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SIAM J. on Mathematical Analysis

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1996

Volume 27, Issue 6, pp. 1503-1815


On a Representation Formula for B. Temple Systems

S. Benzoni-Gavage

SIAM J. Math. Anal. 27, pp. 1503-1519 (17 pages) | Cited 2 times

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The author gives an inf-sup representation formula associated with genuinely nonlinear characteristic fields of B. Temple systems of conservation laws which reduces to the Lax formula in the convex scalar case. The proof is derived by means of geometrical arguments together with a method of characteristics. It holds for piecewise-smooth entropy solutions to the Cauchy problem for a large variety of initial data including Riemann data.

Convergence of the Homogenization Process for a Double-Porosity Model of Immiscible Two-Phase Flow

Alain Bourgeat, Stephan Luckhaus, and Andro Mikelić

SIAM J. Math. Anal. 27, pp. 1520-1543 (24 pages) | Cited 24 times

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In this paper, we justify by periodic homogenization the double-porosity model for immiscible incompressible, two-phase flow. The volume fraction of the fissured part and the nonfissured part are kept positive constants and of the same order. The scaling is such that, in the final homogenized equations, the less permeable part of the matrix contributes as a nonlinear memory term. To prove the convergence of the total velocity and of the “reduced” pressure, we use the two-scale convergence since it seems to be appropriate for the problem, even though it would be possible to work with periodic modulation. However, in the final step, the degenerate ellipticity prevents the use of the two-scale convergence method and leads us to use periodic modulation.

Traveling Waves as Limits of Solutions on Bounded Domains

Giorgio Fusco, Jack K. Hale, and Jianping Xun

SIAM J. Math. Anal. 27, pp. 1544-1558 (15 pages)

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This paper is concerned with the asymptotic behavior as $\epsilon \to 0$ of solutions of the reaction-diffusion equation $u_t = \epsilon ^2 u_{xx} - (u + a)(u^2 - 1)$ defined in $( - 1,1)$ with Neumann boundary conditions. For $a = 0$, this equation has a monotone equilibrium solution $u^\epsilon $ with the property that $u^\epsilon (x) \to - 1$ (resp. $ + 1$) on $[ { - 1,0} )$ (resp. $( {0,1} ]$) as $\epsilon \to 0$; that is, the solution has a sharp transition layer if $a = 0$. Also, it is known that $u^\epsilon $ has a one-dimensional unstable manifold $\mathcal{M}(u^\epsilon )$. Solutions near $\mathcal{M}(u^\epsilon )$ decrease exponentially to $\mathcal{M}(u^\epsilon )$ and move with a speed $O(e^{{{ - c} / \epsilon }} )$ along $\mathcal{M}(u^\epsilon )$.
This paper considers the case where $a$ is small and fixed. For each fixed $\epsilon $, $a \ne 0$, small, there is an equilibrium solution $u^{\epsilon a} $ with unstable manifold of dimension one, but $u^{\epsilon a} $ approaches either the function 1 or $ - 1$ as $\epsilon \to 0$; that is, there is no monotone equilibrium solution with a sharp transition layer. If we rescale $x$ to $\epsilon x$ and consider the rescaled equation on $( - \infty ,\infty )$, then there is a unique (except for translation) monotone traveling-wave solution on $( - \infty ,\infty )$ with wave speed $ - \sqrt 2 a$. Using a geometric approach, we prove that there are positive constants $\epsilon _0 $ and $a_0 $ such that, for $0 < \epsilon < \epsilon _0 $ and $|a| < a_0 $, solutions of the rescaled equations on $( - \frac{1}{\epsilon },\frac{1}{\epsilon })$ in a neighborhood of size $C\sqrt {a_0 } $ of a monotone traveling-wave solution decrease exponentially fast before they enter a neighborhood of size $O(\epsilon ^k )$ of such a solution, where $k$ can be any positive integer. Along the traveling-wave direction, solutions move with the traveling-wave speed plus an error term $O(\epsilon ^k )$. It also is proved that the $L^\infty $-norm between the solution and a translation of the traveling wave is of order $O(\epsilon ^k )$ for $C_1 k\log \frac{1}{\epsilon } < t < \frac{{C_2 }}{\epsilon }$.

