# American Institute of Mathematical Sciences

March  2015, 4(1): 107-113. doi: 10.3934/eect.2015.4.107

## Backward uniqueness for linearized compressible flow

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

Received  September 2014 Revised  January 2015 Published  February 2015

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.
Citation: Michael Renardy. Backward uniqueness for linearized compressible flow. Evolution Equations and Control Theory, 2015, 4 (1) : 107-113. doi: 10.3934/eect.2015.4.107
##### References:
 [1] G. Avalos and T. Clark, Backward uniqueness for a PDE fluid-structure interaction, preprint,, , (). [2] G. Avalos and R. Triggiani, Backward uniqueness of the s.c. semigroup arising in parabolic-hyperbolic fluid-structure interaction, J. Diff. Eq., 245 (2008), 737-761. doi: 10.1016/j.jde.2007.10.036. [3] G. Avalos and R. Triggiani, Backwards uniqueness of the $C_0$-semigroup associated with a parabolic-hyperbolic Stokes-Lamé partial differential equation system, Trans. Amer. Math. Soc., 362 (2010), 3535-3561. doi: 10.1090/S0002-9947-10-04851-8. [4] S. Chowdhury, D. Mitra, M. Ramaswamy and M. Renardy, Null controllability of the linearized compressible Navier Stokes system in one dimension, J. Diff. Eq., 257 (2014), 3813-3849. doi: 10.1016/j.jde.2014.07.010. [5] H. Koch and I. Lasiecka, Backward uniqueness in linear thermoelasticity with time and space variable coefficients, Functional Analysis and Evolution Equations, Birkhäuser, Basel, (2008), 389-403. doi: 10.1007/978-3-7643-7794-6_25. [6] I. Lasiecka, M. Renardy and R. Triggiani, Backward uniqueness for thermoelastic plates with rotational forces, Semigroup Forum, 62 (2001), 217-242. doi: 10.1007/s002330010035.

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##### References:
 [1] G. Avalos and T. Clark, Backward uniqueness for a PDE fluid-structure interaction, preprint,, , (). [2] G. Avalos and R. Triggiani, Backward uniqueness of the s.c. semigroup arising in parabolic-hyperbolic fluid-structure interaction, J. Diff. Eq., 245 (2008), 737-761. doi: 10.1016/j.jde.2007.10.036. [3] G. Avalos and R. Triggiani, Backwards uniqueness of the $C_0$-semigroup associated with a parabolic-hyperbolic Stokes-Lamé partial differential equation system, Trans. Amer. Math. Soc., 362 (2010), 3535-3561. doi: 10.1090/S0002-9947-10-04851-8. [4] S. Chowdhury, D. Mitra, M. Ramaswamy and M. Renardy, Null controllability of the linearized compressible Navier Stokes system in one dimension, J. Diff. Eq., 257 (2014), 3813-3849. doi: 10.1016/j.jde.2014.07.010. [5] H. Koch and I. Lasiecka, Backward uniqueness in linear thermoelasticity with time and space variable coefficients, Functional Analysis and Evolution Equations, Birkhäuser, Basel, (2008), 389-403. doi: 10.1007/978-3-7643-7794-6_25. [6] I. Lasiecka, M. Renardy and R. Triggiani, Backward uniqueness for thermoelastic plates with rotational forces, Semigroup Forum, 62 (2001), 217-242. doi: 10.1007/s002330010035.
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