions $u(y,\tau)\ge 0$ of the fast diffusion equation $u_\tau=\Delta u^m$ in a subrange of exponents $m\in (0,1)$. The equation is posed in $R^n$ for times up to the extinction time $T>0$. The rates take the form $\|u(\cdot,\tau)\|_\infty$ ~ $(T-\tau)^\theta$ for a whole interval of $\theta>0$. These extinction rates depend explicitly on the spatial decay rates of initial data.
A note on a superlinear and periodic elliptic system in the whole space
Shuying He, Rumei Zhang and Fukun Zhao
2011, 10(4): 1149-1163 doi: 10.3934/cpaa.2011.10.1149 +[Abstract](396) +[PDF](431.9KB)
Abstract:
This paper is concerned with the following periodic Hamiltonian elliptic system

$ -\Delta u+V(x)u=g(x,v)$ in $R^N,$

$ -\Delta v+V(x)v=f(x,u)$ in $R^N,$

$ u(x)\to 0$ and $v(x)\to 0$ as $|x|\to\infty,$

where the potential $V$ is periodic and has a positive bound from below, $f(x,t)$ and $g(x,t)$ are periodic in $x$ and superlinear but subcritical in $t$ at infinity. By using generalized Nehari manifold method, existence of a positive ground state solution as well as multiple solutions for odd $f$ and $g$ are obtained.

The inverse Fueter mapping theorem
Fabrizio Colombo, Irene Sabadini and Frank Sommen
2011, 10(4): 1165-1181 doi: 10.3934/cpaa.2011.10.1165 +[Abstract](399) +[PDF](415.3KB)
Abstract:
In a recent paper the authors have shown how to give an integral representation of the Fueter mapping theorem using the Cauchy formula for slice monogenic functions. Specifically, given a slice monogenic function $f$ of the form $f=\alpha+\underline{\omega}\beta$ (where $\alpha$, $\beta$ satisfy the Cauchy-Riemann equations) we represent in integral form the axially monogenic function $\bar{f}=A+\underline{\omega}B$ (where $A,B$ satisfy the Vekua's system) given by $\bar{f}(x)=\Delta^{\frac{n-1}{2}}f(x)$ where $\Delta$ is the Laplace operator in dimension $n+1$. In this paper we solve the inverse problem: given an axially monogenic function $\bar{f}$ determine a slice monogenic function $f$ (called Fueter's primitive of $\bar{f}$ such that $\bar{f}=\Delta^{\frac{n-1}{2}}f(x)$. We prove an integral representation theorem for $f$ in terms of $\bar{f}$ which we call the inverse Fueter mapping theorem (in integral form). Such a result is obtained also for regular functions of a quaternionic variable of axial type. The solution $f$ of the equation $\Delta^{\frac{n-1}{2}}f(x)=\bar{f} (x)$ in the Clifford analysis setting, i.e. the inversion of the classical Fueter mapping theorem, is new in the literature and has some consequences that are now under investigation.
The bifurcation of interfacial capillary-gravity waves under O(2) symmetry
Mark Jones
2011, 10(4): 1183-1204 doi: 10.3934/cpaa.2011.10.1183 +[Abstract](348) +[PDF](509.4KB)
Abstract:
The evolution of an interface between two fluids of different densities is considered. The particular case under examination is when the motion is due to an interaction between the Mth and Nth harmonics of the fundamental mode. By means of a hodograph transformation the problem is cast as an operator equation between two suitable function spaces. Classical techniques are used to reduce the problem to a finite system of algebraic equations. Solutions are found which exhibit a rich variety of behaviour including primary, secondary and multiple bifurcation.
On a general class of free boundary problems for European-style installment options with continuous payment plan
Pierangelo Ciurlia
2011, 10(4): 1205-1224 doi: 10.3934/cpaa.2011.10.1205 +[Abstract](519) +[PDF](428.6KB)
Abstract:
In this paper we present an integral equation approach for the valuation of European-style installment derivatives when the payment plan is assumed to be a continuous function of the asset price and time. The contribution of this study is threefold. First, we show that in the Black-Scholes model the option pricing problem can be formulated as a free boundary problem under very general conditions on payoff structure and payment schedule. Second, by applying a Fourier transform-based solution technique, we derive a recursive integral equation for the free boundary along with an analytic representation of the option price. Third, based on these results, we propose a unified framework which generalizes the existing methods and is capable of dealing with a wide range of monotonic payoff functions and continuous payment plans. Finally, by using the illustrative example of European vanilla installment call options, an explicit pricing formula is obtained for time-varying payment schedules.
Blow-up at space infinity of a solution with a moving singularity for a semilinear parabolic equation
Shota Sato
2011, 10(4): 1225-1237 doi: 10.3934/cpaa.2011.10.1225 +[Abstract](441) +[PDF](365.3KB)
Abstract:
We consider the Cauchy problem for a parabolic partial differential equation with a power nonlinearity. It was shown in our previous paper that in some parameter range, the problem has a solution with a moving singularity that becomes anomalous in finite time. Our concern is a blow-up solution with a moving singularity. In this paper, we show that there exists a solution with a moving singularity such that it blows up at space infinity.
On the support of solutions to the Kadomtsev-Petviashvili (KP-II) equation
Pedro Isaza and Jorge Mejía
2011, 10(4): 1239-1255 doi: 10.3934/cpaa.2011.10.1239 +[Abstract](388) +[PDF](424.2KB)
Abstract:
In this article we prove that sufficiently smooth solutions of the Kadomtsev-Petviashvili (KP-II) equation:

