Archive for Rational Mechanics and Analysis

There are two well known theories to describe the motion and thermodynamics of superfluids when a large number of quantized vortex lines are present and when the phenomena under study are on scales large compared with the vortex line spacing. These works have been criticised on the grounds that their governing equations for the smoothly varying, spatially averaged, fields do not satisfy the accepted invariance principles basic to modern continuum mechanics. This paper demonstrates one way in which such theories can arise from a properly invariant continuum approach and indicates the presence of hitherto unconsidered terms that bring them closer to the generally accepted microscopic picture. The resulting theory has applications both to rotating helium II in the laboratory, and to rotating neutron stars (pulsars).

We consider the asymptotic behaviour of positive solutions u(t, x) of the fast diffusion equation $${u_t=\Delta (u^{m}/m)= {\rm div}\,(u^{m-1} \nabla u)}$$ posed for $${x\in\mathbb R^d}$$ , t > 0, with a precise value for the exponent m = (d − 4)/(d − 2). The space dimension is d ≧ 3 so that m < 1, and even m = −1 for d = 3. This case had been left open in the general study (Blanchet et al. in Arch Rat Mech Anal 191:347–385, 2009) since it requires quite different functional analytic methods, due in particular to the absence of a spectral gap for the operator generating the linearized evolution. The linearization of this flow is interpreted here as the heat flow of the Laplace– Beltrami operator of a suitable Riemannian Manifold $${(\mathbb R^d,{\bf g})}$$ , with a metric g which is conformal to the standard $${\mathbb R^d}$$ metric. Studying the pointwise heat kernel behaviour allows to prove suitable Gagliardo–Nirenberg inequalities associated with the generator. Such inequalities in turn allow one to study the nonlinear evolution as well, and to determine its asymptotics, which is identical to the one satisfied by the linearization. In terms of the rescaled representation, which is a nonlinear Fokker–Planck equation, the convergence rate turns out to be polynomial in time. This result is in contrast with the known exponential decay of such representation for all other values of m.

We study monotonicity and 1-dimensional symmetry for positive solutions with algebraic growth of the following elliptic system: $$\left\{\begin{array}{ll} -\Delta u = -u \upsilon^2 &\quad {\rm in}\, \mathbb{R}^N\\ -\Delta \upsilon= -u^2 \upsilon &\quad {{\rm in}\, \mathbb{R}^N},\end{array}\right.$$ for every dimension $${N \geqq 2}$$ . In particular, we prove a Gibbons-type conjecture proposed by Berestycki et al.

For a large class of variational problems we prove that minimizers are symmetric whenever they are C 1.

Let (M, g) be a complete Riemannian manifold, $${\Omega\subset M}$$ an open subset whose closure is homeomorphic to an annulus. We prove that if ∂Ω is smooth and it satisfies a strong concavity assumption, then there are at least two distinct geodesics in $${\overline\Omega=\Omega\cup\partial\Omega}$$ starting orthogonally to one connected component of ∂Ω and arriving orthogonally onto the other one. Using the results given in Giambò et al. (Adv Differ Equ 10:931–960, 2005), we then obtain a proof of the existence of two distinct homoclinic orbits for an autonomous Lagrangian system emanating from a nondegenerate maximum point of the potential energy, and a proof of the existence of two distinct brake orbits for a class of Hamiltonian systems. Under a further symmetry assumption, the result is improved by showing the existence of at least dim(M) pairs of geometrically distinct geodesics as above, brake orbits and homoclinic orbits. In our proof we shall use recent deformation results proved in Giambò et al. (Nonlinear Anal Ser A: Theory Methods Appl 73:290–337, 2010).

We paralinearize the Muskat equation to extract an explicit parabolic evolution equation having a compact form. This result is applied to give a simple proof of the local well-posedness of the Cauchy problem for rough initial data, in homogeneous Sobolev spaces $$\dot{H}^1(\mathbb {R})\cap \dot{H}^s(\mathbb {R})$$ with $$s>3/2$$. This paper is essentially self-contained and does not rely on general results from paradifferential calculus.