Solutions Of Linear Parabolic Equations Research Articles

Gradient weighted norm inequalities for very weak solutions of linear parabolic equations with BMO coefficients

In this paper, we give a short proof of the Lorentz estimates for gradients of very weak solutions to the linear parabolic equations with the Muckenhoupt class A q -weights u t − div ( A ( x , t ) ∇ u ) = div ( F ) , in a bounded domain Ω × ( 0 , T ) ⊂ R N + 1 , where A has a small mean oscillation, and Ω is a Lipchistz domain with a small Lipschitz constant.

Weighted Parabolic Aleksandrov Estimates: PDE and Stochastic Versions

We prove several pointwise estimates for solutions of linear parabolic equations with measurable coefficients in smooth cylinders through the weighted Ld+1-norm of the free term. The weights allow the free term to blow up near the lateral boundary. These are the first such results for parabolic equations with measurable coefficients. We also present the corresponding weighted estimates for occupation times of diffusion processes.

Generic Transversality of Heteroclinic and Homoclinic Orbits for Scalar Parabolic Equations

In this paper, we consider the scalar reaction–diffusion equations $$\partial _t u= \Delta u+f(x,u, \nabla u)$$ on a bounded domain $$\Omega \subset \mathbb {R}^d$$ of class $$\mathcal {C}^{2,\gamma }$$. We show that the heteroclinic and homoclinic orbits connecting hyperbolic equilibria and hyperbolic periodic orbits are transverse, generically with respect to f. One of the main ingredients of the proof is an accurate study of the singular nodal set of solutions of linear parabolic equations. Our main result is a first step for proving the genericity of Kupka–Smale property, the generic hyperbolicity of periodic orbits remaining unproved.

The Phragmén-Lindelöf theorem for Lp-viscosity solutions of fully nonlinear parabolic equations with unbounded ingredients

The Phragmén-Lindelöf theorem for classical solutions of linear parabolic equations in cones, whose vertex is the origin of Rn+1, was established by Avner Friedman in 1957. It is extended to fully nonlinear parabolic equations with unbounded coefficients.

A class of two- and three-level implicit methods of order two in time and four in space based on half-step discretization for two-dimensional fourth order quasi-linear parabolic equations

Novel compact schemes based on half-step discretization are developed to solve two dimensional fourth order quasi-linear parabolic equations. These discretizations use a single computational cell and the boundary conditions are incorporated in a natural way without the use of any fictitious points or discretization. The schemes have second order accuracy in time and fourth order accuracy in space. The proposed methods are directly applicable to problems constituting singular terms. We also present operator splitting method for the solution of linear parabolic equations which allows the use of the one-dimensional tri-diagonal solver multiple times. It is shown that the operator splitting method is unconditionally stable. The numerical schemes have been successfully applied to the two-dimensional Extended Fisher–Kolmogorov equation, Sivashinsky equation, Boussinesq equation and vibrations problem. The illustrative results confirm the theoretical order of magnitude and accuracy of the proposed methods.

Discrete Maximal Parabolic Regularity for Galerkin Finite Element Methods for Nonautonomous Parabolic Problems

The main goal of the paper is to establish time semidiscrete and space-time fully discrete maximal parabolic regularity for the lowest order time discontinuous Galerkin solution of linear parabolic equations with time-dependent coefficients. Such estimates have many applications. As one of the applications we establish best approximations type results with respect to the $L^p(0,T;L^2(\Omega))$ norm for $1\le p\le \infty$.

Discrete maximal parabolic regularity for Galerkin finite element methods

The main goal of the paper is to establish time semidiscrete and space-time fully discrete maximal parabolic regularity for the time discontinuous Galerkin solution of linear parabolic equations. Such estimates have many applications. They are essential, for example, for establishing optimal a priori error estimates in non- Hilbertian norms without unnatural coupling of spatial mesh sizes with time steps.

Gradient estimates and liouville theorems for linear and nonlinear parabolic equations on riemannian manifolds

In this article, we will derive local elliptic type gradient estimates for positive solutions of linear parabolic equations (Δ−∂∂t)u(x,t)+q(x,t)u(x,t)=0 and nonlinear parabolic equations (Δ−∂∂t)u(x,t)+h(x,t)up(x,t)=0(p>1) on Riemannian manifolds. As applications, we obtain some theorems of Liouville type for positive ancient solutions of such equations. Our results generalize that of Souplet-Zhang ([1], Bull. London Math. Soc. 38(2006), 1045–1053) and the author ([2], Nonlinear Anal. 74 (2011), 5141–5146).

