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Communications on Pure & Applied Analysis

2012 , Volume 11 , Issue 6

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Preface
Noureddine Alaa, Marc Dambrine, Antoine Henrot and Alain Miranville
2012, 11(6): i-ii doi: 10.3934/cpaa.2012.11.6i +[Abstract](251) +[PDF](92.0KB)
Abstract:
The present volume is dedicated to Michel Pierre. An international Workshop for his 60th birthday, entitled "Partial Di erential Equations and Applications", was organized in Vittel (France) from October 22 to October 24, 2009.
Preconditioning operators and $L^\infty$ attractor for a class of reaction-diffusion systems
Boris Andreianov and Halima Labani
2012, 11(6): 2179-2199 doi: 10.3934/cpaa.2012.11.2179 +[Abstract](463) +[PDF](519.0KB)
Abstract:
We suggest an approach for proving global existence of bounded solutions and existence of a maximal attractor in $L^\infty$ for a class of abstract $3\times 3$ reaction-diffusion systems. The motivation comes from the concrete example of ``facilitated diffusion'' system with different non-homogeneous boundary conditions modelling the blood oxigenation reaction $Hb+O_2 \rightleftharpoons HbO_2$.
The method uses classical tools of linear semigroup theory, the $L^p$ techniques developed by Martin and Pierre [16] and B\'enilan and Labani [6] and the hint of ``preconditioning operators'': roughly speaking, the study of solutions of $(\partial_t +A_i)u=f$ is reduced to the study of solutions to

$(\partial_t+B)(B^{-1}u)=B^{-1}f+(I-B^{-1}A_i)u,$

with a conveniently chosen operator $B$. In particular, we need the $L^\infty-L^p$ regularity of $B^{-1}A_i$ and the positivity of the operator $(B^{-1}A_i-I)$ on the domain of $A_i$.
The same ideas can be applied to systems of higher dimension. To give an example, we prove the existence of a maximal attractor in $L^\infty$ for the $5\times 5$ system of facilitated diffusion modelling the coupled reactions $Hb+O_2 \rightleftharpoons HbO_2$, $Hb+CO_2 \rightleftharpoons HbCO_2$.

Friedlander's eigenvalue inequalities and the Dirichlet-to-Neumann semigroup
Wolfgang Arendt and Rafe Mazzeo
2012, 11(6): 2201-2212 doi: 10.3934/cpaa.2012.11.2201 +[Abstract](352) +[PDF](411.2KB)
Abstract:
If $\Omega$ is any compact Lipschitz domain, possibly in a Riemannian manifold, with boundary $\Gamma = \partial \Omega$, the Dirichlet-to-Neumann operator $\mathcal{D}_\lambda$ is defined on $L^2(\Gamma)$ for any real $\lambda$. We prove a close relationship between the eigenvalues of $\mathcal{D}_\lambda$ and those of the Robin Laplacian $\Delta_\mu$, i.e. the Laplacian with Robin boundary conditions $\partial_\nu u =\mu u$. This is used to give another proof of the Friedlander inequalities between Neumann and Dirichlet eigenvalues, $\lambda^N_{k+1} \leq \lambda^D_k$, $k \in N$, and to sharpen the inequality to be strict, whenever $\Omega$ is a Lipschitz domain in $R^d$. We give new counterexamples to these inequalities in the general Riemannian setting. Finally, we prove that the semigroup generated by $-\mathcal{D}_\lambda$, for $\lambda$ sufficiently small or negative, is irreducible.
The maximal regularity operator on tent spaces
Pascal Auscher, Sylvie Monniaux and Pierre Portal
2012, 11(6): 2213-2219 doi: 10.3934/cpaa.2012.11.2213 +[Abstract](356) +[PDF](354.7KB)
Abstract:
Recently, Auscher and Axelsson gave a new approach to non-smooth boundary value problems with $L^2$ data, that relies on some appropriate weighted maximal regularity estimates. As part of the development of the corresponding $L^p$ theory, we prove here the relevant weighted maximal estimates in tent spaces $T^{p, 2}$ for $p$ in a certain open range. We also study the case $p=\infty$.
Harmonic oscillators with Neumann condition on the half-line
Virginie Bonnaillie-Noël
2012, 11(6): 2221-2237 doi: 10.3934/cpaa.2012.11.2221 +[Abstract](278) +[PDF](504.3KB)
Abstract:
We consider the spectrum of the family of one-dimensional self-adjoint operators $-{\mathrm{d}}^2/{\mathrm{d}}t^2+(t-\zeta)^2$, $\zeta\in \mathbb{R}$ on the half-line with Neumann boundary condition. It is well known that the first eigenvalue $\mu(\zeta)$ of this family of harmonic oscillators has a unique minimum when $\zeta\in\mathbb{R}$. This paper is devoted to the accurate computations of this minimum $\Theta_{0}$ and $\Phi(0)$ where $\Phi$ is the associated positive normalized eigenfunction. We propose an algorithm based on finite element method to determine this minimum and we give a sharp estimate of the numerical accuracy. We compare these results with a finite element method.
Abstract reaction-diffusion systems with $m$-completely accretive diffusion operators and measurable reaction rates
Dieter Bothe and Petra Wittbold
2012, 11(6): 2239-2260 doi: 10.3934/cpaa.2012.11.2239 +[Abstract](320) +[PDF](488.4KB)
Abstract:
We consider reaction-diffusion systems with merely measurable reaction terms to cover the possibility of discontinuities. Solutions of such problems are defined as solutions to appropriate differential inclusions which, in an abstract form, lead to evolution inclusions of the form

