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Non-controllability of the viscous Burgers equation and a detour into the well-posedness of unbounded entropy solutions to scalar conservation laws

Abstract : We contribute an answer to the question raised in [Coron, Contemp. Math 2007] concerning the controllability, in the framework of L^2 solutions, of the viscous Burgers equation u_t +(u 2 /2)_x = u_xx for initial and terminal data prescribed for x ∈ (0, 1). 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 an a priori bound of the form ||u||_∞ ≤ L|C|. Further, in order to get closer to the original question we develop a basic well-posedness theory of unbounded entropy solutions to the Cauchy problem for multi-dimensional scalar conservation laws with pure L p data and polynomial growth up to the critical power p of the flux function. The case of the Cauchy-Dirichlet problem for the Burgers equation on an interval is also addressed, in the L^2 solution framework which is considerably simpler than the established L^1 theory of renor-malized solutions to such problems. Local regularity of unbounded entropy solutions is discussed in the one-dimensional case with convex flux. Non-controllability results are then extended to solutions of the viscous Burgers equation in the L^2 setting, under the a priori bound ||u||_2 ≤ LT |C| and an L^2 − L^3_loc regularization assumption.
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Submitted on : Thursday, March 12, 2020 - 1:33:29 PM
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Boris Andreianov, Shyam Sundar Ghoshal, Konstantinos Koumatos. Non-controllability of the viscous Burgers equation and a detour into the well-posedness of unbounded entropy solutions to scalar conservation laws. 2020. ⟨hal-02497181⟩

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