Showing papers in "Siam Journal on Mathematical Analysis in 2007"

Optimally sparse multidimensional representation using shearlets

Abstract: In this paper we show that shearlets, an affine-like system of functions recently introduced by the authors and their collaborators, are essentially optimal in representing 2-dimensional functions f which are $C^2$ except for discontinuities along $C^2$ curves. More specifically, if $f_N^S$ is the N-term reconstruction of f obtained by using the N largest coefficients in the shearlet representation, then the asymptotic approximation error decays as $ orm{f-f_N^S}_2^2 \asymp N^{-2} (\log N)^3, N \to \infty,$ which is essentially optimal, and greatly outperforms the corresponding asymptotic approximation rate $N^{-1}$ associated with wavelet approximations. Unlike curvelets, which have similar sparsity properties, shearlets form an affine-like system and have a simpler mathematical structure. In fact, the elements of this system form a Parseval frame and are generated by applying dilations, shear transformations, and translations to a single well-localized window function.

On the partial differential equations of electrostatic mems devices : Stationary case

Abstract: We analyze the nonlinear elliptic problem $\Delta u =\frac{\lambda f(x)}{(1+u)^2}$ on a bounded domain Ω of $R^N$ with Dirichlet boundary conditions. This equation models a simple electrostatic micro‐electromechanical system (MEMS) device consisting of a thin dielectric elastic membrane with boundary supported at 0 above a rigid ground plate located at ‐1. When a voltage—represented here by λ—is applied, the membrane deflects towards the ground plate, and a snap‐through may occur when it exceeds a certain critical value $\lambda^*$ (pull‐in voltage). This creates a so‐called pull‐in instability, which greatly affects the design of many devices. The mathematical model leads to a nonlinear parabolic problem for the dynamic deflection of the elastic membrane, which will be considered in a forthcoming paper. Here, we focus on the stationary equation and on estimates for $\lambda^*$ in terms of material properties of the membrane, which can be fabricated with a spatially varying dielectric permittivity profile...

Global existence of classical solutions for a haptotaxis model

Abstract: A system of nonlinear partial differential equations modeling haptotaxis is investigated The model arises in cell migration processes involved in tumor invasion The existence of unique global classical solutions is proved

Poisson-nernst-planck systems for ion channels with permanent charges ∗

Abstract: Ionic channels and semiconductor devices use atomic scale structures to control macroscopic flows from one reservoir to another. The one-dimensional steady-state Poisson-Nernst- Planck (PNP) system is a useful representation of these devices, but experience shows that describing the reservoirs as boundary conditions is difficult. We study the PNP system for two types of ions with three regions of piecewise constant permanent charge, assuming the Debye number is large, because the electric field is so strong compared to diffusion. Reservoirs are represented by the outer regions with permanent charge zero. If the reciprocal of the Debye number is viewed as a singular parameter, the PNP system can be treated as a singularly perturbed system that has two limiting systems: inner and outer systems (termed fast and slow systems in geometric singular perturbation theory). A complete set of integrals for the inner system is presented that provides information for boundary and internal layers. Application of the exchange lemma from geometric singular perturbation theory gives rise to the existence and (local) uniqueness of the solution of the singular boundary value problem near each singular orbit. A set of simultaneous equations appears in the construction of singular orbits. Multiple solutions of such equations in this or similar problems might explain a variety of multiple valued phenomena seen in biological channels, for example, some forms of gating, and might be involved in other more complex behaviors, for example, some kinds of active transport.

A new class of entropy solutions of the Buckley-Leverett equation

Abstract: We discuss an extension of the Buckley–Leverett (BL) equation describing two-phase flow in porous media. This extension includes a third order mixed derivatives term and models the dynamic effects in the pressure difference between the two phases. We derive existence conditions for traveling wave solutions of the extended model. This leads to admissible shocks for the original BL equation, which violate the Oleinik entropy condition and are therefore called nonclassical. In this way we obtain nonmonotone weak solutions of the initial-boundary value problem for the BL equation consisting of constant states separated by shocks, confirming results obtained experimentally.

