In this article, we study the uniform null controllability problem for a system of coupled parabolic equations with an oscillating coefficient. This is done in three steps -- first, we study the spectral properties of an elliptic operator; second, we allow the system to evolve freely and obtain the required decay; third, we use a Carleman estimate to prove a suitable observability result. This uniform null controllability property is then used to homogenize the associated coupled parabolic system.
A homogenization result for a family of integral energies is presented, where the fields are subjected to periodic first order oscillating differential constraints in divergence form. The work is based on the theory of A -quasiconvexity with variable coefficients and on two- scale convergence techniques.
This paper is devoted to the homogenization of Shrödinger type equations with periodically oscillating coefficients of the diffusion term, and a rapidly oscillating periodic time-dependent potential. One convergence theorem is proved and we derive the macroscopic homogenized model. Our approach is the well known two-scale convergence method.
In this article we study the homogenization rates of eigenvalues of a Steklov problem with rapidly oscillating periodic weight functions. The results are obtained via a careful study of oscillating functions on the boundary and a precise estimate of the $L^\infty$ bound of eigenfunctions. As an application we provide some estimates on the first nontrivial curve of the Dancer-{F}u{č}{\'ı}k spectrum.
The paper deals with homogenization of Levy-type operators with rapidly oscillating coefficients. We consider cases of periodic and random statistically homogeneous micro-structures and show that in the limit we obtain a Levy-operator. In the periodic case we study both symmetric and non-symmetric kernels whereas in the random case we only investigate symmetric kernels. We also address a nonlinear version of this homogenization problem.
This paper concerns with two issues. The first issue is the existence and the uniqueness of the ergodic type number $d$ which appears in the oblique boundary condition. The second issue is the application of the number for the study of homogenizations of oscillating Neumann boundary conditions.
This paper deals with the homogenization of a mixed boundary value problem for the Laplace operator in a domain with locally periodic oscillating boundary. The Neumann condition is prescribed on the oscillating part of the boundary, and the Dirichlet condition on a separate part. It is shown that the homogenization result holds in the sense of weak $L^2$ convergence of the solutions and their flows, under natural hypothesis on the regularity of the domain. The strong $L^2$ convergence of average preserving extensions of the solutions and their flows is also considered.
We consider an homogenization problem for the second order elliptic equation $- Δu^{\varepsilon} + \dfrac{1}{\varepsilon} V(./\varepsilon) u^{\varepsilon} + νu^{\varepsilon} =f$ when the highly oscillatory potential $V$ belongs to a particular class of non-periodic potentials. We show the existence of an adapted corrector and prove the convergence of $u^{\varepsilon}$ to its homogenized limit.
We consider the scattering of elastic waves by highly oscillating anisotropic periodic media with bounded support. Applying the two-scale homogenization, we first obtain a constant coefficient second-order partial differential elliptic equation that describes the wave propagation of the effective or overall wave field. We study the rate of convergence by introducing complimentary boundary correctors. To account for dispersion induced by the periodic structure, we further pursue a higher-order homogenization. We then investigate the rate of convergence and formally obtain a fourth-order differe…
In this paper, for a family of second-order elliptic equations with rapidly oscillating periodic coefficients and rapidly oscillating periodic potentials, we are interested in the $H^1$ convergence rates and the Dirichlet eigenvalues and bounds of the normal derivatives of Dirichlet eigenfunctions. The $H^1$ convergence rates rely on the Dirichlet correctors and the first-order corrector for the oscillating potentials. And the bound results rely on an $O(\varepsilon)$ estimate in $H^1$ for solutions with Dirichlet condition.
This paper deals with the homogenization of fully nonlinear second order equation with an oscillating Dirichlet boundary data when the operator and boundary data are $\e$-periodic. We will show that the solution $u_\e$ converges to some function $\bar u(x)$ uniformly on every compact subset $K$ of the domain $D$. Moreover, $\bar u$ is a solution to some boundary value problem. For this result, we assume that the boundary of the domain has no (rational) flat spots and the ratio of elliptic constants $Λ/ λ$ is sufficiently large.
