Order Parabolic Operator Research Articles

Analysis of selfsimilar solutions and a comparison principle for an heterogeneous diffusion cooperative system with advection and non-linear reaction

The present analysis introduces a system of cooperative species formulated with a high order parabolic operator, a Fisher-KPP reaction and a linear advection. Firstly, the oscillatory behaviour of solutions is shown to exist with a shooting method approach. It is to be highlighted that the existence of oscillatory patterns (also called instabilities) is an inherent property of high order operators. Afterwards, existence and uniqueness results are provided. The most remarkable result, obtained during the existence exercise, is related with the finding of a particular time-degenerate bound for the advection term that ensures positivity of solutions. This is one of the main results as such positivity property does not hold for high order operators in general. Indeed, high-order operators provide oscillatory solutions that may induce such solutions to be negative in the proximity of the null state introduced by the Fisher-KPP reaction term. As a consequence, a comparison principle does not hold as formulated in order two operators. Further, a positive maximal kernel with similar asymptotic behaviour compared to the high order kernel has been shown to exist and a precise assessment has been done with a computational exercise. Eventually, such a positive maximal kernel permits to show the existence of a comparison principle.

Unique continuation through hyperplane for higher order parabolic and Schrödinger equations

Consider the higher order parabolic operator and the higher order Schrödinger operator in where m and n are any positive integers. Under certain lower order and regularity assumptions, we prove that if solutions to the linear problems vanish when then the solutions vanish in X. Such results are global if n > 1, and we also prove some relevant local results.

The Dirichlet boundary problem for second order parabolic operators satisfying a Carleson condition

We establish L^p , 2\le p\le\infty , solvability of the Dirichlet boundary value problem for a parabolic equation u_t- \mathrm {div}(A\nabla u) - \boldsymbol{B}\cdot\nabla u =0 on time-varying domains with coefficient matrices A=[a_{ij}] and \boldsymbol{B} = [b_{i}] that satisfy a small Carleson condition. The results are sharp in the following sense. For a given value of 1 < p < \infty there exists operators that satisfy Carleson condition but fail to have L^p solvability of the Dirichlet problem. Thus the assumption of smallness is sharp . Our results complements results of Hofmann, Lewis and Rivera-Noriega, where solvability of parabolic L^p (for some large p ) Dirichlet boundary value problem for coefficients that satisfy large Carleson condition was established. We also give a new (substantially shorter) proof of these results.

The Dirichlet problem for second order parabolic operators in divergence form

We study parabolic operators H = ∂t − div λ,x A(x, t)∇ λ,x in the parabolic upper half space R n+2 + = {(λ, x, t) : λ > 0}. We assume that the coefficients are real, bounded, measurable, uniformly elliptic, but not necessarily symmetric. We prove that the associated parabolic measure is absolutely continuous with respect to the surface measure on R n+1 in the sense defined by A∞(dx dt). Our argument also gives a simplified proof of the corresponding result for elliptic measure.

Carleman estimates and Unique Continuation Property for 1-D viscous Camassa-Holm equation

This paper is devoted to studying the 1-D viscous Camassa-Holm equation on a bounded interval. We first deduce the existence and uniqueness of strong solution to the viscous Camassa-Holm equation by using Galerkin method. Then we establish an identity for a second order parabolic operator, by applying this identity we obtain two global Carleman estimates for the linear viscous Camassa-Holm operator. Based on these estimates, we obtain two types of Unique Continuation Property for the viscous Camassa-Holm equation.

BMO solvability and the A∞ condition for second order parabolic operators

We prove that a sharp regularity property (A∞) of parabolic measure for operators in certain time-varying domains is equivalent to a Carleson measure property of bounded solutions. This equivalence was established in the elliptic case by Kenig, Kirchheim, Pipher and Toro, improving an earlier result of Kenig, Dindos and Pipher for solutions with data in BMO. The connection between regularity of the elliptic measure and certain Carleson measure properties of solutions was established in order to study solvability of boundary value problems for non-symmetric divergence form operators (Kenig, Koch, Pipher, and Toro). The extension to the parabolic setting requires an approach to the key estimates of the aforementioned works that primarily exploits the maximum principle. For various classes of parabolic operators ([24]), this criterion also provides an easier route to establish the solvability of the Dirichlet problem with data in Lp for some p, and also to quantify these results in several aspects.

