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Communications on Pure & Applied Analysis

2004 , Volume 3 , Issue 1

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Turbulent channel flow in weighted Sobolev spaces using the anisotropic Lagrangian averaged Navier-Stokes (LANS-$\alpha$) equations
Daniel Coutand and Steve Shkoller
2004, 3(1): 1-23 doi: 10.3934/cpaa.2004.3.1 +[Abstract](198) +[PDF](261.6KB)
Abstract:
Modelling the mean characteristics of turbulent channel flow has been one of the longstanding problems in fluid dynamics. While a great number of mathematical models have been proposed for isotropic turbulence, there are relatively few, if any, turbulence models in the anisotropic wall-bounded regime which hold throughout the entire channel. Recently, the anisotropic Lagrangian averaged Navier-Stokes equations (LANS-$\alpha$) have been derived in [7] and [5]. This paper is devoted to the analysis of this coupled system of nonlinear PDE for the mean velocity and covariance tensor in the channel geometry. The vanishing of the covariance along the walls induces certain degenerate elliptic operators into the model, which require weighted Sobolev spaces to study. We prove that when the no-slip boundary conditions are prescribed for the mean velocity, the LANS-$\alpha$ equations possess unique global weak solutions which converge as time tends to infinity towards the unique stationary solutions. Qualitative properties of the stationary solutions are also established.
Length scales and positivity of solutions of a class of reaction-diffusion equations
Michele V. Bartuccelli, K. B. Blyuss and Y. N. Kyrychko
2004, 3(1): 25-40 doi: 10.3934/cpaa.2004.3.25 +[Abstract](151) +[PDF](230.5KB)
Abstract:
In this paper, the sharpest interpolation inequalities are used to find a set of length scales for the solutions of the following class of dissipative partial differential equations

$u_{t}= -\alpha_{k}(-1)^{k} \nabla^{2k}u+\sum_{j=1}^{k-1}\alpha_{j} (-1)^{j}\nabla^{2j}u+\nabla^{2}(u^{m})+u(1-u^{2p}), $

with periodic boundary conditions on a one-dimensional spatial domain. The equation generalises nonlinear diffusion model for the case when higher-order differential operators are present. Furthermore, we establish the asymptotic positivity as well as the positivity of solutions for all times under certain restrictions on the initial data. The above class of equations reduces for some particular values of the parameters to classical models such as the KPP-Fisher equation which appear in various contexts in population dynamics.