This note is a summary of the recent paper [9]. Here, we study the homogenization of elliptic systems with Dirichlet boundary condition, when both the coefficients and the boundary datum are oscillating. In particular, in the paper [9], we showed that, the solutions converge in L2 with a power rate, and we identified the homogenized limit system and the homogenized boundary data. Due to a boundary layer phenomenon, this homogenized system depends in a non trivial way on the boundary. The analysis in [9] answers a longstanding open problem, raised for instance in [4]
We prove the homogenization of fully nonlinear parabolic equations with periodic oscillating Dirichlet boundary conditions on certain general prescribed space-time domains. It was proved in [9,10] that for elliptic equations, the homogenized boundary data exists at boundary points with irrational normal directions, and it is generically discontinuous elsewhere. However for parabolic problems, on a flat moving part of the boundary, we prove the existence of continuous homogenized boundary data $\bar{g}$. We also show that, unlike the elliptic case, $\bar{g}$ can be discontinuous even if the ope…
In order to have a better description of homogenization for parabolic partial differential equations with periodic coefficients, we define the notion of parametric two-scale convergence. A compactness theorem is proved to justify this notion.
We study two types of asymptotic problems whose common feature - and difficulty- is to exhibit oscillating Dirichlet boundary conditions : the main contribution of this article is to show how to recover the Dirichlet boundary condition for the limiting equation. These two types of problems are (i) periodic homogenization problems for fully nonlinear, second-order elliptic partial differential equations set in a half-space and (ii) parabolic problems with an oscillating in time Dirichlet boundary condition. In order to obtain the Dirichlet boundary condition for the limiting problem, the key st…
For a family of second-order parabolic systems with rapidly oscillating and time-dependent periodic coefficients, we investigate the asymptotic behavior of fundamental solutions and establish sharp estimates for the remainders.
This article establishes an algebraic error estimate for the stochastic homogenization of fully nonlinear uniformly parabolic equations in stationary ergodic spatio-temporal media. The approach is similar to that of Armstrong and Smart in the study of quantitative stochastic homogenization of uniformly elliptic equations.
We study homogenization for fully nonlinear uniformly parabolic equations in stationary ergodic spatio-temporal media from the qualitative and quantitative perspective. We show that under suitable hypotheses, solutions to fully nonlinear uniformly parabolic equations in spatio-temporal media homogenize almost surely. In addition, we obtain a rate of convergence for this homogenization in measure, assuming that the environment is strongly mixing with a prescribed rate. A general methodology to study the stochastic homogenization of uniformly elliptic equations was introduced by Caffarelli, Soug…
For a family of second-order parabolic systems with bounded measurable, rapidly oscillating and time-dependent periodic coefficients, we investigate the sharp convergence rates of weak solutions in $L^2$. Both initial-Dirichlet and initial-Neumann problems are studied.
We consider energies defined as the Dirichlet integral of curves taking values in fast-oscillating manifolds converging to a linear subspace. We model such manifolds as subsets of $R^{m+m'}$ described by a constraint $(x_{m+1},\ldots,x_{m'})= δ\, φ(x_1/ε,\ldots, x_m/ε)$ where $ε$ is the period of the oscillation, $δ$ its amplitude and $φ$ its profile. The interesting case is $ε<\!<δ<\!<1$, in which the limit of the energies is described by a Finsler metric on $R^{m}$ which is defined by optimizing the contribution of oscillations on each level set $\{φ=c\}$. The formulas describing the limit m…
This article studies the homogenization of hyperbolic-parabolic equations in porous media with tiny holes. We assume that the holes are periodically distributed and that the coefficients of the equations are periodic. Using the multi-scale convergence method, we derive a homogenization result whose limit problem is defined on a fixed domain and is of the same type as the problem with oscillating coefficients.
