Log-log convexity and backward uniqueness

Kukavica, Igor

doi:10.1090/S0002-9939-07-08991-5

Log-log convexity and backward uniqueness
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Proc. Amer. Math. Soc. 135 (2007), 2415-2421 Request permission

Abstract:

We study backward uniqueness properties for equations of the form \begin{equation*} u’ + A u = f. \end{equation*} Under mild regularity assumptions on $A$ and $f$, it is shown that $u(0)=0$ implies $u(t)=0$ for $t<0$. The argument is based on $\alpha$-log and log-log convexity. The results apply to mildly nonlinear parabolic equations and systems with rough coefficients and the 2D Navier-Stokes system.

References

Shmuel Agmon, Unicité et convexité dans les problèmes différentiels, Séminaire de Mathématiques Supérieures, No. 13 (Été, vol. 1965, Les Presses de l’Université de Montréal, Montreal, Que., 1966 (French). MR 0252808
S. Agmon and L. Nirenberg, Properties of solutions of ordinary differential equations in Banach space, Comm. Pure Appl. Math. 16 (1963), 121–239. MR 155203, DOI 10.1002/cpa.3160160204
S. Agmon and L. Nirenberg, Lower bounds and uniqueness theorems for solutions of differential equations in a Hilbert space, Comm. Pure Appl. Math. 20 (1967), 207–229. MR 204829, DOI 10.1002/cpa.3160200106
C. Bardos and L. Tartar, Sur l’unicité rétrograde des équations paraboliques et quelques questions voisines, Arch. Rational Mech. Anal. 50 (1973), 10–25 (French). MR 338517, DOI 10.1007/BF00251291
Peter Constantin and Ciprian Foias, Navier-Stokes equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988. MR 972259
Peter Constantin, Ciprian Foias, Igor Kukavica, and Andrew J. Majda, Dirichlet quotients and $2$D periodic Navier-Stokes equations, J. Math. Pures Appl. (9) 76 (1997), no. 2, 125–153. MR 1432371, DOI 10.1016/S0021-7824(97)89948-5
P. Constantin, C. Foias, B. Nicolaenko, and R. Temam, Spectral barriers and inertial manifolds for dissipative partial differential equations, J. Dynam. Differential Equations 1 (1989), no. 1, 45–73. MR 1010960, DOI 10.1007/BF01048790
Luis Escauriaza, Carleman inequalities and the heat operator, Duke Math. J. 104 (2000), no. 1, 113–127. MR 1769727, DOI 10.1215/S0012-7094-00-10415-2
Luis Escauriaza and Luis Vega, Carleman inequalities and the heat operator. II, Indiana Univ. Math. J. 50 (2001), no. 3, 1149–1169. MR 1871351, DOI 10.1512/iumj.2001.50.1937
Jean-Michel Ghidaglia, Some backward uniqueness results, Nonlinear Anal. 10 (1986), no. 8, 777–790. MR 851146, DOI 10.1016/0362-546X(86)90037-4
Igor Kukavica, Backward uniqueness for solutions of linear parabolic equations, Proc. Amer. Math. Soc. 132 (2004), no. 6, 1755–1760. MR 2051137, DOI 10.1090/S0002-9939-03-07355-6
P. D. Lax, A stability theorem for solutions of abstract differential equations, and its application to the study of the local behavior of solutions of elliptic equations, Comm. Pure Appl. Math. 9 (1956), 747–766. MR 86991, DOI 10.1002/cpa.3160090407
M. Lees and M. H. Protter, Unique continuation for parabolic differential equations and inequalities, Duke Math. J. 28 (1961), 369–382. MR 140840
Hajimu Ogawa, Lower bounds for solutions of differential inequalities in Hilbert space, Proc. Amer. Math. Soc. 16 (1965), 1241–1243. MR 185291, DOI 10.1090/S0002-9939-1965-0185291-6
M. H. Protter, Properties of solutions of parabolic equations and inequalities, Canadian J. Math. 13 (1961), 331–345. MR 153982, DOI 10.4153/CJM-1961-028-1
James Serrin, The initial value problem for the Navier-Stokes equations, Nonlinear Problems (Proc. Sympos., Madison, Wis., 1962) Univ. Wisconsin Press, Madison, Wis., 1963, pp. 69–98. MR 0150444