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This investigation is a part of a research program aiming to characterize theextreme behavior possible in hydrodynamic models by analyzing the maximumgrowth of certain fundamental quantities. We consider here the rate of growthof the classical and fractional enstrophy in the fractional Burgers equation inthe subcritical and supercritical regimes. Since solutions to this equationexhibit, respectively, globally well-posed behavior and finite-time blow-up inthese two regimes, this makes it a useful model to study the maximuminstantaneous growth of enstrophy possible in these two distinct situations.First, we obtain estimates on the rates of growth and then show that theseestimates are sharp up to numerical prefactors. This is done by numericallysolving suitably defined constrained maximization problems and thendemonstrating that for different values of the fractional dissipation exponentthe obtained maximizers saturate the upper bounds in the estimates as theenstrophy increases. We conclude that the power-law dependence of the enstrophyrate of growth on the fractional dissipation exponent has the same global formin the subcritical, critical and parts of the supercritical regime. Thisindicates that the maximum enstrophy rate of growth changes smoothly as globalwell-posedness is lost when the fractional dissipation exponent attainssupercritical values. In addition, nontrivial behavior is revealed for themaximum rate of growth of the fractional enstrophy obtained for small values ofthe fractional dissipation exponents. We also characterize the structure of themaximizers in different cases.

Maximum Rate of Growth of Enstrophy in Solutions of the Fractional Burgers Equation