This paper is concerned with quantitative homogenization of second-order parabolic systems with periodic coefficients varying rapidly in space and time, in different scales. We obtain large-scale interior and boundary Lipschitz estimates as well as interior $C^{1, α}$ and $C^{2, α}$ estimates by utilizing higher-order correctors. We also investigate the problem of convergence rates for initial-boundary value problems.
In this article, we study homogenization of a parabolic linear problem governed by a coefficient matrix with rapid spatial and temporal oscillations in periodically perforated domains with homogeneous Neumann data on the boundary of the holes. We prove results adapted to the problem for a characterization of multiscale limits for gradients and very weak multiscale convergence.
This paper is concerned with the optimal convergence rate in homogenization of higher order parabolic systems with bounded measurable, rapidly oscillating periodic coefficients. The sharp $O(\va)$ convergence rate in the space $L^2(0,T; H^{m-1}(\Om))$ is obtained for both the initial-Dirichlet problem and the initial-Neumann problem. The duality argument inspired by \cite{suslinaD2013} is used here.
In this paper, for a family of second-order parabolic equation with rapidly oscillating and time-dependent periodic coefficients, we are interested in an approximate two-sphere one-cylinder inequality for these solutions in parabolic periodic homogenization, which implies an approximate quantitative propagation of smallness. The proof relies on the asymptotic behavior of fundamental solutions and the Lagrange interpolation technique.
We consider a family of second-order parabolic systems in divergence form with rapidly oscillating and time-dependent coefficients, arising in the theory of homogenization. We obtain uniform interior $W^{1,p}$, Hölder, and Lipschitz estimates as well as boundary $W^{1,p}$ and Hölder estimates, using compactness methods. As a consequence, we establish uniform $W^{1,p}$ estimates for the initial-Dirichlet problems in $C^{1}$ cylinders.
In the homogenization of monotone parabolic partial differential equations with oscillations in both the space and time variables the gradients converges only weakly in $L^p$. In the present paper we construct a family of correctors, such that, up to a remainder which converges to zero strongly in $L^p$, we obtain strong convergence of the gradients in $L^p$.
In this paper we prove a general homogenization result for monotone parabolic problems with an arbitrary number of microscopic scales in space as well as in time, where the scale functions are not necessarily powers of epsilon. The main tools for the homogenization procedure are multiscale convergence and very weak multiscale convergence, both adapted to evo lution problems. At the end of the paper an example is given to concretize the use of the main result.
This paper considers a family of second-order periodic parabolic equations with highly oscillating potentials, which have been considered many times for the time-varying potentials in stochastic homogenization. Following a standard two-scale expansions illusion, we can guess and succeed in determining the homogenized equation in different cases that the potentials satisfy the corresponding assumptions, based on suitable uniform estimates of the $L^2(0,T;H^1(Ω))$-norm for the solutions. To handle the more singular case and obtain the convergence rates in $L^\infty(0,T;L^2(Ω))$, we need to estim…
We prove the homogenization of the Dirichlet problem for fully nonlinear elliptic operators with periodic oscillation in the operator and of the boundary condition for a general class of smooth bounded domains. This extends the previous results of Barles and Mironescu in half spaces. We show that homogenization holds despite a possible lack of continuity in the homogenized boundary data. The proof is based on a comparison principle with partial Dirichlet boundary data which is of independent interest.
In this paper we study the quantitative homogenization of second-order parabolic systems with locally periodic (in both space and time) coefficients. The $O(\varepsilon)$ scale-invariant error estimate in $L^2(0, T; L^{\frac{2d}{d-1}}(Ω))$ is established in $C^{1, 1}$ cylinders under minimum smoothness conditions on the coefficients. This process relies on critical estimates of smoothing operators. We also develop a new construction of flux correctors in the parabolic manner and a sharp estimate for temporal boundary layers.