Asymptotic Behavior of Two Interreacting Chemicals in a Chromatography Reactor

Daniel N. Ostrov

SIAM J. Math. Anal. 27, pp. 1559-1596 (38 pages)

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The chromatographic separation of two chemical species ($c_1 $ and $c_2 $) that transform into each other with first-order kinetics as they pass through a Langmuir isotherm reactor is governed by the following system of nonlinear hyperbolic conservation equations: \[\begin{gathered} \frac{{\partial c_1 }}{{\partial x}} + \frac{\partial }{{\partial t}}\left( {\frac{{c_1 }}{{1 + c_1 + c_2 }}} \right) = - kc_1 + k'c_2 \hfill \\ {\text{and}}\quad \frac{{\partial c_2 }}{{\partial x}} + \frac{\partial }{{\partial t}}\left( {\frac{{\gamma c_2 }}{{1 + c_1 + c_2 }}} \right) = \gamma \left( {kc_1 + k'c_2 } \right), \hfill \\ {\text{where }}t \in ( - \infty ,\infty ). \hfill \\ \end{gathered} \]An analysis is presented of the two species’ asymptotic behavior as they progress down a semiinfinite (i.e., $x \in [ {0,\infty } )$) separation reactor with cyclic (periodic) entering feed concentrations. First it is shown that the method of generalized characteristics can be extended to describe the above system of equations. Then generalized characteristics are applied to show that the $\omega $-limit set for the species concentrations is comprised of a single determined point on the curve of chemical equilibrium and that this point is approached at an exponential rate.

On the Solution of Time-Harmonic Scattering Problems for Maxwell’s Equations

Christophe Hazard and Marc Lenoir

SIAM J. Math. Anal. 27, pp. 1597-1630 (34 pages) | Cited 32 times

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This paper deals with the scattering of a monochromatic electromagnetic wave by a perfect conductor surrounded by a locally inhomogeneous medium. The direct numerical solution of this problem by a finite-element method requires special edge elements. The aim of the present paper is to give an equivalent formulation of the problem well suited for both easy theoretical investigation and numerical implementation. Following a well-known idea, this formulation is obtained by adding a regularizing term such as “grad div” in the time-harmonic Maxwell equations, which leads us to solve an elliptic problem similar to the vector Helmholtz equation instead of Maxwell’s equation. The numerical treatment of this new formulation requires only standard Lagrange finite elements.
A unified approach, which is valid for the equations satisfied by either the electric or the magnetic field, is presented. It applies for a conductor with a Lipschitz-continuous boundary surrounded by a dissipative or nondissipative medium whose electromagnetic coefficients (permittivity and permeability) may be irregular. A family of scattering problems is defined, that is, the classical problem (which follows from Maxwell’s equations) and the so-called “regularized problem” obtained by adding a regularizing term in Maxwell’s equations. These problems are shown to be well posed and to have the same solution. An integral representation technique is described.

Spectral Analysis of a Multistratified Acoustic Strip Part II: Asymptotic Behavior of Solutions for a Simple Stratification

