Service interruption on Monday 11 July from 12:30 to 13:00: all the sites of the CCSD (HAL, Epiciences, SciencesConf, AureHAL) will be inaccessible (network hardware connection).
Skip to Main content Skip to Navigation
Preprints, Working Papers, ...

Lack of controllability of the viscous Burgers equation.Part I: The L^∞ setting.

Abstract : We contribute an answer to a quantitative variant of the question raised in [Coron, Contemp. Math 2007] concerning the controllability of the viscous Burgers equation u_t+(u2/2)_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 >= 1, for sufficiently large |C| and T < (1 + Delta)/|C| (where Delta > 0 depends on L), the constant terminal state u(.; T) := C is not attainable at time T, starting from zero data, by weak solutions of the viscous Burgers equation satisfying a bounded ampli cation restriction of the form ||u||_ ∞ <= L|C|. Our focus on L^∞ 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 L^2 setting, upon extending the Kruzhkov theory appropriately
Document type :
Preprints, Working Papers, ...
Complete list of metadata

https://hal.archives-ouvertes.fr/hal-02497181
Contributor : Boris Andreianov Connect in order to contact the contributor
Submitted on : Friday, May 27, 2022 - 2:58:37 PM
Last modification on : Wednesday, June 1, 2022 - 3:37:25 AM

File

AGK-CoronQ-Part I - Linfty.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : hal-02497181, version 3

Collections

Citation

Boris Andreianov, Shyam Sundar Ghoshal, Konstantinos Koumatos. Lack of controllability of the viscous Burgers equation.Part I: The L^∞ setting.. 2022. ⟨hal-02497181v3⟩

Share

Metrics

Record views

256

Files downloads

77