Backward uniqueness for parabolic equations
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.
scientific article; zbMATH DE number 2021066