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. © 2013

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. © 2013

Sharp constants for the l°°-norm on the torus and applications to dissipative partial differential equations

Sharp estimates are obtained for the constants appearing in the Sobolev embedding theorem for the L°° norm on the d-dimensioned torus for d = 1,2,3. The sharp constants are expressed in terms of the Riemann zeta-function, the Dirichlet beta-series and various lattice sums. We then provide some applications including the two dimensional Navier-Stokes equations

Bifurcation curves of subharmonic solutions and Melnikov theory under degeneracies

We revisit a problem considered by Chow and Hale on the existence of subharmonic solutions for perturbed systems. In the analytic setting, under more general (weaker) conditions, we prove their results on the existence of bifurcation curves from the nonexistence to the existence of subharmonic solutions. In particular our results apply also when one has degeneracy to first order -- i.e. when the subharmonic Melnikov function vanishes identically. Moreover we can deal as well with the case in which degeneracy persists to arbitrarily high orders, in the sense that suitable generalisations to higher orders of the subharmonic Melnikov function are also identically zero. In general the bifurcation curves are not analytic, and even when they are smooth they can form cusps at the origin: we say in this case that the curves are degenerate as the corresponding tangent lines coincide. The technique we use is completely different from that of Chow and Hale, and it is essentially based on rigorous perturbation theory

Asymptotic expansions and extremals for the critical Sobolev and Gagliardo-Nirenberg inequalities on a torus

We give a comprehensive study of interpolation inequalities for periodic functions with zero mean, including the existence of and the asymptotic expansions for the extremals, best constants, various remainder terms, etc. Most attention is paid to the critical (logarithmic) Sobolev inequality in the two-dimensional case, although a number of results concerning the best constants in the algebraic case and different space dimensions are also obtained