Liouville's formula under the viewpoint of minimal surfaces
Francisco Brito, Maria Luiza Leite and Vicente de Souza Neto
2004, 3(1): 41-51 doi: 10.3934/cpaa.2004.3.41 +[Abstract](235) +[PDF](228.4KB)
Abstract:
We study the equation $\Delta u=e^{-2u}$ in dimension two and review the Liouville's formula for a solution $u$ in terms of the Weierstrass representation of a minimal surface in $\mathbb R^3.$ We list minimal surfaces corresponding to classical solutions and point out a gap of $100$ years from Bonnet's family of minimal surfaces to solutions discovered by physicists in the sixties. We also prove Chen-Li's symmetry theorem in the context of minimal surfaces theory.
On the Riemann problem for some discontinuous systems of conservation laws describing phase transitions
João-Paulo Dias and Mário Figueira
2004, 3(1): 53-58 doi: 10.3934/cpaa.2004.3.53 +[Abstract](274) +[PDF](164.7KB)
Abstract:
For a special class of discontinuous flux functions that can be associated to the limit case of a phase transition it has been introduced in [2] an appropriate notion of entropy weak solution to the Cauchy problem and some existence results were proved. In this paper, for the discontinuous scalar case, we give a counter-example to uniqueness and we prove an estimate based in Kruskov's method. Then, for a class of discontinuous $p$-systems, we prove, by applying a variant of the regularization method introduced by Dafermos in [1], an existence result for the Riemann problem.
Asymptotic behavior of solutions of the mixed problem for semilinear hyperbolic equations
Akisato Kubo
2004, 3(1): 59-74 doi: 10.3934/cpaa.2004.3.59 +[Abstract](171) +[PDF](243.9KB)
Abstract:
We discuss the optimality of the decay estimate of the mixed problem (MP) for semilinear hyperbolic equations of the type of the Euler-Poisson-Darboux equation. For this purpose we investigate decay properties and the lower bounds of the solutions to a boundary value problem related to (MP) as $t \rightarrow \infty $.
Nonlinear functionals in oscillation theory of matrix differential systems
Angelo B. Mingarelli
2004, 3(1): 75-84 doi: 10.3934/cpaa.2004.3.75 +[Abstract](319) +[PDF](187.8KB)
Abstract:
General oscillation criteria for second order two-term linear differential systems and, as a consequence, a more general class of Hamiltonian systems with symmetric coefficients are established using nonlinear functionals on a suitable matrix space. This extends and unifies most known results dealing with oscillation criteria using the particular maximum eigenvalue functional.
On a unilateral reaction-diffusion system and a nonlocal evolution obstacle problem
José-Francisco Rodrigues and João Lita da Silva
2004, 3(1): 85-95 doi: 10.3934/cpaa.2004.3.85 +[Abstract](187) +[PDF](199.7KB)
Abstract:
We prove the existence of solution to a free boundary problem of obstacle type with a diffusion coefficient depending on a function whose equation has a discontinuous reaction term. Our method uses the continuous dependence properties of the coincidence set of the evolution obstacle problem under a general non-degenerating condition. Motivated by the oxygen consumption problem with, for instance, temperature dependent diffusion, we obtain in a limit case a nonlocal problem of new type, which involves the measure of the domain occupied by the oxygen at each instant.
On the slightly compressible MHD system in the half-plane
Paola Trebeschi
2004, 3(1): 97-113 doi: 10.3934/cpaa.2004.3.97 +[Abstract](213) +[PDF](226.7KB)
Abstract:
In this paper we prove the existence of a smooth compressible solution for the MHD system in the half-plane. It is well-known that, as the Mach number goes to zero, the compressible MHD problem converges to the incompressible one, which has a global solution in time. Hence, it is natural to expect that, for Mach number sufficiently small, the compressible solution exists on any arbitrary time interval, with no restriction on the size of the initial velocity. In order to obtain the existence result, we decompose the solution as the sum of the solution of the irrotational Euler problem, the solution of the incompressible MHD system and the solution of the remainder problem which describes the interaction between the first two components. We show that the solution of the remainder part exists on any arbitrary time interval. Since this holds also for the solution of the irrotational Euler problem, this yields the existence of the smooth compressible solution for the MHD system.
Existence and regularity results for the primitive equations in two space dimensions
M. Petcu, Roger Temam and D. Wirosoetisno
2004, 3(1): 115-131 doi: 10.3934/cpaa.2004.3.115 +[Abstract](200) +[PDF](241.1KB)
Abstract:
Our aim in this article is to present some existence, uniqueness and regularity results for the Primitive Equations of the ocean in space dimension two with periodic boundary conditions. We prove the existence of weak solutions for the PEs, the existence and uniqueness of strong solutions and the existence of more regular solutions, up to $\mathcal C^\infty$ regularity.
A Newton-type method for computing best segment approximations
Hans J. Wolters
2004, 3(1): 133-148 doi: 10.3934/cpaa.2004.3.133 +[Abstract](168) +[PDF](225.7KB)
Abstract:
This paper presents a new method for computing best segment approximations. It is based on Newton iteration, but modified to obtain global convergence. The method is described in detail and a thorough convergence analysis is given. Polynomials are used as approximating functions. Not that the basic method will produce approximations that are not smooth and coninuity is not guaranteed. Howwever, we will describe applications to produce smooth spline approximation with free knots.
Asymptotic behavior of the $L^1$ norm of solutions to nonlinear parabolic equations
P. R. Zingano
2004, 3(1): 151-159 doi: 10.3934/cpaa.2004.3.151 +[Abstract](187) +[PDF](209.7KB)
Abstract:
We examine the large time behavior of the $L^{1}$ norm of solutions $ u(\cdot,t) $ to nonlinear parabolic equations $u_{t} + f(u)_{x} = (\kappa(u) u_{x})_{x}$ in 1-D with (arbitrary) initial states $ u(\cdot,0) $ in $ L^{1}(\mathbb{R}) $, where $ \kappa(u) $ is positive. If $ u(\cdot,t) $, ũ$(\cdot,t) $ are any solutions having the same mass, say $m$, then one has $\| u(\cdot,t) -$ ũ$(\cdot,t) \|_{L^{1}(\mathbb{R})} \rightarrow 0$ as $t \rightarrow \infty $, and the limiting value for the $L^{1}$ norm of either solution is the absolute value of $m$. Other results of interest are also discussed.

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