A mathematical analysis of the optimal exercise boundary for american put options

Abstract: We study a free boundary problem arising from American put options. In particular we prove existence and uniqueness for this problem, and we derive and rigorously prove high order asymptotic expansions for the early exercise boundary near expiry. We provide four approximations for the boundary: one is explicit and is valid near expiry (weeks); two others are implicit involving inverse functions and are accurate for longer time to expiry (months); the fourth is an ODE initial value problem which is very accurate for all times to expiry, is extremely stable, and hence can be solved instantaneously on any computer. We further provide an ode iterative scheme which can reach its numerical fixed point in five iterations for all time to expiry. We also provide a large time (equivalent to regular expiration times but large interest rate and/or volatility) behavior of the exercise boundary. To demonstrate the accuracy of our approximations, we present the results of a numerical simulation.

Nonuniform Average Sampling and Reconstruction of Signals with Finite Rate of Innovation

Abstract: From an average (ideal) sampling/reconstruction process, the question arises whether the original signal can be recovered from its average (ideal) samples and, if so, how. We consider the above que...

Elliptic Differential Equations with Coefficients Measurable with Respect to One Variable and VMO with Respect to the Others

Abstract: We prove the unique solvability of second order elliptic equations in nondivergence form in Sobolev spaces. The coefficients of the second order terms are measurable in one variable and VMO in other variables. From this result, we obtain the weak uniqueness of the Martingale problem associated with the elliptic equations.

Effective Transmission Conditions for Reaction-Diffusion Processes in Domains Separated by an Interface

Abstract: In this paper, we develop multiscale methods appropriate for the homogenization of processes in domains containing thin heterogeneous layers. Our model problem consists of a nonlinear reaction-diffusion system defined in such a domain, and properly scaled in the layer region. Both the period of the heterogeneities and the thickness of the layer are of order $\varepsilon.$ By performing an asymptotic analysis with respect to the scale parameter $\varepsilon$ we derive an effective model which consists of the reaction-diffusion equations on two domains separated by an interface together with appropriate transmission conditions across this interface. These conditions are determined by solving local problems on the standard periodicity cell in the layer. Our asymptotic analysis is based on weak and strong two-scale convergence results for sequences of functions defined on thin heterogeneous layers. For the derivation of the transmission conditions, we develop a new method based on test functions of boundary l...

Two-scale homogenization for evolutionary variational inequalities via the energetic formulation ∗

Abstract: This paper is devoted to two-scale homogenization for a class of rate-independent systems described by the energetic formulation or equivalently by an evolutionary variational inequality. In particular, we treat the classical model of linearized elastoplasticity with hardening. The associated nonlinear partial differential inclusion has periodically oscillating coefficients, and the aim is to find a limit problem for the case in which the period tends to 0. Our approach is based on the notion of energetic solutions, which is phrased in terms of a stability condition and an energy balance of an energy-storage functional and a dissipation functional. Using the recently developed method of weak and strong two-scale convergence via periodic unfolding, we show that these two functionals have a suitable two-scale limit, but now involving the macroscopic variable in the physical domain as well as the microscopic variable in the periodicity cell. Moreover, relying on an abstract theory of $\Gamma$-convergence for...

Elastic Energy Stored in a Crystal Induced by Screw Dislocations: From Discrete to Continuous

Abstract: This paper deals with the passage from discrete to continuous in modeling the static elastic properties of vertical screw dislocations in a cylindrical crystal, in the setting of antiplanar linear ...

