Explicit estimates on the torus for the sup-norm and the dissipative length scale of solutions of the Swift-Hohenberg Equation in one and two space dimensions

In this work we have obtained explicit and accurate estimates of the sup-norm for solutions of the Swift-Hohenberg Equation (SHE) in one and two space dimensions. By using the best (so far) available estimates of the embedding constants which appear in the classical functional interpolation inequalities used in the study of solutions of dissipative partial differential equations, we have evaluated in an explicit manner the values of the sup-norm of the solutions of the SHE. In addition we have calculated the so-called time-averaged dissipative length scale associated to the above solutions.