Elisabeth Croc and Yves Dermenjian

SIAM J. Math. Anal. 27, pp. 1631-1652 (22 pages) | Cited 4 times

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We consider the acoustic propagator $A = - \nabla \cdot c^2 \nabla $ in $\Omega = \{ {(x,z) \in {{\mathbb{R}^2 } / {0 < z < H}}} \}$. The velocity $c$, which describes the stratification of the strip $\Omega $, depends only on the variable $z$: it is assumed to be a function in $L^\infty ((0,H))$ bounded from below by $c_m > 0$. Let $A$ be the self-adjoint operator associated with the Neumann or Dirichlet condition at $z = 0$ and $z = H$; let $\mu $ be a real number in the spectrum of $A$; and let $u$ be the solutions of the equation $(A - \mu I)u = f$ locally in the domain of $A$, which are determined by the limiting absorption principle in [E. Croc and Y. Dermenjian, SIAM J. Math. Anal., 26 (1995), pp. 880–924] and made explicit with trace operators. Thanks to accurate Hölder properties for the trace operators, we control the asymptotic behavior of $u$ with so-called “zero-trace” conditions for $f$.

Semilinear Elliptic Equations in $\mathbb{R}^N $ with Almost Periodic or Unbounded Forcing Term

Gilles Fournier, Andrzej Szulkin, and Michel Willem

SIAM J. Math. Anal. 27, pp. 1653-1660 (8 pages) | Cited 1 time

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This work is devoted to the existence and uniqueness of almost periodic solutions in $\mathbb{R}^N $ of the equation $ - \Delta u + \sum\nolimits_{j = 1}^N {c_j \partial _j u + g(u) = h(x)} $. We also prove the existence of solutions with the same growth as some unbounded forcing terms. Under a local monotonicity assumption, the method of upper and lower solutions is used.

What is the Subdifferential of the Closed Convex Hull of a Function?

J. Benoist and J.-B. Hiriart-Urruty

SIAM J. Math. Anal. 27, pp. 1661-1679 (19 pages) | Cited 11 times

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Given a function $f:\mathbb{R}^n \to ( { - \infty , + \infty } ]$ and its closed convex hull $\overline {{\operatorname{co}}} f$, we consider the question of expressing the subdifferential of $\overline {{\operatorname{co}}} f$ in terms of the subdifferential of $f$. Under a fairly general assumption on the behavior of $f$ at infinity, we obtain an explicit formula of the subdifferential of $\overline {{\operatorname{co}}} f$ from that of $f$ and its asymptotic function.

Inversion de Certains Opérateurs Elliptiques à Coefficients Variables

Philippe Tchamitchian

SIAM J. Math. Anal. 27, pp. 1680-1703 (24 pages) | Cited 1 time

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We consider elliptic operators in divergence form with variable coefficients defined through an accretive sesquilinear form on the whole space. The coefficients of leading order are supposed to be lipschitzian. We show how wavelets bases allow us to explicitly compute the inverse of such operators. The first main ingredient is a detailed study, of independant interest, of the convergence of the usual Galerkin approximations. The second main ingredient is the notion of paraproduct, suitably adapted to our context.

On Quasi-Periodic Perturbations of Elliptic Equilibrium Points

Àngel Jorba and Carles Simó

SIAM J. Math. Anal. 27, pp. 1704-1737 (34 pages) | Cited 17 times

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This work focuses on quasi-periodic time-dependent perturbations of ordinary differential equations near elliptic equilibrium points. This means studying\[ \dot x = (A + \varepsilon Q(t,\varepsilon ))x + \varepsilon g(t,\varepsilon ) + h(x,t,\varepsilon ), \]where $A$ is elliptic and $h$ is $\mathcal{O}(x^2 )$. It is shown that, under suitable hypothesis of analyticity, nonresonance and nondegeneracy with respect to $\varepsilon $, there exists a Cantorian set $\mathcal{E}$ such that for all $\varepsilon \in \mathcal{E}$ there exists a quasi-periodic solution such that it goes to zero when $\varepsilon $ does. This quasi-periodic solution has the same set of basic frequencies as the perturbation. Moreover, the relative measure of the set $[0,\varepsilon _0 ]\backslash \mathcal{E}$ in $[0,\varepsilon _0 ]$ is exponentially small in $\varepsilon _0 $. The case $g \equiv 0$, $h \equiv 0$ (quasi-periodic Floquet theorem) is also considered.
Finally, the Hamiltonian case is studied. In this situation, most of the invariant tori that are near the equilibrium point are not destroyed but only slightly deformed and “shaken” in a quasi-periodic way. This quasi-periodic “shaking” has the same basic frequencies as the perturbation.

