Perturbations of Parabolic Equations and Diffusion Processes with Degeneration: Boundary Problems, Metastability, and Homogenization (original) (raw)

Homogenization of solutions of initial boundary value problems for parabolic systems

Functional Analysis and Its Applications, 2015

Let O ⊂ R d be a bounded C 1,1 domain. In L 2 (O; C n) we consider strongly elliptic operators A D,ε and A N,ε given by the differential expression b(D) * g(x/ε)b(D), ε > 0, with Dirichlet and Neumann boundary conditions, respectively. Here g(x) is a bounded positive definite matrixvalued function assumed to be periodic with respect to some lattice and b(D) is a first-order differential operator. We find approximations of the operators exp(−A D,ε t) and exp(−A N,ε t) for fixed t > 0 and small ε in the L 2 → L 2 and L 2 → H 1 operator norms with error estimates depending on ε and t. The results are applied to homogenize the solutions of initial boundary value problems for parabolic systems.

Abstract approach of degenerate parabolic equations with dynamic boundary conditions

arXiv: Analysis of PDEs, 2017

An initial boundary value problem of the nonlinear diffusion equation with a dynamic boundary condition is treated. The existence problem of the initial-boundary value problem is discussed. The main idea of the proof is an abstract approach from the evolution equation governed by the subdifferential. To apply this, the setting of suitable function spaces, more precisely the mean-zero function spaces, is important. In the case of a dynamic boundary condition, the total mass, which is the sum of volumes in the bulk and on the boundary, is a point of emphasis. The existence of a weak solution is proved on this basis.

HOMOGENIZATION OF INITIAL BOUNDARY VALUE PROBLEMS FOR PARABOLIC SYSTEMS WITH PERIODIC COEFFICIENTS

arXiv: 1503.05892 (2015); Applicable Analysis, 95:8 (2016), 1736–1775

Let mathcalOsubsetmathbbRd\mathcal{O} \subset \mathbb{R}^dmathcalOsubsetmathbbRd be a bounded domain of class C1,1C^{1,1}C1,1. In the Hilbert space L2(mathcalO;mathbbCn)L_2(\mathcal{O};\mathbb{C}^n)L2(mathcalO;mathbbCn), we consider matrix elliptic second order differential operators mathcalAD,varepsilon\mathcal{A}_{D,\varepsilon}mathcalAD,varepsilon and mathcalAN,varepsilon\mathcal{A}_{N,\varepsilon}mathcalAN,varepsilon with the Dirichlet or Neumann boundary condition on partialmathcalO\partial \mathcal{O}partialmathcalO, respectively. Here varepsilon>0\varepsilon>0varepsilon>0 is the small parameter. The coefficients of the operators are periodic and depend on mathbfx/varepsilon\mathbf{x}/\varepsilonmathbfx/varepsilon. The behavior of the operator e−mathcalA†,varepsilonte^{-\mathcal{A}_{†,\varepsilon}t}emathcalA,varepsilont, †=D,N†=D,N=D,N, for small varepsilon\varepsilonvarepsilon is studied. It is shown that, for fixed t>0t>0t>0, the operator e−mathcalA†,varepsilonte^{-\mathcal{A}_{†,\varepsilon}t}emathcalA,varepsilont converges in the L2L_2L2-operator norm to e−mathcalA†0te^{-\mathcal{A}_†^0 t}emathcalA0t, as varepsilonto0\varepsilon \to 0varepsilonto0. Here mathcalA†0\mathcal{A}_†^0mathcalA0 is the effective operator with constant coefficients. For the norm of the difference of the operators e−mathcalA†,varepsilonte^{-\mathcal{A}_{† ,\varepsilon}t}emathcalA,varepsilont and e−mathcalA†0te^{-\mathcal{A}_†^0 t}emathcalA0t a sharp order estimate (of order O(varepsilon)O(\varepsilon)O(varepsilon)) is obtained. Also, we find approximation for the exponential e−mathcalA†,varepsilonte^{-\mathcal{A}_{†,\varepsilon}t}emathcalA,varepsilont in the (L_2rightarrowH1)(L_2\rightarrow H^1)(L_2rightarrowH1)-norm with error estimate of order O(varepsilon1/2)O(\varepsilon ^{1/2})O(varepsilon1/2); in this approximation, a corrector is taken into account. The results are applied to homogenization of solutions of initial boundary value problems for parabolic systems.