Aim of this paper is to extend the continuous dependence estimates proved in \cite{JK1} to quasi-monotone systems of fully nonlinear second-order parabolic equations. As by-product of these estimates, we get an Hölder estimate for bounded solutions of systems and a rate of convergence estimate for the vanishing viscosity approximation. In the second part of the paper we employ similar techniques to study the periodic homogenization of quasi-monotone systems of fully nonlinear second-order uniformly parabolic equations. Finally, some examples are discussed.
This paper is concerned with a family of second-order elliptic systems in divergence form with rapidly oscillating periodic coefficients. We initiate the study of homogenization and boundary layers for Neumann problems with first-order oscillating boundary data. We identify the homogenized system and establish the sharp rate of convergence in $L^2$ in dimension three or higher. Sharp regularity estimates are also obtained for the homogenized boundary data in both Dirichlet and Neumann problems. The results are used to obtain a higher-order convergence rate for Neumann problems with non-oscilla…
In this paper we establish compactness results of multiscale and very weak multiscale type for sequences bounded in $L^{2}(0,T;H_{0}^{1}(Ω))$, fulfilling a certain condition. We apply the results in the homogenization of the parabolic partial differential equation $\varepsilon ^{p}\partial_{t}u_{\varepsilon }\left( x,t\right) -\nabla \cdot \left( a\left( x/\varepsilon ,x/\varepsilon ^{2},t/\varepsilon^{q},t/\varepsilon ^{r}\right) \nabla u_{\varepsilon }\left( x,t\right)\right) = f\left( x,t\right) $, where $0<p<q<r$. The homogenization result reveals two special phenomena, namely that the hom…
Pseudo-parabolic equations have been used to model unsaturated fluid flow in porous media. In this paper it is shown how a pseudo-parabolic equation can be upscaled when using a spatio-temporal decomposition employed in the Peszyn'ska-Showalter-Yi paper [8]. The spatial-temporal decomposition transforms the pseudo-parabolic equation into a system containing an elliptic partial differential equation and a temporal ordinary differential equation. To strengthen our argument, the pseudo-parabolic equation has been given advection/convection/drift terms. The upscaling is done with the technique of …
The aim of this paper is twofold. The first is to study the asymptotics of a parabolically scaled, continuous and space-time stationary in time version of the well-known Funaki-Spohn model in Statistical Physics. After a change of unknowns requiring the existence of a space-time stationary eternal solution of a stochastically perturbed heat equation, the problem transforms to the qualitative homogenization of a uniformly elliptic, space-time stationary, divergence form, nonlinear partial differential equation, the study of which is the second aim of the paper. An important step is the construc…
In this paper, we characterize the asymptotic behavior of a first-order stationary mean-field game (MFG) with a logarithm coupling, a quadratic Hamiltonian, and a periodically oscillating potential. This study falls into the realm of the homogenization theory, and our main tool is the two-scale convergence. Using this convergence, we rigorously derive the two-scale homogenized and the homogenized MFG problems, which encode the so-called macroscopic or effective behavior of the original oscillating MFG. Moreover, we prove existence and uniqueness of the solution to these limit problems.
We study the large scale behavior of elliptic systems with stationary random coefficient that have only slowly decaying correlations. To this aim we analyze the so-called corrector equation, a degenerate elliptic equation posed in the probability space. In this contribution, we use a parabolic approach and optimally quantify the time decay of the semigroup. For the theoretical point of view, we prove an optimal decay estimate of the gradient and flux of the corrector when spatially averaged over a scale R larger than 1. For the numerical point of view, our results provide convenient tools for …
This paper is concerned with homogenization of systems of linear elasticity with rapidly oscillating periodic coefficients. We establish sharp convergence rates in $L^2$ for the mixed boundary value problems with bounded measurable coefficients.
