Carleman estimate for solutions to a degenerate convection-diffusion equation

This paper concerns a control system governed by a convection-diffusion equation, which is weakly degenerate at the boundary. In the governing equation, the convection is independent of the degeneracy of the equation and cannot be controlled by the diffusion. The Carleman estimate is established by means of a suitable transformation, by which the diffusion and the convection are transformed into a complex union, and complicated and detailed computations. Then the observability inequality is proved and the control system is shown to be null controllable.

In this paper, we study the null controllability of the problem (5), (2), (3) in the weakly degenerate case (0 < α < 1). In order to treat the convection term, we have to assume that b ∈ W 2,1 ∞ ((0, 1) × (0, T )), i.e. b, b x , b xx , b t ∈ L ∞ ((0, 1) × (0, T )). Since b x ∈ L ∞ ((0, 1) × (0, T )), the convection term in (5) can be rewritten into a nondivergence form. Precisely, we investigate the null controllability of the following problem As many studies on null controllability, the key to prove the null controllability of the problem (6)-(8) is the Carleman estimate for its conjugate problem where v T ∈ L 2 (0, 1). In [15], the authors established the Carleman estimate in the same way as the case without convection term since the convection term can be controlled by the diffusion term in the governing equation. As to the equation with a general convection term, [27] treated the case where 0 < α < 1/2. In [27], in order to get the Carleman estimate, both the reaction term and the convection term were regarded as a source term in the governing equation and the similar auxiliary functions to the case without convection term were used. The key for the Carleman estimate is to estimate the effect of the convection term. Furthermore, to show that the restriction 0 < α < 1/2 is optimal when one establishes the Carleman estimate in such a way, the authors choose auxiliary functions for the Carleman estimate by the method of undetermined coefficients. Therefore, to establish the Carleman estimate for the problem (9)-(11) with 0 < α < 1, one must treat the convection in a different way but not regard it as a source term as done in [27]. For the problem (9)-(11), the classical solution may not exist and weak solution should be considered since (9) is degenerate. Therefore, it is more convenient to establish the uniform Carleman estimate for the regularized problem with 0 < η < 1. As mentioned above, one must treat the convection in a different way but not regard it as a source term as done in [27]. To get the uniform Carleman estimate, we introduce an auxiliary function by which the diffusion and the convection are transformed into a union. The transformation depends on the coefficient of the convection and we have to assume that b ∈ W 2,1 ∞ ((0, 1) × (0, T )) for the desired uniform Carleman estimate. Furthermore, the union transformed from the diffusion and the convection is so complex that the computations for the desired uniform Carleman estimate are more complicated and detailed than the ones in [15,27]. After the Carleman estimate, one can get the observability inequality and further prove the null controllability of the problem (6)-(8) in a standard way. It is noted that the strongly degenerate case (1 ≤ α < 2) still remains open. Indeed, the key auxiliary function in the paper, by which the diffusion and the convection are transformed into a union, is bounded in the weakly degenerate case while unbounded in the strongly degenerate case. Therefore, the method in the paper is not suitable for the strongly degenerate case.
The paper is organized as follows. In §2 we recall the well-posedness of the problem (6)-(8) and some a priori estimates. Some uniform estimates for smooth solutions to a regularized equation are established in §3. By means of these estimates, we first prove the Carleman estimate for the problem (12)- (14) in §4, and subsequently, the observability inequality for the problem (12)- (14) and the null controllability of the problem (6)-(8) are shown.
2. Well-posedness and some a priori estimates. Let us recall the well-posedness of the problem (6)- (8) and some a priori estimates in [27].
Consider the nondegenerate linear problem ). Moreover, u η satisfies the following energy estimates.
where N > 0 depends only on K, T and α.
Recall the following Hardy inequality proved in [10].
Proposition 1. There exist two positive constants s 1 and M 1 depending only on b W 2,1 ∞ (Q T ) , T and α, but independent of η, such that for each s ≥ s 1 , Proof. Let us estimate the terms on the right side of (25). First, it follows from the definition of ϕ and (23) that Second, the definitions of ϕ and ς, (24), the Hölder inequality and Lemma 3.2 lead to Third, for each 0 < κ ≤ 1, it follows from the definition of A , (23), Lemma 3.2 and by choosing κ = min 1, .