Periodic Monotone Systems with an Invariant Function

Jiang Ji-Fa

SIAM J. Math. Anal. 27, pp. 1738-1744 (7 pages) | Cited 2 times

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The author studies the periodic time-dependent type-$K$ monotone system\[ \dot x_i = F_i \left( {t,x_1 , \ldots ,x_n } \right)\quad (i = 1, \ldots ,n) \]in the interior of the nonnegative orthant in $n$-space satisfying the following conditions: (i) if $x \ne y$, $x_i = y_i $ and $x_j \leq y_j $ for $j \ne i$, then $F_i (t,x) \leq F_i (t,y)$; (ii) $F(t,x)$ is periodic in $t$ of period $\tau > 0$; (iii) $F$ possesses an invariant function with positive gradient. It is proved that every solution to such a system either converges to a periodic solution or eventually leaves any compact set. This result gives an affirmative answer to the conjecture recently proposed by B. R. Tang, Y. Kuang, and H. Smith in [SIAM J. Math. Anal., 24 (1993), pp. 1331–1339] for periodic type-$K$ monotone systems.

Nonstationary Subdivision Schemes and Multiresolution Analysis

Albert Cohen and Nira Dyn

SIAM J. Math. Anal. 27, pp. 1745-1769 (25 pages) | Cited 19 times

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Nonstationary subdivision schemes consist of recursive refinements of an initial sparse sequence with the use of masks that may vary from one scale to the next finer one. This paper is concerned with both the convergence of nonstationary subdivision schemes and the properties of their limit functions. We first establish a general result on the convergence of such schemes to $C^\infty $ compactly supported functions. We show that these limit functions allow us to define a multiresolution analysis that has the property of spectral approximation. Finally, we use these general results to construct $C^\infty $ compactly supported cardinal interpolants and also $C^\infty $ compactly supported orthonormal wavelet bases that constitute Riesz bases for Sobolev spaces of any order.

A Nonlinear Operator Related to Scaling Functions and Wavelets

Ying Huang

SIAM J. Math. Anal. 27, pp. 1770-1790 (21 pages) | Cited 1 time

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This paper studies a certain nonlinear operator $T$ from $L^2 (\mathbb{R})$ to itself under which every scaling function is a fixed point. The iterations $T^n f$ of $T$ on any $L^2 $-function $f$ with the Riesz basis property are investigated; they turn out to be the subdivision-scheme iterates of $f$ with weights depending on $f$ only. The paper gives conditions for convergence of $T^n f$ to a limit in different topologies and studies the regularity of the limit functions. The results are illustrated with examples.

Intertwining Multiresolution Analyses and the Construction of Piecewise-Polynomial Wavelets

George C. Donovan, Jeffrey S. Geronimo, and Douglas P. Hardin

SIAM J. Math. Anal. 27, pp. 1791-1815 (25 pages) | Cited 26 times

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Let $(V_p )$ be a local multiresolution analysis of $L^2 ({\bf R})$ of multiplicity $r \geq 1$, i.e., $V_0 $ is generated by $r$ compactly supported scaling functions. If the scaling functions generate an orthogonal basis of $V_0 $, then $(V_p )$ is called an orthogonal multiresolution analysis. We prove that there exists an orthogonal local multiresolution analysis $(V'_p )$ of multiplicity $r'$ such that \[V_q \subset V'_0 \subset V_{q + n} \] for some integers $q \geq 0$, $n \geq 1$, and $r' > 1$.
In particular, this shows that compactly supported orthogonal polynomial spline wavelets and scaling functions (of multiplicity $r' > 1$) of arbitrary regularity exist, and we give several such examples.
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