Non-autonomous Parabolic Problems Research Articles

Adaptive Space-Time Finite Element Methods for Non-autonomous Parabolic Problems with Distributional Sources

Abstract We consider locally stabilized, conforming finite element schemes on completely unstructured simplicial space-time meshes for the numerical solution of parabolic initial-boundary value problems with variable coefficients that are possibly discontinuous in space and time. Distributional sources are also admitted. Discontinuous coefficients, non-smooth boundaries, changing boundary conditions, non-smooth or incompatible initial conditions, and non-smooth right-hand sides can lead to non-smooth solutions. We present new a priori and a posteriori error estimates for low-regularity solutions. In order to avoid reduced rates of convergence that appear when performing uniform mesh refinement, we also consider adaptive refinement procedures based on residual a posteriori error indicators and functional a posteriori error estimators. The huge system of space-time finite element equations is then solved by means of GMRES preconditioned by space-time algebraic multigrid. In particular, in the 4d space-time case, simultaneous space-time parallelization can considerably reduce the computational time. We present and discuss numerical results for several examples possessing different regularity features.

Closedness and invertibility for the sum of two closed operators

We show a Kalton–Weis type theorem for the general case of noncommuting operators. More precisely, we consider sums of two possibly noncommuting linear operators defined in a Banach space such that one of the operators admits a bounded $H^\infty$-calculus, the resolvent of the other one satisfies some weaker boundedness condition and the commutator of their resolvents has certain decay behavior with respect to the spectral parameters. Under this consideration, we show that the sum is closed and that after a sufficiently large positive shift it becomes invertible and moreover sectorial. As an application we recover a classical result on the existence, uniqueness, and maximal $L^{p}$-regularity for solutions of the abstract linear nonautonomous parabolic problem.

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 $1\le p\le \infty$.

On maximal parabolic regularity for non-autonomous parabolic operators

We consider linear inhomogeneous non-autonomous parabolic problems associated to sesquilinear forms, with discontinuous dependence of time. We show that for these problems, the property of maximal parabolic regularity can be extrapolated to time integrability exponents r≠2. This allows us to prove maximal parabolic Lr-regularity for discontinuous non-autonomous second-order divergence form operators in very general geometric settings and to prove existence results for related quasilinear equations.

Hölder-estimates for non-autonomous parabolic problems with rough data

In this paper we establish Hölder estimates for solutions to nonautonomous parabolic equations on non-smooth domains which are complemented with mixed boundary conditions. The corresponding elliptic operators are of divergence type, the coefficient matrix of which depends only measurably on time. These results are in the tradition of the classical book of Ladyshenskaya et al. [40], which also serves as the starting point for our investigations.

On existence and uniqueness for non-autonomous parabolic Cauchy problems with rough coefficients

We consider existence and uniqueness issues for the initial value problem of parabolic equations ∂ t u = divA∇u on the upper half space, with initial data in L p spaces. The coefficient matrix A is assumed to be uniformly elliptic, but merely bounded measurable in space and time. For real coefficients and a single equation, this is an old topic for which a comprehensive theory is available, culminating in the work of Aronson. Much less is understood for complex coefficients or systems of equations except for the work of Lions, mainly because of the failure of maximum principles. In this paper, we come back to this topic with new methods that do not rely on maximum principles. This allows us to treat systems in this generality when p ≥ 2, or under certain assumptions such as bounded variation in the time variable (a much weaker assumption that the usual Holder continuity assumption) when p < 2. We reobtain results for real coefficients, and also complement them. For instance, we obtain uniqueness for arbitrary L p data, 1 ≤ p ≤ ∞, in the class L ∞ (0, T ; L p (R n)). Our approach to the existence problem relies on a careful construction of propagators for an appropriate energy space, encompassing previous constructions. Our approach to the uniqueness problem, the most novel aspect here, relies on a parabolic version of the Kenig-Pipher maximal function, used in the context of elliptic equations on non-smooth domains. We also prove comparison estimates involving conical square functions of Lusin type and prove some Fatou type results about non-tangential convergence of solutions. Recent results on maximal regularity operators in tent spaces that do not require pointwise heat kernel bounds are key tools in this study.

