AGK-CoronQ-Part I - Linfty.pdf (412.6 kB)
Lack of controllability of the viscous Burgers equation. Part I: the L8 setting
journal contribution
posted on 2023-06-10, 04:22 authored by Konstantinos KoumatosKonstantinos Koumatos, Boris Andreianov, Shyam Sundar GoshalWe contribute an answer to a quantitative variant of the question raised in [Coron, Contemp. Math 2007] concerning the controllability of the viscous Burgers equation ut+(u 2 /2)x = uxx for initial and terminal data prescribed for x ? (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 + ?)/|C| (where ? > 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 amplification restriction of the form kuk8 = L|C|. Our focus on L8 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 L2 setting, upon extending the Kruzhkov theory appropriately.
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Publication status
- Published
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- Accepted version
Journal
Journal of Evolution EquationsISSN
1424-3199Publisher
Springer NatureExternal DOI
Issue
70Volume
20Page range
1-24Department affiliated with
- Mathematics Publications
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- Yes
Legacy Posted Date
2022-08-02First Compliant Deposit (FCD) Date
2022-07-31Usage metrics
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