This paper is concerned with space-time homogenization problems for damped wave equations with spatially periodic oscillating elliptic coefficients and temporally (arithmetic) quasi-periodic oscillating viscosity coefficients. Main results consist of a homogenization theorem, qualitative properties of homogenized matrices which appear in homogenized equations and a corrector result for gradients of solutions. In particular, homogenized equations and cell problems will turn out to deeply depend on the quasi-periodicity as well as the log ratio of spatial and temporal periods of the coefficients…
In a Lipschitz cylinder, this paper is devoted to establish an almost sharp error estimate $O(\varepsilon\log_2(1/\varepsilon))$ in $L^2$-norm for parabolic systems of elasticity with initial-Dirichlet conditions, arising in the homogenization theory. To achieve the goal, with the parabolic distance function being a weight, we first developed some new weighted-type inequalities for the smoothing operator at scale $\varepsilon$ in terms of t-anisotropic Sobolev spaces, and then all the problems may be reduced to three kinds of estimate for the homogenized system, in which a weighted-type Caccio…
In this paper, we develop a viscosity method for Homogenization of Nonlinear Parabolic Equations constrained by highly oscillating obstacles or Dirichlet data in perforated domains. The Dirichlet data on the perforated domain can be considered as a constraint or an obstacle. Homogenization of nonlinear eigen value problems has been also considered to control the degeneracy of the Porous medium equation in perforated domains. For the simplicity, we consider obstacles that consist of cylindrical columns distributed periodically and perforated domains with punctured balls. If the decay rate of th…
In this paper, we determine the convergence speed of an upscaling of a pseudo-parabolic system containing drift terms with scale separation of size $ε\ll 1$. Both the upscaling and convergence speed determination exploit a natural spatial-temporal decomposition, which splits the pseudo-parabolic system into a spatial elliptic partial differential equation and a temporal ordinary differential equation. We extend the applicability to space-time domains that are a product of spatial and temporal domains, such as a time-independent perforated spatial domain. Finally, for special cases we show conv…
This paper studies the convergence rates in $L^2$ and $H^1$ of Dirichelt problems for Stokes systems with rapidly oscillating periodic coefficients, without any regularity assumptions on the coefficients.
The paper deals with homogenization problem for nonlinear elliptic and parabolic equations in a periodically perforated domain, a nonlinear Fourier boundary conditions being imposed on the perforation border. Under the assumptions that the studied differential equation satisfies monotonicity and 2-growth conditions and that the coefficient of the boundary operator is centered at each level set of unknown function, we show that the problem under consideration admits homogenization and derive the effective model.
This paper is concerned with boundary regularity estimates in the homogenization of elliptic equations with rapidly oscillating and high-contrast coefficients. We establish uniform nontangential-maximal-function estimates for the Dirichlet, regularity, and Neumann problems with $L^2$ boundary data in a periodically perforated Lipschitz domain.
We investigate quantitative estimates in homogenization of the locally periodic parabolic operator with multiscales $$ \partial_t- \text{div} (A(x,t,x/\varepsilon,t/κ^2) \nabla ),\qquad \varepsilon>0,\, κ>0. $$ Under proper assumptions, we establish the full-scale interior and boundary Lipschitz estimates. These results are new even for the case $κ=\varepsilon$, and for the periodic operators $ \partial_t-\text{div}(A(x/\varepsilon, t/\varepsilon^{\ell}) \nabla ),$ $0<\varepsilon,\ell<\infty, $ of which the large-scale Lipschitz estimate down to $\varepsilon+\varepsilon^{\ell/2}$ was recently …
The paper deals with homogenization of divergence form second order parabolic operators whose coefficients are periodic in spatial variables and random stationary in time. Under proper mixing assumptions, we study the limit behaviour of the normalized difference between solutions of the original and the homogenized problems. The asymptotic behaviour of this difference depends crucially on the ratio between spatial and temporal scaling factors. Here we study the case of self-similar parabolic diffusion scaling.