A Landesman–Lazer type result for periodic parabolic problems on [formula omitted] at resonance

We are concerned with T-periodic solutions of nonautonomous parabolic problem of the form ut=Δu+V(x)u+f(t,x,u), t>0, x∈RN, with V∈L∞(RN)+Lp(RN), where 2<p<∞ if N=1,2 and N≤p<∞ for N≥3 and T-periodic continuous perturbation f:RN×R→R. The so-called resonant case is considered, i.e. when N:=Ker(Δ+V)≠{0} and f is bounded. We derive a formula for the fixed point index of the associated translation along trajectories operator in terms of the Brouwer topological degree of the time averaging of the mapping fˆ:N→N, fˆ being the restriction of f to N. By use of the formula and continuation techniques we show that Landesman–Lazer type conditions imply the existence of T-periodic solutions.

Topological theory of non-autonomous parabolic evolution inclusions on a noncompact interval and applications

In this paper we consider the topological structure of the solution set of non-autonomous parabolic evolution inclusions with time delay, defined on non-compact intervals. The result restricted to compact intervals is then extended to non-autonomous parabolic control problems with time delay. Moreover, as the applications of the information about the structure, we establish the existence result of global integral solutions for non-autonomous Cauchy problems subject to nonlocal condition, and prove the invariance of a reachability set for non-autonomous control problems under single-valued nonlinear perturbations. Finally, some illustrating examples are supplied.

Dynamics of non-autonomous parabolic problems with subcritical and critical nonlinearities

This is a subsequent work of our previous one in [25]. Let the nonlinear terms of non-autonomous parabolic problems with singular initial data satisfy subcritical and critical growth conditions. We first establish the existence of uniform attractors in W1,r(Ω), 1<r<N, for the family of processes corresponding to the equations with external forces being translation bounded but not translation compact. Then, we prove the existence of pullback attractors in Lr(Ω) and W1,r(Ω), respectively, for the process corresponding to the equation with the weaker assumption on the external force than previous one. Finally, we investigate the robust of attractors and establish the existence of pullback exponential attractors for the process acting in Lr(Ω) and W1,r(Ω), respectively.

Non-autonomous parabolic problems with singular initial data in weighted spaces

Non-autonomous parabolic problems with singular initial data in weighted spaces

Attractors for non-autonomous parabolic problems with singular initial data

In this paper, we study the asymptotic behavior of solutions of non-autonomous parabolic problems with singular initial data. We first establish the well-posedness of the equation when the initial data belongs to L r ( Ω ) ( 1 < r < ∞ ) and W 1 , r ( Ω ) ( 1 < r < N ), respectively. When the initial data belongs to L r ( Ω ) , we establish the existence of uniform attractors in L r ( Ω ) for the family of processes with external forces being translation bounded but not translation compact in L loc p ( R ; L r ( Ω ) ) . When we consider the existence of uniform attractors in H 0 1 ( Ω ) , the solution of equation lacks the higher regularity, so we introduce a new type of solution and prove the existence result. For the long time behavior of solutions of the equation in W 1 , r ( Ω ) , we only obtain the uniform attracting property in the weak topology.

Non-autonomous Ornstein-Uhlenbeck equations in exterior domains

In this paper, we consider non-autonomous Ornstein-Uhlenbeck operators in smooth exterior domains $\Omega\subset \mathbb R^d$ subject to Dirichlet boundary conditions. Under suitable assumptions on the coefficients, the solution of the corresponding non-autonomous parabolic Cauchy problem is governed by an evolution system $\{P_\Omega(t,s)\}_{0\le s\le t}$ on $L^p(\Omega)$ for $1< p < \infty$. Furthermore, $L^p$-estimates for spatial derivatives and $L^p$-$L^q$ smoothing properties of $P_\Omega(t,s),\,0\le s\le t,$ are obtained.

