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Backward uniqueness for linearized compressible flow

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  • 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.
    Mathematics Subject Classification: 47D06, 93B99.


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    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.


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