Global existence for strong solutions of viscous Burgers equation. (1) The bounded case

• Autorzy: Unterberger J.
• Języki publikacji: EN

Abstrakty

EN

We prove that the viscous Burgers equation (∂t−∆)u(t, x)+( u •∇)u(t, x) = g(t, x), (t, x) ∈ R+ × Rd (d ≥ 1) has a globally defined smooth solution in all dimensions provided the initial condition and the forcing term g are smooth and bounded together with their derivatives. Such solutions may have infinite energy. The proofdoes not rely on energy estimates, but on a combinationof the maximumprinciple and quantitative Schauder estimates. We obtain precise bounds on the sup norm of the solution and its derivatives, making it plain that there is no exponential increase in time. In particular, these bounds are time-independent if g is zero. To get a classical solution, it suffices to assume that the initial condition and the forcing term have bounded derivatives up to order two.

Słowa kluczowe

EN

viscous Burgers equation; conservation laws; maximum principle; Schauder estimates

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2017
• Tom: Vol. 46, no. 2
• Strony: 109--136
• Opis fizyczny: Bibliogr. 26 poz., rys.

Twórcy

autor

Unterberger J.

• Institut Elie Cartan∗, Universite´ de Lorraine, B.P. 239, F – 54506 Vandœuvre-le`s-Nancy Cedex, France

Bibliografia

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Uwagi

PL

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).