$ \partial _t u+\partial^3_x u+\partial^{-1}_x\partial^2_y u+u\partial_x u =0, $

that have compact support for two different times are identically zero.

On the similarity of Hamiltonian and reversible vector fields in 4D
Ricardo Miranda Martins and Marco Antonio Teixeira
2011, 10(4): 1257-1266 doi: 10.3934/cpaa.2011.10.1257 +[Abstract](387) +[PDF](316.2KB)
Abstract:
We study the existence of formal conjugacies between reversible vector fields and Hamiltonian vector fields in 4D around a generic singularity. We construct conjugacies for a generic class of reversible vector fields. We also show that reversible vector fields are formally orbitally equivalent to polynomial decoupled Hamiltonian vector fields. The main tool we employ is the normal form theory.
Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system
Yanqin Fang and Jihui Zhang
2011, 10(4): 1267-1279 doi: 10.3934/cpaa.2011.10.1267 +[Abstract](524) +[PDF](401.4KB)
Abstract:
In this paper, we study the following system

$-\epsilon^2\Delta v+V(x)v+\psi(x)v=v^p, \quad x\in R^3,$

$-\Delta\psi=\frac{1}{\epsilon}v^2,\quad \lim_{|x|\rightarrow\infty}\psi(x)=0,\quad x\in R^3,$

where $\epsilon>0$, $p\in (3,5)$, $V$ is positive potential. We relate the number of solutions with topology of the set where $V$ attain their minimum value. By applying Ljusternik-Schnirelmann theory, we prove the multiplicity of solutions.

Averaging of a 3D Lagrangian averaged Navier-Stokes-$\alpha$ model with oscillating external forces
T. Tachim Medjo
2011, 10(4): 1281-1305 doi: 10.3934/cpaa.2011.10.1281 +[Abstract](460) +[PDF](518.1KB)
Abstract:
In this article, we consider a non-autonomous three-dimensional Lagrangian averaged Navier-Stokes-$\alpha$ equations with a singularly 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 [18] in the case of the two-dimensional Navier-Stokes systems, we study the convergence of $A^\epsilon$ as $\epsilon$ goes to zero.
Alternative proof for the existence of Green's function
Sungwon Cho
2011, 10(4): 1307-1314 doi: 10.3934/cpaa.2011.10.1307 +[Abstract](542) +[PDF](337.9KB)
Abstract:
We present a new method for the existence of a Green's function of nod-divergence form parabolic operator with Hölder continuous coefficients. We also derive a Gaussian estimate. Main ideas involve only basic estimates and known results without a potential approach, which is used by E.E. Levi.

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