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We prove that the viscous Burgers equation has a globally defined smoothsolution in all dimensions provided the initial condition and the forcing termare smooth and bounded together with their derivatives. Such solutions may haveinfinite energy. The proof does not rely on energy estimates, but on acombination of the maximum principle and quantitative Schauder estimates. Weobtain 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 the forcing term is zero. To get aclassical solution, it suffices to assume that the initial condition and theforcing term have bounded derivatives up to order two.

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