$u' \in - A u + F(t,u)$ on $[0,T], u(0)=u_{0},$

where $A$ is $m$-accretive and $F$ is of upper semicontinuous type. While such problems, in general, can exhibit non-existence of solutions, the present paper shows that especially for $m$-completely accretive $A$, and under reasonable assumptions on $F$, mild solutions do exist.

Long time behavior of the Caginalp system with singular potentials and dynamic boundary conditions
Laurence Cherfils, Stefania Gatti and Alain Miranville
2012, 11(6): 2261-2290 doi: 10.3934/cpaa.2012.11.2261 +[Abstract](303) +[PDF](1452.0KB)
Abstract:
This paper is devoted to the study of the well-posedness and the long time behavior of the Caginalp phase-field model with singular potentials and dynamic boundary conditions. Thanks to a suitable definition of solutions, coinciding with the strong ones under proper assumptions on the bulk and surface potentials, we are able to get dissipative estimates, leading to the existence of the global attractor with finite fractal dimension, as well as of an exponential attractor.
The annulus as a K-spectral set
Michel Crouzeix
2012, 11(6): 2291-2303 doi: 10.3934/cpaa.2012.11.2291 +[Abstract](292) +[PDF](378.2KB)
Abstract:
We consider the annulus $\mathcal{A}_R$ of complex numbers with modulus and inverse of modulus bounded by $R>1$. We present some situations, in which this annulus is a K-spectral set for an operator $A$, and some related estimates.
Diffusion limit for a stochastic kinetic problem
Arnaud Debussche and Julien Vovelle
2012, 11(6): 2305-2326 doi: 10.3934/cpaa.2012.11.2305 +[Abstract](286) +[PDF](454.7KB)
Abstract:
We study the limit of a kinetic evolution equation involving a small parameter and perturbed by a smooth random term which also involves the small parameter. Generalizing the classical method of perturbed test functions, we show the convergence to the solution of a stochastic diffusion equation.
Some inverse problems around the tokamak Tore Supra
Yannick Fischer, Benjamin Marteau and Yannick Privat
2012, 11(6): 2327-2349 doi: 10.3934/cpaa.2012.11.2327 +[Abstract](308) +[PDF](1105.0KB)
Abstract:
We consider two inverse problems related to the tokamak Tore Supra through the study of the magnetostatic equation for the poloidal flux. The first one deals with the Cauchy issue of recovering in a two dimensional annular domain boundary magnetic values on the inner boundary, namely the limiter, from available overdetermined data on the outer boundary. Using tools from complex analysis and properties of genereralized Hardy spaces, we establish stability and existence properties. Secondly the inverse problem of recovering the shape of the plasma is addressed thank tools of shape optimization. Again results about existence and optimality are provided. They give rise to a fast algorithm of identification which is applied to several numerical simulations computing good results either for the classical harmonic case or for the data coming from Tore Supra.
Flow invariance for nonautonomous nonlinear partial differential delay equations
Seyedeh Marzieh Ghavidel and Wolfgang M. Ruess
2012, 11(6): 2351-2369 doi: 10.3934/cpaa.2012.11.2351 +[Abstract](338) +[PDF](443.0KB)
Abstract:
Several fundamental results on existence and flow-invariance of solutions to the nonlinear nonautonomous partial differential delay equation $ \dot{u}(t) + B(t)u(t) \ni F(t; u_t), 0 \leq s \leq t, u_s = \varphi, $ with $ B(t)\subset X\times X$ $\omega-$accretive, are developed for a general Banach space $X.$ In contrast to existing results, with the history-response $F(t;\cdot)$ globally defined and, at least, Lipschitz on bounded sets, the results are tailored for situations with $F(t;\cdot)$ defined on -- possibly -- thin subsets of the initial-history space $E$ only, and are applied to place several classes of population models in their natural $L^1-$setting. The main result solves the open problem of a subtangential condition for flow-invariance of solutions in the fully nonlinear case, paralleling those known for the cases of (a) no delay, (b) ordinary delay equations with $B(\cdot)\equiv 0,$ and (c) the semilinear case.
Compactness of discrete approximate solutions to parabolic PDEs - Application to a turbulence model
T. Gallouët and J.-C. Latché
2012, 11(6): 2371-2391 doi: 10.3934/cpaa.2012.11.2371 +[Abstract](335) +[PDF](486.7KB)
Abstract:
In this paper, we prove an adaptation of the classical compactness Aubin-Simon lemma to sequences of functions obtained through a sequence of discretizations of a parabolic problem. The main difficulty tackled here is to generalize the classical proof to handle the dependency of the norms controlling each function $u^{(n)}$ of the sequence with respect to $n$. This compactness result is then used to prove the convergence of a numerical scheme combining finite volumes and finite elements for the solution of a reduced turbulence problem.
Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous second-order gradient-like systems
Maurizio Grasselli and Morgan Pierre
2012, 11(6): 2393-2416 doi: 10.3934/cpaa.2012.11.2393 +[Abstract](487) +[PDF](529.8KB)
Abstract:
Following a result of Chill and Jendoubi in the continuous case, we study the asymptotic behavior of sequences $(U^n)_n$ in $R^d$ which satisfy the following backward Euler scheme:

