scholarly journals Propagation of uniform Gevrey regularity of solutions to evolution equations

2003 ◽  
Author(s):  
Todor Gramchev ◽  
Ya-Guang Wang
Keyword(s):  
Evolution Equations ◽  
Regularity Of Solutions ◽  
Gevrey Regularity

Petrov–Galerkin space-time hp-approximation of parabolic equations in H1/2

2019 ◽  
Vol 40 (4) ◽  
pp. 2717-2745
Author(s):  
Denis Devaud
Keyword(s):  
Evolution Equations ◽  
Exponential Convergence ◽  
Broad Class ◽  
Infinite Horizon ◽  
Initial Boundary ◽  
Space Time ◽  
Time Step ◽  
Gevrey Regularity ◽  
Liouville Derivative ◽  
Temporal Variable

Abstract We analyse a class of variational space-time discretizations for a broad class of initial boundary value problems for linear, parabolic evolution equations. The space-time variational formulation is based on fractional Sobolev spaces of order $1/2$ and the Riemann–Liouville derivative of order $1/2$ with respect to the temporal variable. It accommodates general, conforming space discretizations and naturally accommodates discretization of infinite horizon evolution problems. We prove an inf-sup condition for $hp$-time semidiscretizations with an explicit expression of stable test functions given in terms of Hilbert transforms of the corresponding trial functions; inf-sup constants are independent of temporal order and the time-step sequences, allowing quasi-optimal, high-order discretizations on graded time-step sequences, and also $hp$-time discretizations. For solutions exhibiting Gevrey regularity in time and taking values in certain weighted Bochner spaces, we establish novel exponential convergence estimates in terms of $N_t$, the number of (elliptic) spatial problems to be solved. The space-time variational setting allows general space discretizations and, in particular, for spatial $hp$-FEM discretizations. We report numerical tests of the method for model problems in one space dimension with typical singular solutions in the spatial and temporal variable. $hp$-discretizations in both spatial and temporal variables are used without any loss of stability, resulting in overall exponential convergence of the space-time discretization.

scholarly journals Besov regularity of parabolic and hyperbolic PDEs

2019 ◽  
Vol 17 (02) ◽  
pp. 235-291 ◽  
Author(s):  
Stephan Dahlke ◽  
Cornelia Schneider
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Approximation ◽  
Approximation Order ◽  
Nonlinear Evolution Equations ◽  
Approximation Schemes ◽  
Regularity Of Solutions ◽  
Nonsmooth Domains ◽  
Besov Regularity ◽  
Specific Scale

This paper is concerned with the regularity of solutions to linear and nonlinear evolution equations on nonsmooth domains. In particular, we study the smoothness in the specific scale [Formula: see text] of Besov spaces. The regularity in these spaces determines the approximation order that can be achieved by adaptive and other nonlinear approximation schemes. We show that for all cases under consideration the Besov regularity is high enough to justify the use of adaptive algorithms.

scholarly journals On the Gevrey regularity of solutions to the 3D ideal MHD equations

2019 ◽  
Vol 39 (11) ◽  
pp. 6485-6506
Author(s):  
Feng Cheng ◽  
◽  
Chao-Jiang Xu ◽  
◽  
Keyword(s):  
Mhd Equations ◽  
Regularity Of Solutions ◽  
Gevrey Regularity ◽  
Ideal Mhd

scholarly journals If time were a graph, what would evolution equations look like?

2021 ◽  
Author(s):  
Amru Hussein ◽  
Delio Mugnolo
Keyword(s):  
Evolution Equations ◽  
Initial Conditions ◽  
Coupled Systems ◽  
Regularity Of Solutions ◽  
Parabolic Problems ◽  
Well Posedness ◽  
Cauchy Problems ◽  
Unified Framework ◽  
Time Graph ◽  
Branching Points

AbstractLinear evolution equations are considered usually for the time variable being defined on an interval where typically initial conditions or time periodicity of solutions is required to single out certain solutions. Here, we would like to make a point of allowing time to be defined on a metric graph or network where on the branching points coupling conditions are imposed such that time can have ramifications and even loops. This not only generalizes the classical setting and allows for more freedom in the modeling of coupled and interacting systems of evolution equations, but it also provides a unified framework for initial value and time-periodic problems. For these time-graph Cauchy problems questions of well-posedness and regularity of solutions for parabolic problems are studied along with the question of which time-graph Cauchy problems cannot be reduced to an iteratively solvable sequence of Cauchy problems on intervals. Based on two different approaches—an application of the Kalton–Weis theorem on the sum of closed operators and an explicit computation of a Green’s function—we present the main well-posedness and regularity results. We further study some qualitative properties of solutions. While we mainly focus on parabolic problems, we also explain how other Cauchy problems can be studied along the same lines. This is exemplified by discussing coupled systems with constraints that are non-local in time akin to periodicity.

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