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Galerkin approximation with proper orthogonal decomposition : new error estimates and illustrative examples

Dominique Chapelle, Asven Gariah, Jacques Sainte-Marie (2012)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We propose a numerical analysis of proper orthogonal decomposition (POD) model reductions in which a priori error estimates are expressed in terms of the projection errors that are controlled in the construction of POD bases. These error estimates are derived for generic parabolic evolution PDEs, including with non-linear Lipschitz right-hand sides, and for wave-like equations. A specific projection continuity norm appears in the estimates and – whereas a general uniform continuity bound seems out...

Galerkin approximation with proper orthogonal decomposition : new error estimates and illustrative examples

Dominique Chapelle, Asven Gariah, Jacques Sainte-Marie (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

Galerkin approximation with proper orthogonal decomposition : new error estimates and illustrative examples

Dominique Chapelle, Asven Gariah, Jacques Sainte-Marie (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

Gaussian estimates for fundamental solutions to certain parabolic systems.

Steve Hofmann, Seick Kim (2004)

Publicacions Matemàtiques

Auscher proved Gaussian upper bound estimates for the fundamental solutions to parabolic equations with complex coefficients in the case when coefficients are time-independent and a small perturbation of real coefficients. We prove the equivalence between the local boundedness property of solutions to a parabolic system and a Gaussian upper bound for its fundamental matrix. As a consequence, we extend Auscher's result to the time dependent case.

Generalizations of Melin's inequality to systems

Raymond Brummelhuis (2001)

Journées équations aux dérivées partielles

We discuss a recent necessary and sufficient condition for Melin's inequality for a class of systems of pseudodifferential operators.

Generalized Cahn-Hilliard equations based on a microforce balance.

Miranville, Alain (2003)

Journal of Applied Mathematics

Generalized n-dimensional Gronwall's inequality and its applications to non-selfadjoint and non-linear hyperbolic equations

Sudhanshu K. Ghoshal, Abha Ghoshal, M. Abu-Masood (1977)

Annales Polonici Mathematici

Global and exponential attractors for a Caginalp type phase-field problem

Brice Bangola (2013)

Open Mathematics

We deal with a generalization of the Caginalp phase-field model associated with Neumann boundary conditions. We prove that the problem is well posed, before studying the long time behavior of solutions. We establish the existence of the global attractor, but also of exponential attractors. Finally, we study the spatial behavior of solutions in a semi-infinite cylinder, assuming that such solutions exist.

Global Caccioppoli-type and Poincaré inequalities with Orlicz norms.

Agarwal, Ravi P., Ding, Shusen (2010)

Journal of Inequalities and Applications [electronic only]

Global Classical Solutions of Nonlinear Wave Equations.

Wolf von Wahl, Philip Brenner (1981)

Mathematische Zeitschrift

Global existence and $L^{\infty}$ estimates of solutions for a quasilinear parabolic system.

Zhou, Jun (2010)

The Journal of Nonlinear Sciences and its Applications

Global existence and regularity of solutions for complex Ginzburg-Landau equations

Stéphane Descombes, Mohand Moussaoui (2000)

Bollettino dell'Unione Matematica Italiana

Si considerano equazioni di Ginzburg-Landau complesse del tipo $u_{t} - α Δ u + P ({|u|}^{2}) u = 0$ in $R^{N}$ dove $P$ è polinomio di grado $K$ a coefficienti complessi e $α$ è un numero complesso con parte reale positiva $ℜ α$ . Nell'ipotesi che la parte reale del coefficiente del termine di grado massimo $P$ sia positiva, si dimostra l'esistenza e la regolarità di una soluzione globale nel caso $|α| < C ℜ α$ , dove $C$ dipende da $K$ e $N$ .

Global existence for a quasilinear wave equation outside of star-shaped domains

Hart F. Smith (2001)

Journées équations aux dérivées partielles

This talk describes joint work with Chris Sogge and Markus Keel, in which we establish a global existence theorem for null-type quasilinear wave equations in three space dimensions, where we impose Dirichlet conditions on a smooth, compact star-shaped obstacle $𝒦 \subset ℝ^{3}$ . The key tool, following Christodoulou [1], is to use the Penrose compactification of Minkowski space. In the case under consideration, this reduces matters to a local existence theorem for a singular obstacle problem. Full details will...

Global existence for the nonlinear equations of crystal optics

Otto Liess (1989)

Journées équations aux dérivées partielles

Global existence of solutions for reaction-diffusion systems with a full matrix of diffusion coefficients and nonhomogeneous boundary conditions.

Kouachi, Said (2002)

Electronic Journal of Qualitative Theory of Differential Equations [electronic only]

Global existence of strong solutions to the one-dimensional full model for phase transitions in thermoviscoelastic materials

Elisabetta Rocca, Riccarda Rossi (2008)

Applications of Mathematics

This paper is devoted to the analysis of a one-dimensional model for phase transition phenomena in thermoviscoelastic materials. The corresponding parabolic-hyperbolic PDE system features a strongly nonlinear internal energy balance equation, governing the evolution of the absolute temperature $ϑ$ , an evolution equation for the phase change parameter $χ$ , including constraints on the phase variable, and a hyperbolic stress-strain relation for the displacement variable $𝐮$ . The main novelty of the model...

Global Hölder estimates for equations of Monge-Ampère type.

John I.E. Urbas (1988)

Inventiones mathematicae

Global $L^{\infty}$ bounds for a class of quasilinear parabolic equations.

Horn, Werner (2002)

Southwest Journal of Pure and Applied Mathematics [electronic only]

Global pointwise a priori bounds and large time behaviour for a nonlinear system describing the spread of infectious disease

M. Kirane (1993)

Applicationes Mathematicae

This paper considers a reaction-diffusion system with biatic diffusion.Existence of a globally bounded solution is proved and its large timebehaviour is given.

Global Schauder estimates for a class of degenerate Kolmogorov equations

Enrico Priola (2009)

Studia Mathematica

We consider a class of possibly degenerate second order elliptic operators on ℝⁿ. This class includes hypoelliptic Ornstein-Uhlenbeck type operators having an additional first order term with unbounded coefficients. We establish global Schauder estimates in Hölder spaces both for elliptic equations and for parabolic Cauchy problems involving . The Hölder spaces in question are defined with respect to a possibly non-Euclidean metric related to the operator . Schauder estimates are deduced by sharp...

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