Traveling fronts of pyramidal shapes in the Allen-Cahn equations

Abstract: This paper studies pyramidal traveling fronts in the Allen–Cahn equation or in the Nagumo equation. For the nonlinearity we are concerned mainly with the bistable reaction term with unbalanced energy density. Two-dimensional V-form waves and cylindrically symmetric waves in higher dimensions have been recently studied. Our aim in this paper is to construct truly three-dimensional traveling waves. For a pyramid that satisfies a condition, we construct a traveling front for which the contour line has a pyramidal shape. We also construct generalized pyramidal fronts and traveling waves of a hybrid type between pyramidal waves and planar V-form waves. We use the comparison principles and construct traveling fronts between supersolutions and subsolutions.

Bifurcation for a Free Boundary Problem Modeling Tumor Growth by Stokes Equation

Abstract: We consider a free boundary problem modeling tumor growth in fluid‐like tissue. The model equations include a diffusion equation for the nutrient concentration, and the Stokes equation with a source which represents the proliferation density of the tumor cells. The proliferation rate μ and the cell‐to‐cell adhesiveness γ which keeps the tumor intact are two parameters which characterize the “aggressiveness” of the tumor. For any positive radius R there exists a unique radially symmetric stationary solution with radius $r=R$. We prove that for a sequence $\mu/\gamma = M_n(R)$ there exist symmetry‐breaking bifurcation branches of solutions with free boundary $r=R+\varepsilon Y_{n,0}(\theta)+O(\varepsilon^2)$ (n $\text{even} \ge 2$) for small |e|, where $Y_{n,0}$ is the spherical harmonic of mode $(n,0)$. Furthermore, the smallest $M_n(R)$, say, $M_{n_*}(R)$, is such that $n_*=n_*(R)\to\infty$ as $R\to\infty$. The biological implications of this result are discussed at the end of the paper.

Lp-THEORY FOR A CLASS OF NON-NEWTONIAN FLUIDS

Abstract: Local-in-time well-posedness of the initial-boundary value problem for a class of non-Newtonian Navier–Stokes problems on domains with compact $C^{\mbox{3-}}$-boundary is proven in an $L_p$-setting for any space dimension $n\geq2$. The stress tensor is assumed to be of the generalized Newtonian type, i.e., $\cS=2\mu(|{\mathcal E}|_2^2){\mathcal E} -\pi I$, ${\mathcal E}=\frac{1}{2}( abla u+ abla u^{\sf T}),$ where $|{\mathcal E}|_2^2=\sum_{i,j=1}^n \varepsilon_{ij}^2$ denotes the Hilbert–Schmidt norm of the rate of strain tensor ${\mathcal E}$. The viscosity function $\mu\in C^{2-}({\mathbb R}_+)$ is subject only to the condition $\mu(s)>0$, $\mu(s)+2s\mu^\prime(s)>0$, $s\geq 0,$ which for the standard power-law–like function $\mu(s)=\mu_0(1+s)^{\frac{d-2}{2}}$ merely means $\mu_0>0$ and $d\geq 1$. This result is based on maximal regularity theory for a suitable linear problem and a contraction argument.

Navier–Stokes Equations Interacting with a Nonlinear Elastic Biofluid Shell

Abstract: We study a moving boundary value problem consisting of a viscous incompressible fluid moving and interacting with a nonlinear elastic fluid shell. The fluid motion is governed by the Navier–Stokes equations, while the fluid shell is modeled by a bending energy which extremizes the Willmore functional and a membrane energy with density given by a convex function of the local area ratio. The fluid flow and shell deformation are coupled together by continuity of displacements and tractions (stresses) along the moving surface defining the shell. We prove the existence and uniqueness of solutions in Sobolev spaces for a short time.

Existence of Solutions for Supply Chain Models Based on Partial Differential Equations

Abstract: We consider a model for supply chains governed by partial differential equations. The mathematical properties of a continuous model are discussed and existence and uniqueness are proven. Moreover, Lipschitz continuous dependence on the initial data is proven. We make use of the front tracking method to construct approximate solutions. The obtained results extend the preliminary work of [S. Gottlich, M. Herty, and A. Klar, Commun. Math. Sci., 3 (2005), pp. 545–559].

