Some backward uniqueness results. (English) Zbl 0622.35029

This paper establishes backward uniqueness and unique continuation results for nonlinear parabolic equations and inequalities. There are two principal results. One establishes backward uniqueness for inequalities of the type \[ \| du/dt+\mu A(t)u(t)\|_ H\leq n(t)\| u(t)\|_{D[A^{1/2}(t)]}\quad, \] where \(\{A(t)\}_{t\geq 0}\) is a family of self-adjoint unbounded linear operators on a Hilbert space H and n(t) is a square integrable function. The second result shows that for u not identically zero then there exist \(a,b>0\), such that \(| u(t)| \geq C \exp [-Re \mu (1+b)at]\); this is the unique continuation result.
The results are applied to the Navier-Stokes equations, equations for a heat conducting viscous fluid, MHD equations, and the Kuramoto- Sivashinsky equations on both bounded and unbounded spatial domains; \(L^ 2\) integrability is always assumed.
Reviewer: B.Straughan


35K55 Nonlinear parabolic equations
35Q30 Navier-Stokes equations
35B60 Continuation and prolongation of solutions to PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
34G10 Linear differential equations in abstract spaces
35B40 Asymptotic behavior of solutions to PDEs
Full Text: DOI


[1] Agmon, S.; Nirenberg, L., Lower bounds and uniqueness theorems for solutions of differential equations in a Hilbert space, Communs pure appl. math., 20, 207-229, (1967) · Zbl 0147.34603
[2] Bardos, C.; Tartar, L., Sur l’unicité rétrograde des équations paraboliques et quelques questions voisines, Archs ration. mech. analysis, 50, 10-25, (1973) · Zbl 0258.35039
[3] Chandrasekhar, S., Hydrodynamic and hydrodynamic stability, (1961), Clarendon Press Oxford · Zbl 0142.44103
[4] Cohen, P.J.; Lees, M., Asymptotic decay of solutions of differential inequalities, Pacif. J. math., 11, 1235-1249, (1961) · Zbl 0171.35002
[5] Dyer, R.H.; Edmunds, D., Lower bounds for solutions of the Navier-Stokes equations, Proc. lond. math. soc., 3, 169-178, (1968) · Zbl 0157.57005
[6] Edmunds, D.E., Asymptotic behavior of solutions of the Navier-Stokes equations, Archs ration. mech. analysis, 22, 15-21, (1966) · Zbl 0147.45202
[7] Foias, C.; Saut, J.C., Asymptotic behaviour, as t → + ∞, of solutions of Navier-Stokes equations and nonlinear spectral manifolds, Indiana univ. math. J., 33, 459-477, (1984) · Zbl 0565.35087
[8] Ghidaglia, J.M., Long time behavior of solutions of abstract inequalities. applications to thermo-hydraulic and magnetohydrodynamic equations, J. diff. eqns., 61, 268-294, (1986) · Zbl 0549.35102
[9] Ghidaglia, J.M., On the fractal dimension of attractors for viscous incompressible fluid flows, SIAM J. math. analysis, (1986), (in press) · Zbl 0626.35078
[10] Lax, P.D., A stability theorem for solutions of abstract differential equations, and its application to the study of the local behavior of solutions of elliptic equations, Communs pure appl. math., 9, 747-766, (1956) · Zbl 0072.33004
[11] Lees, M., Asymptotic behavior of solutions of parabolic differential inequalities, Can. J. math., 13, 331-345, (1961)
[12] Lions, J.L.; Malgrange, B., Sur l’unicité rétrograde dans LES problèmes mixtes paraboliques, Math. scand., 8, 277-286, (1960) · Zbl 0126.12202
[13] Nicolaenko, B.; Scheurer, B.; Temam, R., Some global dynamical properties of the Kuramoto-Sivashinsky equations: nonlinear stability and attractors, Physica, 16D, 155-183, (1985) · Zbl 0592.35013
[14] Nozaki, K.; Bekki, N., Chaos in a perturbed nonlinear Schrödinger equation, Phys. rev. letts, 50, 1226-1229, (1983)
[15] Ogawa, H., Lower bounds for solutions of parabolic differential inequalities, Can. J. math., 19, 667-672, (1967) · Zbl 0152.10202
[16] Ogawa, H., On lower bounds and uniqueness for solutions of the Navier-Stokes equations, J. math. mech., 18, 445-452, (1968) · Zbl 0176.40102
[17] Ogawa, H., On the maximum rate of decay of solutions of parabolic differential inequalities, Archs ration. mech. analysis, 38, 173-177, (1970) · Zbl 0198.14701
[18] Protter, M.H., Properties of solutions of parabolic equations and inequalities, Can. J. math., 13, 331-345, (1961) · Zbl 0099.30001
[19] Sermange, M.; Temam, R., Some mathematical questions related to the M.H.D. equations, Communs pure appl. math., 36, 634-664, (1983) · Zbl 0524.76099
[20] Straugham, B., Backward uniqueness and unique continuation for solutions to the Navier-Stokes equations on an exterior domain, J. math. pures appl., 62, 49-62, (1983)
[21] Temam, R., Navier-Stokes equations, theory and numerical analysis, (1984), North-Holland Amsterdam · Zbl 0568.35002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.