Local exact controllability to the trajectories of the Swift–Hohenberg equation

This paper is addressed to study the local exact controllability to the trajectories of the Swift–Hohenberg equation. We first establish a new global Carleman estimate for the fourth order parabolic operator, and we consider the exact controllability to the trajectories of the linearized system. Then, by Local inversion theorem, we show that the same result holds for the Swift–Hohenberg equation.

Square function estimates and the Kato problem for second order parabolic operators in [formula omitted

In [4] the Kato square root problem for second order complex uniformly elliptic operators of the form L=−div(A∇f), with only bounded and measurable coefficients, was solved. The solution is a consequence of a square function estimate for the operator (1+λ2L)−1λL. This and related square function estimates have recently spurred a wave of new and ground breaking results in the area of elliptic PDEs. In this paper we establish similar square function estimates for second order parabolic operators in Rn+1 of the form ∂t+L paving the way for important developments in the area of parabolic PDEs.

Explicit Form of the Fundamental Solution to a Second Order Parabolic Operator

We propose and justify an explicit representation of the fundamental solution to a system of parabolic equations with special initial conditions. Bibliography: 5 titles.

Homogenization of random parabolic operators. Diffusion approximation

This paper deals with homogenization of divergence form second order parabolic operators whose coefficients are periodic with respect to the 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.

Insensitizing controls for the Cahn-Hilliard type equation

This paper is addressed to showing the existence of insensitizing controls for the one-dimensional Cahn-Hilliard type equation with a superlinear nonlinearity. We solve this problem by reducing the original problem to a controllability problem. The crucial point in this paper is an observability estimate for a linearized cascade system of the Cahn-Hilliard type equation. In order to obtain this observability estimate, we establish a global Carleman estimate for a fourth order parabolic operator.

Pointwise gradient estimates for evolution operators associated with Kolmogorov operators

We determine sufficient conditions for the occurrence of a pointwise gradient estimate for the evolution operators associated with nonautonomous second order parabolic operators with (possibly) unbounded coefficients. Moreover, we exhibit a class of operators which satisfy our conditions.

Diffusion effects in a superconductive model

A superconductive model characterized by a third order parabolic operator $ {\mathcal L}_\varepsilon $ is analyzed. When the viscous terms, represented by higher-order derivatives, tend to zero, a hyperbolic operator $ {\mathcal L}_0 $ appears. Furthermore, if ${\mathcal P}_\varepsilon$ is the Dirichlet initial-boundary value problem for $ {\mathcal L}_\varepsilon$, when ${\mathcal L} _\varepsilon $ turns into ${\mathcal L}_0 , $ ${\mathcal P}_\varepsilon$ turns into a problem ${\mathcal P}_0$ with the same initial-boundary conditions of ${\mathcal P}_\varepsilon $. As long as the higher-order derivatives of the solution of ${\mathcal P}_0$ are bounded, an estimate of solution for the nonlinear problem related to the remainder term $ r, $ is achieved. Moreover, some classes of explicit solutions related to $ {\mathcal P}_0 $ are determined, proving the existence of at least one motion whose derivatives are bounded. The estimate shows that the diffusion effects are bounded even when time tends to infinity.

The Dirichlet problem for second order parabolic operators in non‐cylindrical domains

AbstractIn this paper we develope a perturbation theory for second order parabolic operators in non‐divergence form. In particular we study the solvability of the Dirichlet problem in non cylindrical domains with Lp ‐data on the parabolic boundary (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Homogenization of nonlinear reaction-diffusion equation with a large reaction term

This paper deals with the homogenization of a second order parabolic operator with a large nonlinear potential and periodically oscillating coefficients of both spatial and temporal variables. Under a centering condition for the nonlinear zero-order term, we obtain the effective problem and prove a convergence result. The main feature of the homogenized equation is the appearance of a non-linear convection term.