Inverse Source Problems in Transport Equations

Abstract: This paper proposes an iterative technique to reconstruct the source term in transport equations, which account for scattering effects, from boundary measurements. In the two‐dimensional setting, the full outgoing distribution in the phase space (position and direction) needs to be measured. In three space dimensions, we show that measurements for angles that are orthogonal to a given direction are sufficient. In both cases, the derivation is based on a perturbation of the inversion of the two‐dimensional attenuated Radon transform and requires that (the anisotropic part of) scattering be sufficiently small. We present an explicit iterative procedure, which converges to the source term we want to reconstruct. Applications of the inversion procedure include optical molecular imaging, an increasingly popular medical imaging modality.

Stable solutions for the bilaplacian with exponential nonlinearity.

Abstract: Let $\lambda^*>0$ denote the largest possible value of $\lambda$ such that {$\Delta^2 u = \la e^u \ \text{in B, } u = \pd{u}{n} = 0 \ \text{on $ B $}\}$ has a solution, where B is the unit ball in $\R^N$ and n is the exterior unit normal vector. We show that for $\lambda=\lambda^*$ this problem possesses a unique weak solution $u^*$. We prove that $u^*$ is smooth if $N\le 12$ and singular when $N\ge 13$, in which case $ u^*(r) = - 4 \log r + \log ( 8(N-2)(N-4) / \lambda^*) + o(1)$ as $r\to 0$. We also consider the problem with general constant Dirichlet boundary conditions.

Strong Instability of Standing Waves for the Nonlinear Klein–Gordon Equation and the Klein–Gordon–Zakharov System

Abstract: The orbital instability of ground state standing waves $e^{i\omega t}\phi_{\omega}(x)$ for the nonlinear Klein–Gordon equation has been known in the domain of all frequencies ω for the supercritical case and for frequencies strictly less than a critical frequency $\omega_c$ in the subcritical case. We prove the strong instability of ground state standing waves for the entire domain above. For the case when the frequency is equal to the critical frequency $\omega_c$ we prove strong instability for all radially symmetric standing waves $e^{i\omega_c t}\varphi(x)$. We prove similar strong instability results for the Klein–Gordon–Zakharov system.

Bifurcation analysis of an elliptic free boundary problem modelling the growth of avascular tumors

Abstract: We study bifurcations from radially symmetric solutions of a free boundary problem modelling the dormant state of nonnecrotic avascular tumors. This problem consists of two semilinear elliptic equations with a Dirichlet and a Neumann boundary condition, respectively, and a third boundary condition coupling surface tension effects on the free interface to the internal pressure. By reducing the full problem to an abstract bifurcation equation in terms of the free boundary only and by characterizing the linearization as a Fourier multiplication operator, we carry out a precise analysis of local bifurcations of this problem.

A Losing Estimate for the Ideal MHD Equations with Application to Blow‐up Criterion

Abstract: In this paper we study the blow‐up criterion of smooth solution to the ideal MHD equations in $\R^n$. By means of the Fourier frequency localization and Bony paraproduct decomposition, we show a losing estimate for the ideal MHD equations and apply it to establish an improved blow‐up criterion of smooth solutions. As a special case, we recover a previous result of Planchon for the incompressible Euler equations.

Diffusion Limit of a Semiconductor Boltzmann–Poisson System

Abstract: The paper deals with the diffusion limit of the initial‐boundary value problem for the multidimensional semiconductor Boltzmann–Poisson system. Here, we generalize the one‐dimensional results obtained in [5] to the case of several dimensions using global renormalized solutions. The method of moments and a velocity averaging lemma are used to prove the convergence of the renormalized solutions to the semiconductor Boltzmann–Poisson system towards a global weak solution of the drift‐diffusion‐Poisson model.