$\varepsilon\frac{(U^{n+1}-2U^n+U^{n-1}}{\Delta t^2} +\frac{U^{n+1}-U^n}{\Delta t}+\nabla F(U^{n+1})=G^{n+1}, n\ge 0, $

where $\Delta t>0$ is the time step, $\varepsilon\ge 0$, $(G^{n+1})_n$ is a sequence in $ R^d$ which converges to $0$ in a suitable way, and $F\in C^{1,1}_{l o c}(R^d, R)$ is a function which satisfies a Łojasiewicz inequality. We prove that the above scheme is Lyapunov stable and that any bounded sequence $(U^n)_n$ which complies with it converges to a critical point of $F$ as $n$ tends to $\infty$. We also obtain convergence rates. We assume that $F$ is semiconvex for some constant $c_F\ge 0$ and that $1/\Delta t

Some applications of the Łojasiewicz gradient inequality
Alain Haraux
2012, 11(6): 2417-2427 doi: 10.3934/cpaa.2012.11.2417 +[Abstract](361) +[PDF](120.7KB)
Abstract:
In the present survey paper, basic convergence results for gradient-like systems relying on the Łojasiewicz gradient inequality are recalled in a self-contained way. A uniform version of the gradient inequality is used to get directly convergence and the rate of convergence in one step and a new technical trick, consisting in the evaluation of the integral of the velocity norm from $t$ to $2t$ is introduced. A short idea of the state of the art without technical details is also given.
On some spectral problems arising in dynamic populations
Antoine Henrot, El-Haj Laamri and Didier Schmitt
2012, 11(6): 2429-2443 doi: 10.3934/cpaa.2012.11.2429 +[Abstract](301) +[PDF](363.7KB)
Abstract:
We study a spectral problem related to a reaction-diffusion model where preys and predators do not live on the same area. We are interested in the optimal zone where a control should take place. First, we prove existence of an optimal domain in a natural class. Then, it seems plausible that the optimal domain is localized in the intersection of the living areas of the two species. We prove this fact in one dimension for small sized domains.
The rate of attraction of super-critical waves in a Fisher-KPP type model with shear flow
Patrick Martinez and Jean-Michel Roquejoffre
2012, 11(6): 2445-2472 doi: 10.3934/cpaa.2012.11.2445 +[Abstract](258) +[PDF](520.5KB)
Abstract:
We consider in this paper the thermo-diffusive model for flame propagation, which is a reaction-diffusion equation of the KPP (Kolmogorov, Petrovskii, Piskunov) type, posed on an infinite cylinder. Such a model has a family of travelling waves of constant speed, larger than a critical speed $c_*$. The family of all supercritical waves attract a large class of initial data, and we try to understand how. We describe in this paper the fate of an initial datum trapped between two supercritical waves of the same velocity: the solution will converge to a whole set of translates of the same wave, and we identify the convergence dynamics as that of an effective drift, around which an effective diffusion process occurs.
On dual dynamic programming in shape control
Andrzej Nowakowski and Jan Sokolowski
2012, 11(6): 2473-2485 doi: 10.3934/cpaa.2012.11.2473 +[Abstract](323) +[PDF](357.9KB)
Abstract:
We propose a new method for analysis of shape optimization problems. The framework of dual dynamic programming is introduced for a solution of the problems. The shape optimization problem for a linear elliptic boundary value problem is formulated in terms of characteristic functions which define the suport of control. The optimal solution of such a problem can be obtained by solving the sufficient optimality conditions.

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