On Monte Carlo algorithms applied to Dirichlet problems for parabolic operators in the setting of time-dependent domains

Dirichlet problems for second order parabolic operators in space-time domains Ω ⊂ ℝn+1 are of paramount importance in analysis, partial differential equations and applied mathematics. These problems can be approached in many different ways using techniques from partial differential equations, potential theory, stochastic differential equations, stopped diffusions and Monte Carlo methods. The performance of any technique depends on the structural assumptions on the operator, the geometry and smoothness properties of the space-time domain Ω, the smoothness of the Dirichlet data and the smoothness of the coefficients of the operator under consideration. In this paper, which mainly is of numerical nature, we attempt to further understand how Monte Carlo methods based on the numerical integration of stochastic differential equations perform when applied to Dirichlet problems for uniformly elliptic second order parabolic operators and how their performance vary as the smoothness of the boundary, Dirichlet data and coefficients change from smooth to non-smooth. Our analysis is set in the genuinely parabolic setting of time-dependent domains, which in itself adds interesting features previously only modestly discussed in the literature. The methods evaluated and discussed include elaborations on the non-adaptive method proposed by Gobet [ESAIM: Probability and Statistics 5: 261–297, 2001] based on approximation by half spaces and exit probabilities and the adaptive method proposed in [Lecture Notes in Computational Science and Engineering 44: 59–88, 2005] for weak approximation of stochastic differential equations.

Carleman Estimates, Optimal Three Cylinder Inequality, and Unique Continuation Properties for Solutions to Parabolic Equations

We prove a Carleman estimate with singular weight for general second order parabolic operators. As a consequence we get an optimal three cylinder inequality for any solution, u, of a second order parabolic equation Lu=0, in D× (−T, T), where Dis a domain of and Tis a positive number. Further, we prove that if , for every and every t∈(−T, T), then uvanishes in D× (−T, T).

Diffusion and wave behaviour in linear Voigt model

A boundary value problem P ε related to a third order parabolic equation with a small parameter ε is analized. This equation models the one-dimensional evolution of many dissipative media as viscoelastic fluids or solids, viscous gases, superconducting materials, incompressible and electrically conducting fluids. Moreover, the third order parabolic operator regularizes various nonlinear second order wave equations. In this paper, the hyperbolic and parabolic behaviour of the solution of P ε is estimated by means of slow time τ= εt and fast time θ= t/ ε. As consequence, a rigorous asymptotic approximation for the solution of P ε is established. To cite this article: M. De Angelis, P. Renno, C. R. Mecanique 330 (2002) 21–26

Heat Kernel Estimates for Operators with Boundary Conditions

We prove Gaussian upper bounds for kernels associated with non – symmetric, non – autonomous second order parabolic operators of divergence form subject to various boundary conditions. The growth of the kernel in time is determined by theboundary conditions and the geometric properties of the domain. The theory gives a unified treatment for Dirichlet, Neumann and Robin boundary conditions, and the existence of a Gaussian type bound is essentially reduced to verifying some properties of the Hilbert space in the weak formulation of the problem.

On removable lateral singularities for quasilinear parabolic PDE's

Suppose that L is a second order parabolic differential operator with smooth coefficients in a bounded smooth domain D and let Q T = [ 0 , T ] × D = [0, T] × D. Let 1 < α ≤ 2, and let Γ be a compact subset of the lateral boundary ∂ ∂ Q of Q T . We show that the following four properties are equivalent: o (a) Γ is a removable lateral singularity for the equation K + Lu = u α in Q T ; (b) The Poisson capacity of Γ is equal to zero; (c) The Besov capacity, Cap 1/α,2/α,α′(Γ), is equal to 0; (d) Γ is not hit by the graph of the ( L,α)-superdiffusion in Q T