Homogenization Results for Parabolic Problems with Dynamical Boundary Conditions

2004

The aim of this paper is to study the asymptotic behavior of the solution of a parabolic dynamical boundary-value problem in a periodically perforated domain. The domain is considered to be a fixed bounded open subset Ω⊂R N , in which identical and periodically distributed perforations (holes) of size e are made. In the perforated domain we consider a heat equation, with a Dirichlet condition on the exterior boundary and a dynamical boundary condition on the surface of the holes. The limit equation, as e→0, is a heat equation with constant coefficients, but with extra-terms coming from the influence of the non-homogeneous dynamical boundary condition.

Singular Perturbations for a Class of Degenerate Parabolic Equations with Mixed Dirichlet-Neumann Boundary Conditions

Annales mathématiques Blaise Pascal, 2003

We study the limit as goes to 0 + for the sequence (u) >0 of solutions to the Dirichlet problem for the quasilinear parabolic operators H (t, x, .) : u → ∂ t u + p i=1 ∂ x i ϕ i (t, x, u) + ψ(t, x, u) − ∆φ(u), where φ is a nondecreasing function, associated with a positiveness condition in an open bounded domain of R p , 1 ≤ p < +∞. The positive parameter being fixed, we first propose the definition of a weak entropy solution, the boundary conditions being expressed through the mathematical framework of the Divergence-Measure fields. Then, the uniqueness proof for refers to the technique of doubling the variables and the existence property is obtained through the artificial viscosity method. Lastly, a BV ∩ L ∞-estimate for the sequence (u) >0 is used to take the limit with .

Asymptotics in the Dirichlet problem for second order elliptic equations with degeneration on the boundary

Journal of Differential Equations

We study small perturbations of the Dirichlet problems for second order elliptic equations that degenerate on the boundary. The limit of the solution, as the perturbation tends to zero, is calculated. The result is based on a certain asymptotic self-similarity near the boundary, which holds in the generic case. In the last section, we briefly consider the stabilization of solutions to the corresponding parabolic equations with a small parameter. Metastability effects arise in this case: the asymptotics of the solution depends on the time scale. Initial-boundary value problem with the Neumann boundary condition is discussed in the last section as well.

On stabilization of the solutions of parabolic equations with small parameter

Proceedings of the National Academy of Sciences, 1984

We consider two classes of quasi-linear parabolic equations depending on a small parameter E. The asymptotic behavior of the solutions as t -X00 and E -O 0 is investigated by studying the associated Markov family. We find its dependence on the way t and E 1 go to infinity and on the initial point.

Stability for degenerate parabolic equations

Advances in Calculus of Variations, 2010

We show that an initial and boundary value problem related to the parabolic p-Laplace equation is stable with respect to p if the complement of the cylindrical domain satisfies a uniform capacity density condition. This condition is essentially optimal for our stability results.

Linear Stochastic Parabolic Equations, Degenerating on the Boundary of a Domain

Electronic Journal of Probability, 2001

A class of linear degenerate second-order parabolic equations is considered in arbitrary domains. It is shown that these equations are solvable using special weighted Sobolev spaces in essentially the same way as the non-degenerate equations in R d are solved using the usual Sobolev spaces. The main advantages of this Sobolev-space approach are less restrictive conditions on the coefficients of the equation and near-optimal space-time regularity of the solution. Unlike previous works on degenerate equations, the results cover both classical and distribution solutions and allow the domain to be bounded or unbounded without any smoothness assumptions about the boundary. An application to nonlinear filtering of diffusion processes is discussed.