Explicit Density Approximations for Local Volatility Models Using Heat Kernel Expansions

Heat kernel perturbation theory is a tool for constructing explicit approximation formulas for the solutions of linear parabolic equations. We review the crux of this perturbative formalism and then apply it to differential equations which govern the transition densities of several local volatility processes. In particular, we compute all the heat kernel coefficients for the CEV and quadratic local volatility models; in the later case, we are able to use these to construct an exact explicit formula for the processes' transition density. We then derive low order approximation formulas for the cubic local volatility model, an affine-affine short rate model, and a generalized mean reverting CEV model. We finally demonstrate that the approximation formulas are accurate in certain model parameter regimes via comparison to Monte Carlo simulations.

Quasi-analytic solutions of linear parabolic equations

Uniqueness of solutions of the Cauchy problem of a parabolic equation, and the related question of analyticity with respect to time, depend on global properties of the solution. We demonstrate that if the growth of the initial function (and, if relevant, of the inhomogeneous part and its derivatives) is not too great, solutions of non-stationary parabolic equations whose coefficients belong to quasi-analytic classes are quasi-analytic with respect to all variables and hence are unique. This study is motivated by the problem of endogenous completeness in continuous-time financial markets.

DOMAIN PERTURBATION FOR PARABOLIC EQUATIONS

We study the effect of domain perturbation on the behaviour of parabolic equations. The first aspect considered in this thesis is the behaviour of solutions under changes of the domain. We show how solutions of linear and semilinear parabolic equations behave as a sequence of domains Ωn converges to an open set Ω in a certain sense. In particular, we are interested in singular domain perturbations so that a change of variables is not possible on these domains. For autonomous linear equations, it is known that convergence of solutions under domain perturbation is closely related to the corresponding elliptic equations via a standard semigroup theory. We show that there is also a relation between domain perturbation for non-autonomous linear parabolic equations and domain perturbation for elliptic equations. The key result for this is the equivalence of Mosco convergences between various closed and convex subsets of Banach spaces. An important consequence is that the same conditions for a sequence of domains imply convergence of solutions under domain perturbation for both parabolic and elliptic equations. By applying variational methods, we obtain the convergence of solutions of initial value problems under Dirichlet or Neumann boundary conditions. A similar technique can be applied to obtain the convergence of weak solutions of parabolic variational inequalities when the underlying convex set is perturbed. Using the linear theory, we then study domain perturbation for initial boundary value problems of semilinear type. We are also interested in the behaviour of bounded entire solutions of parabolic equations defined on the whole real line. We establish a convergence result for bounded entire solutions of linear parabolic equations under L2 and Lp-norms. For the Lp-theory, we also prove Holder regularity of bounded entire solutions with respect to time. In addition, the persistence of some classes of bounded entire solutions is given for semilinear equations using the Leray-Schauder degree theory. The second aspect is to study the dynamics of parabolic equations under domain perturbation. In this part, we consider parabolic equation as a dynamical system in an

Persistence of bounded solutions of parabolic equations under domain perturbation

The aim of this paper is to study the behavior of bounded solutions of parabolic equations on the whole real line under perturbation of the underlying domain. We give the convergence of bounded solutions of linear parabolic equations in the L 2 and the L p -settings. For the L p -theory, we also prove the Holder regularity of bounded solutions with respect to time. In addition, we study the persistence of a class of bounded solutions which decay to zero at t → ±∞ of semilinear parabolic equations under domain perturbation.

Convergence of fundamental solutions of linear parabolic equations under Cheeger–Gromov convergence

In this note we show the convergence of the fundamental solutions of the parabolic equations assuming the Cheeger–Gromov convergence of the underlying manifolds and the uniform L1-bound of the solutions. We also prove a local integral estimate of fundamental solutions.

Explicit Density Approximations for Local Volatility Models Using Heat Kernel Expansions

Heat kernel perturbation theory is a tool for constructing explicit approximation formulas for the solutions of linear parabolic equations. We review the crux of this perturbative formalism and then apply it to differential equations which govern the transition densities of several local volatility processes. In particular, we compute all the heat kernel coefficients for the CEV and quadratic local volatility models; in the later case, we are able to use these to construct an exact explicit formula for the processes' transition density. We then derive low order approximation formulas for the cubic local volatility model, an affine-affine short rate model, and a generalized mean reverting CEV model. We finally demonstrate that the approximation formulas are accurate in certain model parameter regimes via comparison to Monte Carlo simulations.

