Backward uniqueness for linearized compressible flow

Michael Renardy

doi:10.3934/eect.2015.4.107

Article Contents

2015, Volume 4, Issue 1: 107-113. Doi: 10.3934/eect.2015.4.107

This issue Previous Article Global stabilization of the Navier-Stokes equations around an unstable equilibrium state with a boundary feedback controller Next Article

Backward uniqueness for linearized compressible flow

Michael Renardy^1,

1.
Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123

Received: August 31, 2014

Revised: December 31, 2014

Published: February 2015

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Abstract

We prove that a $C_0$-semigroup of operators $\exp(At)$ satisfies backward uniqueness if the resolvent of $A$ exists on a ray $z=re^{i\theta}$ in the left half plane ($\pi/2<\theta\le \pi$) and satisfies a bound $\|(A-z I)^{-1}\|\le C\exp(|z|^\alpha)$, $\alpha<1$ on this ray. The proof of this result is based on the Phragmen-Lindelöf theorem. The result is applied to the linearized compressible Navier-Stokes equations in one space dimension and to the wave equation with linear damping and absorbing boundary condition.

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Mathematics Subject Classification: 47D06, 93B99.

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References

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