Lack of controllability of the viscous Burgers equation: part I-the $$\mathrm{L}^\infty $$ setting

We contribute an answer to a quantitative variant of the question raised in Coron (in: Perspectives in nonlinear partial differential equations. Contemporary mathematics, vol 446, American Mathematical Society, Providence, pp 215–243, 2007) concerning the controllability of the viscous Burgers equation $$u_t+(u^2/2)_x=u_{xx}$$ for initial and terminal data prescribed for $$x\in (0,1)$$ . We investigate the (non)-controllability under the additional a priori bound imposed on the (nonlinear) operator that associates the solution to the terminal state. In contrast to typical techniques on the controllability of the viscous Burgers equation invoking the heat equation, we combine scaling and compensated compactness arguments along with observations on the non-controllability of the inviscid Burgers equation to point out wide sets of terminal states non-attainable from zero initial data by solutions of restricted size. We prove in particular that, given $$L\ge 1$$ , for sufficiently large |C| and $$T< (1+\Delta )/|C|$$ (where $$\Delta >0$$ depends on L), the constant terminal state $$u(\cdot ,T):=C$$ is not attainable at time T, starting from zero data, by weak solutions of the viscous Burgers equation satisfying a bounded amplification restriction of the form $$\Vert u\Vert _\infty \le L|C|$$ . Our focus on $$\mathrm{L}^\infty $$ solutions is due to the fact that we rely upon the classical theory of Kruzhkov entropy solutions to the inviscid equation. In Part II of this paper, we will extend the non-controllability results to solutions of the viscous Burgers equation in the $$\mathrm{L}^2$$ setting, upon extending the Kruzhkov theory appropriately.