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

Unterberger J.

Control and Cybernetics

2017

Vol. 46, no. 2

109--136

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.