## Backward uniqueness for parabolic equations.(English)Zbl 1039.35052

The authors show that any solution of the parabolic equation $u_ t +\Delta u + b\cdot Du +cu =0$ in the unbounded domain $$(\mathbb R^ n\setminus B_ R)\times (0,T)$$ must vanish identically if it vanishes at $$t=0$$ and if it grows no faster than $$M\exp(M| x| ^2)$$ for some positive constant $$M$$ as $$| x| \to\infty$$. The only assumptions on the coefficients $$b$$ and $$c$$ are that they are bounded. The basic idea of the proof is to show that two Carleman inequalities are satisfied; these inequalities give weighted $$L^2$$ estimates on $$u$$ and its gradient in terms of corresponding weighted estimates of $$u_ t +\Delta u$$. Several applications are mentioned, with references for details: first, the uniqueness result implies smoothness of weak solutions to the Navier-Stokes equations in a borderline case; and, second, it is related (in a negative way) to the controllability of such problems.

### MSC:

 35K55 Nonlinear parabolic equations 35R45 Partial differential inequalities and systems of partial differential inequalities

### Keywords:

exterior domain; two Carleman inequalities; controllability
Full Text:

### References:

 [1] Ahlfors, L.V.: Complex Analysis. McGraw-Hill, 1966 · Zbl 0154.31904 [2] Chen, X.Y.: A strong unique continuation theorem for parabolic equations. Math. Ann. 311, 603–630 (1996) · Zbl 0990.35028 [3] Escauriaza, L.: Carleman inequalities and the heat operator. Duke Math. J. 104, 113–127 (2000) · Zbl 0979.35029 [4] Escauriaza, L., Vega, L.: Carleman inequalities and the heat operator II. Indiana U. Math. J. 50, 1149–1169 (2001) · Zbl 1029.35046 [5] Escauriaza, L., Fernández, F.J.: Unique continuation for parabolic operators. (to appear) · Zbl 1028.35052 [6] Fernández, F.J.: Unique continuation for parabolic operators II. (to appear) · Zbl 1028.35052 [7] Hopf, E.: Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4, 213–231 (1950) · Zbl 0042.10604 [8] Hörmander, L.: Linear Partial Differential Operators. Springer, 1963 · Zbl 0108.09301 [9] Hörmander, L.: Uniqueness theorems for second order elliptic differential equations. Communications in PDE 8, 21–64 (1983) · Zbl 0546.35023 [10] Jones, B.F.: A fundamental solution of the heat equation which is supported in a strip. J. Math. Anal. Appl. 60, 314–324 (1977) · Zbl 0357.35043 [11] Kato, T.: Strong L p -solutions of the Navier-Stokes equations in $$\mathbb{R}$$ m with applications to weak solutions. Math. Zeit. 187, 471–480 (1984) · Zbl 0545.35073 [12] Ladyzhenskaya, O.A.: Mathematical problems of the dynamics of viscous incompressible fluids. Gordon and Breach, 1969 · Zbl 0184.52603 [13] Ladyženskaja, O.A., Solonnikov, V.A., Uralceva, N.N.: Linear and quasilinear equations of parabolic type. Translations of Mathematical Monographs, Amer. Math. Soc., 1968 [14] Lin, F.H.: A uniqueness theorem for parabolic equations. Comm. Pure Appl. Math. 42, 125–136 (1988) [15] Littman, W.: Boundary control theory for hyperbolic and parabolic partial differential equations with constant coefficients. Annali Scuola Norm. Sup. Pisa Serie IV 3, 567–580 (1978) · Zbl 0395.35007 [16] Micu, S., Zuazua, E.: On the lack of null-controllability of the heat equation on the half space. Portugaliae Mathematica 58, 1–24 (2001) · Zbl 0991.35010 [17] Poon, C.C.: Unique continuation for parabolic equations. Comm. Partial Differential Equations 21, 521–539 (1996) · Zbl 0852.35055 [18] Saut, J.C., Scheurer, E.: Unique continuation for evolution equations. J. Differential Equations 66, 118–137 (1987) · Zbl 0631.35044 [19] Seregin, G., Šverák, V.: The Navier-Stokes equations and backward uniqueness. (to appear) · Zbl 1024.76011 [20] Sogge, C.D.: A unique continuation theorem for second order parabolic differential operators. Ark. Mat. 28, 159–182 (1990) · Zbl 0704.35016 [21] Treves, F.: Linear Partial Differential Equations. Gordon and Breach, 1970 · Zbl 0129.06905
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.