Non-autonomous semilinear evolution equations with almost sectorial operators

Inspired by the theory of semigroups of growth α, we construct an evolution process of growth α. The abstract theory is applied to study semilinear singular non-autonomous parabolic problems. We prove that, under natural assumptions, a reasonable concept of solution can be given to such semilinear singularly non-autonomous problems. Applications are considered to non-autonomous parabolic problems in space of Holder continuous functions and to a parabolic problem in a domain \(\Omega \subset {\mathbb{R}}^n\) with a one dimensional handle.

Stability of linear multistep methods and applications to nonlinear parabolic problems

In the present paper, stability and convergence properties of linear multistep methods are investigated. The attention is focused on parabolic problems and variable stepsizes. Under weak assumptions on the method and the stepsize sequence an asymptotic stability result is shown. Further, stability bounds for linear nonautonomous parabolic problems with Hölder continuous operator are given. With the help of these results, convergence estimates for semilinear and fully nonlinear parabolic problems are derived.

Time-dependent parabolic problems on non-cylindrical domains with inhomogeneous boundary conditions

We study the relationship between the Dirichlet problem and the Cauchy problem with inhomogeneous boundary conditions for local operators. Our results are applied to non-autonomous parabolic problems on non-cylindrical domains.

Shadowing for Nonautonomous Parabolic Problems with Applications to Long-Time Error Bounds

Shadowing results for time discretizations of nonautonomous parabolic evolution equations z'(t) = A(t) z(t) + f(t) are given. The analysis is performed in an abstract Banach space setting of uniformly sectorial operators A(t). Under natural assumptions on the operators A(t), it is shown that the backward Euler discretization of a smooth solution, obtained with constant stepsize h, is Ch-shadowed by a true solution of the system and, vice versa, every smooth trajectory is Ch-shadowed by a suitable backward Euler discretization. This validates numerical computations over long time intervals. It is further explained how these results extend to higher order Runge--Kutta methods.

Optimal Convergence Results for Runge-Kutta Discretizations of Linear Nonautonomous Parabolic Problems

We analyze the convergence properties of implicit Runge-Kutta methods for the time-discretization of linear nonautonomous parabolic problems. We work in an abstract Banach space setting that covers standard parabolic initial-boundary value problems with time-dependent smooth coefficients. The Runge-Kutta methods are only assumed to be A(θ)-stable, and variable stepsizes are allowed in the analysis. As the main result, we derive optimal convergence rates for Runge-Kutta discretizations of temporally smooth solutions.

Stability of Time-Stepping Methods for Abstract Time-Dependent Parabolic Problems

We consider an abstract nonautonomous parabolic problem, $$ u'(t) = A(t)u(t); \quad u(t_0) = u_0, $$ where $A(t) : D_t \subset X \to X$, $t_0 \le t \le t_1$, is a family of sectorial operators in a Banach space X. This problem is discretized in time by means of either an A($\theta$)-stable Runge--Kutta or an A($\theta$)-stable linear multistep method. We prove that the resulting discretization is stable, under some natural assumptions on the relative total variation of A(t) with respect to t. For strongly A($\theta$)-stable Runge--Kutta methods, stability holds even for variable time step-sizes. Our results are applicable to the analysis of time-dependent parabolic problems in the Lp norms.

On the abstract nonautonomous parabolic cauchy problem in the case of constant domains

We study existence, uniqueness and regularity of the strict, classical and strong solution u ∈ C([0,T],E) of the non-autonomous evolution equation u′(t)−A(t)u(t)= f(t), with the initial datum, u(0)=x, in a Banach space E, where {A(t)} is a family of infinitesimal generators of analytic semi-groups whose domains are constant in t and possibly not dense in E. We prove necessary and sufficient conditions for existence and Holder regularity of the solutions and their first derivative.