We study the singular ordinary differential equation dU dt = 1 ζ(U) φs(U) + φns(U), (0.1) where U ∈ R N , the functions φs ∈ R N and φns ∈ R N are of class C 2 and ζ is a real valued C 2 function. The equation is singular because ζ(U) can attain the value 0. We focus on the solutions of (0.1) that belong to a small neighbourhood of a point U¯ such that φs(U¯ ) = φns(U¯ ) = ~0 and ζ(U¯ ) = 0. We investigate the existence of manifolds that are locally invariant for (0.1) and that contain orbits with a prescribed asymptotic behaviour. Under suitable hypotheses on the set {U : ζ(U) = 0}, we extend to the case of the singular ODE (0.1) the definitions of center manifold, center-stable manifold and of uniformly stable manifold. We prove that the solutions of (0.1) lying on each of these manifolds are regular: this is not trivial since we provide examples showing that, in general, a solution of (0.1) is not continuously differentiable. Finally, we show a decomposition result for a center-stable manifold and for the uniformly stable manifold. An application of our analysis concerns the study of the viscous profiles with small total variation for a class of mixed hyperbolic-parabolic systems in one space variable. Such a class includes the compressible Navier Stokes equation.

Invariant manifolds for a singular ordinary differential equation / Bianchini, S.; Spinolo, L. V.. - In: JOURNAL OF DIFFERENTIAL EQUATIONS. - ISSN 0022-0396. - 250:4(2011), pp. 1788-1827. [10.1016/j.jde.2010.11.010]

Invariant manifolds for a singular ordinary differential equation

Bianchini, S.;Spinolo, L. V.
2011

Abstract

We study the singular ordinary differential equation dU dt = 1 ζ(U) φs(U) + φns(U), (0.1) where U ∈ R N , the functions φs ∈ R N and φns ∈ R N are of class C 2 and ζ is a real valued C 2 function. The equation is singular because ζ(U) can attain the value 0. We focus on the solutions of (0.1) that belong to a small neighbourhood of a point U¯ such that φs(U¯ ) = φns(U¯ ) = ~0 and ζ(U¯ ) = 0. We investigate the existence of manifolds that are locally invariant for (0.1) and that contain orbits with a prescribed asymptotic behaviour. Under suitable hypotheses on the set {U : ζ(U) = 0}, we extend to the case of the singular ODE (0.1) the definitions of center manifold, center-stable manifold and of uniformly stable manifold. We prove that the solutions of (0.1) lying on each of these manifolds are regular: this is not trivial since we provide examples showing that, in general, a solution of (0.1) is not continuously differentiable. Finally, we show a decomposition result for a center-stable manifold and for the uniformly stable manifold. An application of our analysis concerns the study of the viscous profiles with small total variation for a class of mixed hyperbolic-parabolic systems in one space variable. Such a class includes the compressible Navier Stokes equation.
250
4
1788
1827
https://doi.org/10.1016/j.jde.2010.11.010
Bianchini, S.; Spinolo, L. V.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/14598
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