Synchronization of a stochastic reaction-diffusion system on a thin two-layer domain

Abstract: A system of semilinear parabolic stochastic partial differential equations with additive space‐time noise is considered on the union of thin bounded tubular domains $D_{1,\eps}:=\Gamma\times(0,\eps)$ and $D_{2,\eps}:=\Gamma\times(-\eps,0)$ joined at the common base $\Gamma \subset {\R}^{d}$, where $d\ge1$. The equations are coupled by an interface condition on $\Gamma$ which involves a reaction intensity $k(x',\eps)$, where $x = (x',x_{d+1}) \in \mathbb{R}^{d+1}$ with $x' \in \Gamma$ and $|x_{d+1}| < \eps$. Random influences are included through additive space‐time Brownian motion, which depend only on the base spatial variable $x' \in \Gamma$ and not on the spatial variable $x_{d+1}$ in the thin direction. Moreover, the noise is the same in both layers $D_{1,\eps}$ and $D_{2,\eps}$. Limiting properties of the global random attractor are established as the thinness parameter of the domain $\eps \to 0$, i.e., as the initial domain becomes thinner, when the intensity function possesses the property $\lim_{\...

Continuity estimates for the Monge-Ampere equation

Abstract: In this paper, we study the regularity of solutions to the Monge–Ampere equation. We prove the log-Lipschitz continuity for the gradient under certain assumptions. We also give a unified treatment ...

Nonlinear asymptotic stability of the semistrong pulse dynamics in a regularized Gierer-Meinhardt model

Abstract: We use renormalization group (RG) techniques to prove the nonlinear asymptotic stability for the semistrong regime of two‐pulse interactions in a regularized Gierer–Meinhardt system. In the semistrong limit the localized activator pulses interact strongly through the slowly varying inhibitor. The interaction is not tail‐tail as in the weak interaction limit, and the pulse amplitudes and speeds change as the pulse separation evolves on algebraically slow time scales. In addition the point spectrum of the associated linearized operator evolves with the pulse dynamics. The RG approach employed here validates the interaction laws of quasi‐steady two‐pulse patterns obtained formally in the literature, and establishes that the pulse dynamics reduce to a closed system of ordinary differential equations for the activator pulse locations. Moreover, we fully justify the reduction to the nonlocal eigenvalue problem (NLEP) showing that the large difference between the quasi‐steady NLEP operator and the operator arisi...

Spatial Dynamics Methods for Solitary Gravity-Capillary Water Waves with an Arbitrary Distribution of Vorticity

Abstract: This paper presents existence theories for several families of small-amplitude solitary-wave solutions to the classical two-dimensional water-wave problem in the presence of surface tension and with an arbitrary distribution of vorticity. Moreover, the established local bifurcation diagram for irrotational solitary waves is shown to remain qualitatively unchanged for any choice of vorticity distribution. The hydrodynamic problem is formulated as an infinite-dimensional Hamiltonian system in which the horizontal spatial direction is the timelike variable. A center-manifold reduction technique is employed to reduce the system to a locally equivalent Hamiltonian system with a finite number of degrees of freedom. Homoclinic solutions to the reduced system, which correspond to solitary water waves, are detected by a variety of dynamical systems methods.

Existence, Uniqueness, and Regularity of Optimal Transport Maps

Abstract: Adapting some techniques and ideas of McCann [Duke Math. J., 80 (1995), pp. 309–323], we extend a recent result with Fathi [Optimal Transportation on Manifolds, preprint] to yield existence and uniqueness of a unique transport map in very general situations, without any integrability assumption on the cost function. In particular this result applies for the optimal transportation problem on an n‐dimensional noncompact manifold M with a cost function induced by a $C^2$‐Lagrangian, provided that the source measure vanishes on sets with σ‐finite $(n-1)$‐dimensional Hausdorff measure. Moreover we prove that in the case $c(x,y)=d^2(x,y)$, the transport map is approximatively differentiable a.e.with respect to the volume measure, and we extend some results of [D. Cordero‐Erasquin, R. J. McCann, and M. Schmuckenschlager, Invent. Math., 146 (2001), pp. 219–257] about concavity estimates and displacement convexity.