An elementary approach to nonexistence of solutions of linear parabolic equations

This note presents an elementary approach to the nonexistence of solu- tions of linear parabolic initial-boundary value problems considered in the Feller test. 1. Introduction. Our goal is to study the solvability of the initial- boundary value problem for a linear partial dierential equation of parabolic type in one space dimension of the form ut = a(x)uxx +b(x)ux; (1) u(0;t) = 0; u(1;t) = 1; (2) u(x;0) = u0(x); (3) where x 2 (0;1), t > 0, a and b are given measurable nonnegative func- tions on (0;1), and u0 is a continuous nonnegative function satisfying the compatibility conditions u0(0) = 0, u0(1) = 1. The question of the existence of a solution satisfying the boundary con- ditions (2) is delicate. William Feller developed tools which permitted him to give a complete answer to that question in his seminal 1952 paper (5). The theory presented there, culminating in the famous Feller test, is by no means elementary, and is based on a fine probabilistic analysis of stochastic processes associated with linear parabolic equations with diusion

Maximum Principle and Convergence of Fundamental Solutions for the Ricci Flow

In this paper we will prove a maximum principle for the solutions of linear parabolic equation on complete non-compact manifolds with a time varying metric. We will prove the convergence of the Neumann Green function of the conjugate heat equation for the Ricci flow in $B_k\times (0,T)$ to the minimal fundamental solution of the conjugate heat equation as $k\to\infty$. We will prove the uniqueness of the fundamental solution under some exponential decay assumption on the fundamental solution. We will also give a detail proof of the convergence of the fundamental solutions of the conjugate heat equation for a sequence of pointed Ricci flow $(M_k\times (-\alpha,0],x_k,g_k)$ to the fundamental solution of the limit manifold as $k\to\infty$ which was used without proof by Perelman in his proof of the pseudolocality theorem for Ricci flow [P].

Adaptive Crank-Nicolson methods with dynamic finite-element spaces for parabolic problems

We construct a posteriori error estimators for approximate solutions of linear parabolic equations. We consider discretizations of the problem by modified discontinuous Galerkin schemes in time and continuous Galerkin methods in space. Especially, finite element spaces are permitted to change at different time levels. Exploiting Crank-Nicolson reconstruction idea introduced by Akrivis, Makridakis & Nochetto [2], we derive space-time a posteriori error estimators of second order in time for the Crank-Nicolson-Galerkin finite element method.

Differentiability with respect to parameters of weak solutions of linear parabolic equations

We consider the differentiability of weak solutions of linear parabolic equations with respect to parameters and initial data. Under natural assumptions, it is shown that solutions possess as much differentiability with respect to the data as do the terms appearing in the equation. The derivatives are shown to satisfy the appropriate sensitivity equations. The theoretical results are illustrated with an example.

Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems

We derive a posteriori error estimates for fully discrete approximations to solutions of linear parabolic equations. The space discretization uses finite element spaces that are allowed to change in time. Our main tool is an appropriate adaptation of the elliptic reconstruction technique, introduced by Makridakis and Nochetto. We derive novel a posteriori estimates for the norms of L ∞ ⁡ ( 0 , T ; L 2 ⁡ ( Ω ) ) \operatorname {L}_\infty (0,T;\operatorname {L}_2(\Omega )) and the higher order spaces, L ∞ ⁡ ( 0 , T ; H 1 ⁡ ( Ω ) ) \operatorname {L}_\infty (0,T;\operatorname {H}^1(\Omega )) and H 1 ⁡ ( 0 , T ; L 2 ⁡ ( Ω ) ) \operatorname {H}^1(0,T;\operatorname {L}_2(\Omega )) , with optimal orders of convergence.

A domain decomposition and mixed method for a linear parabolic boundary value problem

In this paper, we discuss the convergence of a domain decomposition method for the solution of linear parabolic equations in their mixed formulations. The subdomain meshes need not be quasi‐uniform; they are composed of triangles or quadrilaterals that do not match at interfaces. For the ease of computation, this lack of continuity is compensated by a mortar technique based on piecewise constant (discontinuous) multipliers. It is shown that the method on triangles, parallelograms or slightly distorted parallelograms is convergent at the expense of a half‐order loss of accuracy compared with mortar methods based on piecewise linear multipliers.