TITLE

# Globalâ€“inâ€“time Uniform Convergence for Linear Hyperbolicâ€“Parabolic Singular Perturbations

AUTHOR(S)
Ghisi, Marina; Gobbino, Massimo
PUB. DATE
July 2006
SOURCE
Acta Mathematica Sinica;Jul2006, Vol. 22 Issue 4, p1161
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
We consider the Cauchy problem $$\varepsilon {u}\ifmmode{''}\else''\fi_{\varepsilon } + \delta {u}\ifmmode{'}\else'\fi_{\varepsilon } + Au_{\varepsilon } = 0,\;u_{\varepsilon } (0) = u_{0} ,\;{u}\ifmmode{'}\else'\fi_{\varepsilon } (0) = u_{1} ,$$ where âˆˆ > 0, Î´ > 0, H is a Hilbert space, and A is a selfâ€“adjoint linear nonâ€“negative operator on H with dense domain D( A). We study the convergence of { u âˆˆ} to the solution of the limit problem Î´ u' + Au = 0, u(0) = u 0. For initial data ( u 0, u 1) âˆˆ D( A 1/2) Ã— H, we prove globalâ€“inâ€“time convergence with respect to strong topologies. Moreover, we estimate the convergence rate in the case where ( u 0, u 1) âˆˆ D( A 3/2) Ã— D( A 1/2), and we show that this regularity requirement is sharp for our estimates. We give also an upper bound for $${\left| {{u}\ifmmode{'}\else'\fi_{\varepsilon } (t)} \right|}$$ which does not depend on âˆˆ.
ACCESSION #
21690502

## Related Articles

• A system of differential equations with an analytic nonlinearity. Zavizion, G.; Klyuchnik, I. // Mathematical Notes;Apr2012, Vol. 91 Issue 3/4, p579

The article discusses the local linearization and derived convergence conditions for a linear transformation of a singular perturbed first-order differential equation. It says that a regularization method was used in the construction of asymptotic solutions of singular first-order differential...

• On the asymptotics of the solution to a singularly perturbed system of first-order partial differential equations with small nonlinearity in the critical case. Nesterov, A. // Computational Mathematics & Mathematical Physics;Jul2012, Vol. 52 Issue 7, p1035

A complete asymptotic expansion of the solution to an initial value problem for a singularly perturbed system of hyperbolic equations is constructed and justified. A specific feature of the problem is that its solution has a wavelet zone in a neighborhood of which the asymptotics is described by...

• On Ïµ-uniform convergence of exponentially fitted methods. MARUŠIĆ, MILJENKO // Mathematical Communications;2014, Vol. 19 Issue 3, p545

A class of methods constructed to numerically approximate the solution of two-point singularly perturbed boundary value problems of the form Ïµu'' + bu' + cu = f use exponentials to mimic exponential behavior of the solution in the boundary layer(s). We refer to them as exponentially fitted...

• Singular perturbation theory applied to magnetohydrostatic equilibria: Proof of convergence. Kopp, A.; Schindler, K. // Journal of Mathematical Physics;Jun91, Vol. 32 Issue 6, p1437

Magnetohydrostatic equilibria with two relevant length scales are considered. Their ratio Î• is assumed to be small. Regular solutions in the sense of singular perturbation theory are discussed and the magnetic flux function A (2-D case) and Euler potentials Î±, Î² (3-D case) are shown...

• Difference Scheme on a Uniform Grid for the Singularly Perturbed Cauchy Problem. Zadorin, A. I.; Tikhovskaya, S. V. // Journal of Mathematical Sciences;Dec2013, Vol. 195 Issue 6, p865

We consider the Cauchy problem for a singularly perturbed second order ordinary differential equation. Based on the maximum principle, we estimate the solution and its derivatives. We construct an exponential fitted scheme generalizing the well-known scheme due to A. M. Ilâ€™in. We also...

• GLOBAL EXISTENCE OF WEAKLY DISCONTINUOUS SOLUTIONS TO THE CAUCHY PROBLEM WITH A KIND OF NON-SMOOTH INITIAL DATA FOR QUASILINEAR HYPERBOLIC SYSTEMS. LI, TATSIEN; WANG, LIBIN // Chinese Annals of Mathematics;Jul2004, Vol. 25 Issue 3, p319

The authors consider the Cauchy problem with a kind of non-smooth initial data for quasilinear hyperbolic systems and obtain a necessary and sufficient condition to guarantee the existence and uniqueness of global weakly discontinuous solution.

• A PDE APPROACH TO FINITE TIME INDICATORS IN ERGODIC THEORY. Bernardi, Olga; Cardin, Franco; Guzzo, Massimiliano; Zanelli, Lorenzo // Journal of Nonlinear Mathematical Physics (World Scientific Publ;Jun2009, Vol. 16 Issue 2, p195

For dynamical systems defined by vector fields over a compact invariant set, we introduce a new class of approximated first integrals based on finite time averages and satisfying an explicit first order partial differential equation. These approximated first integrals can be used as finite time...

• SINGULAR PERTURBATION AND BIFURCATION OF DIFFUSE TRANSITION LAYERS IN INHOMOGENEOUS MEDIA, PART II. CHAOQUN HUANG; NUNG KWAN YIP // Networks & Heterogeneous Media;Dec2015, Vol. 10 Issue 4, p897

In this paper, we study the connection between the bifurcation of diffuse transition layers and that of the underlying limit interfacial problem in a degenerate spatially inhomogeneous medium. In dimension one, we prove the existence of bifurcation of diffuse interfaces in a pitchfork spatial...

• Derivation of Asymptotic Dynamical Systems with Partial Lie Symmetry Groups. Iwasa, Masatomo // Journal of Applied Mathematics;9/29/2015, Vol. 2015, p1

Lie group analysis has been applied to singular perturbation problems in both ordinary differential and difference equations and has allowed us to find the reduced dynamics describing the asymptotic behavior of the dynamical system. The present study provides an extended method that is also...

Share