Asymptotics of the Poisson Problem in Domains with Curved Rough Boundaries

Abstract: Effective boundary conditions (wall laws) are commonly employed to approximate PDEs in domains with rough boundaries, but it is neither easy to design such laws nor to estimate the related approximation error. A two-scale asymptotic expansion based on a domain decomposition result is used here to mitigate such difficulties, and as an application we consider the Poisson equation. The proposed scheme considers rough curved boundaries and allows a complete asymptotic expansion for the solution, highlighting the influence of the boundary curvature. The derivation and estimation of high order effective conditions is a corollary of such development. Sharp estimates for first and second order wall law approximations are considered for different Sobolev norms and show superior convergence rates in the interior of the domain. A numerical test illustrates several of the results obtained here.

The effect of inhibitor on the plasmid-bearing and plasmid-free model in the unstirred chemostat

Abstract: This paper deals with a chemostat model with an inhibitor in the context of competi- tion between plasmid-bearing and plasmid-free organisms. First, sufficient conditions for coexistence of the steady-state are determined. Second, the effects of the inhibitor are considered. It turns out that the parameter μ, which represents the effect of the inhibitor, plays a very important role in deciding the number of the coexistence solutions. The results show that if μ is sufficiently large this model has at least two coexistence solutions provided that the maximal growth rate a of u lies in a certain range and has only one unique asymptotically stable coexistence solution when a belongs to another range. Finally, extensive simulations are done to complement the analytic results. The main tools used here include degree theory in cones, bifurcation theory, and perturbation technique. 1. Introduction. The chemostat is a common model in microbial ecology. It is used as an ecological model of a simple lake, as a model of waste treatment, and as a model for commercial production of fermentation processes. It is important in ecology because the parameters are readily measurable and, thus, the mathematical results are readily testable. For a general discussion of competitive systems see (29), while a de- tailed mathematical description of competition in the chemostat can be found in (30). Our study focuses on a chemostat model in the context of competition between plasmid-bearing and plasmid-free organisms. This issue has recently received consid- erable attention. The theoretical literature on this model includes Ryder and DiBiaso (25), Stephanopoulos and Lapidus (28), Hsu, Waltman, and Wolkowicz (17), Lu and Hadeler (22), Levin (20), Luo and Hsu (18), and Macken, Levin, and Waldstatter (23). In industry, genetically altered organisms are frequently used to manufacture a desired product, for instance, a pharmaceutical. The alteration is accomplished by introducing a piece of DNA into the cell in the form of a plasmid. The burden imposed on the cell by the task of production can result in the genetically altered (the plasmid-bearing) organism being a less able competitor than the plasmid-free organ- ism. Unfortunately, the plasmid can be lost in the reproductive process. Thus, it is possible for the plasmid-free organism to take over the culture. To avoid "capture" of the process by the plasmid-free organism, the obvious choice is to alter the medium in such a way as to favor the plasmid-bearing organism. An example of this would be to introduce an antibiotic into the feed bottle. See (10, 15, 16) for a detailed biological and chemical background. Models in this direction have been studied in Lenski and Hattingh (21), Hsu and Waltman (13, 15, 16), Hsu, Luo, and Waltman (12), Nie and Wu (24), and the references therein.

Sharp Estimates on Minimum Travelling Wave Speed of Reaction Diffusion Systems Modelling Autocatalysis

Abstract: This article studies propagating wave fronts in an isothermal chemical reaction $ A + 2B \rightarrow 3B$ involving two chemical species, a reactant A and an autocatalyst B, whose diffusion coefficients, $D_A$ and $D_B$, are unequal due to different molecular weights and/or sizes. Explicit bounds $v_*$ and $v^*$ that depend on $D_B/D_A$ are derived such that there is a unique travelling wave of every speed $v \geq v^*$ and there does not exist any travelling wave of speed $v