Archive for Rational Mechanics and Analysis最新文献

We consider a family of vectorial models for cohesive fracture, which may incorporate (textrm{SO}(n))-invariance. The deformation belongs to the space of generalized functions of bounded variation and the energy contains an (elastic) volume energy, an opening-dependent jump energy concentrated on the fractured surface, and a Cantor part representing diffuse damage. We show that this type of functional can be naturally obtained as (Gamma )-limit of an appropriate phase-field model. The energy densities entering the limiting functional can be expressed, in a partially implicit way, in terms of those appearing in the phase-field approximation.

In this paper we prove the well-posedness of the generalized Dean–Kawasaki equation driven by noise that is white in time and colored in space. The results treat diffusion coefficients that are only locally ({1}/{2})-Hölder continuous, including the square root. This solves several open problems, including the well-posedness of the Dean–Kawasaki equation and the nonlinear Dawson–Watanabe equation with correlated noise.

A longstanding belief has been that the semimajor axes, in the Newtonian planetary problem, are stable. Our the course of the XIX century, Laplace, Lagrange and others gave stronger and stronger arguments in this direction, thus culminating in what has commonly been referred to as the first Laplace–Lagrange stability theorem. In the problem with 3 planets, we prove the existence of orbits along which the semimajor axis of the outer planet undergoes large random variations thus disproving the conclusion of the Laplace–Lagrange theorem. The time of instability varies as a negative power of the masses of the planets. The orbits we have found fall outside the scope of the theory of Nekhoroshev–Niederman because they are not confined by the conservation of angular momentum and because the Hamiltonian is not (uniformly) convex with respect to the Keplerian actions.

The aim of this paper is to prove the existence of optimal shapes in bilinear parabolic optimal control problems. We consider a parabolic equation (partial _tu_m-Delta u_m=f(t,x,u_m)+mu_m). The set of admissible controls is given by (A={min L^infty ,, m_-leqq mleqq m_+{text { almost everywhere, }}int _Omega m(t,cdot )=V_1(t)}), where (m_pm =m_pm (t,x)) are two reference functions in (L^infty ({(0,T)times {Omega }})), and where (V_1=V_1(t)) is a reference integral constraint. The functional to optimise is (J:mmapsto iint _{(0,T)times {Omega }} j_1(u_m)+int _{Omega }j_2(u_m(T))). Roughly speaking, we prove that, if (j_1) and (j_2) are non-decreasing and if one is increasing, then any solution of (max _A J) is bang-bang: any optimal (m^*) writes (m^*=mathbb {1}_E m_-+mathbb {1}_{E^c}m_+) for some (Esubset {(0,T)times {Omega }}). From the point of view of shape optimization, this is a parabolic analog of the Buttazzo-Dal Maso theorem in shape optimisation. The proof is based on second-order criteria and on an approximation-localisation procedure for admissible perturbations. This last part uses the theory of parabolic equations with measure data.

In this paper we establish the existence of the locally stable Meissner solutions of the three dimensional full Ginzburg–Landau system of superconductivity.

We show that propagation speeds in invasion processes modeled by reaction–diffusion systems are determined by marginal spectral stability conditions, as predicted by the marginal stability conjecture. This conjecture was recently settled in scalar equations; here we give a full proof for the multi-component case. The main new difficulty lies in precisely characterizing diffusive dynamics in the leading edge of invasion fronts. To overcome this, we introduce coordinate transformations which allow us to recognize a leading order diffusive equation relying only on an assumption of generic marginal pointwise stability. We are then able to use self-similar variables to give a detailed description of diffusive dynamics in the leading edge, which we match with a traveling invasion front in the wake. We then establish front selection by controlling these matching errors in a nonlinear iteration scheme, relying on sharp estimates on the linearization about the invasion front. We briefly discuss applications to parametrically forced amplitude equations, competitive Lotka–Volterra systems, and a tumor growth model.

We obtain a new formula for the Loewner energy of Jordan curves on the sphere, which is a Kähler potential for the essentially unique Kähler metric on the Weil–Petersson universal Teichmüller space, as the renormalised energy of moving frames on the two domains of the sphere delimited by the given curve.

We consider the Boltzmann equation in a convex domain with a non-isothermal boundary of diffuse reflection. For both unsteady/steady problems, we construct solutions belonging to (W^{1,p}_x) for any (p<3). We prove that the unsteady solution converges to the steady solution in the same Sobolev space exponentially quickly as (t rightarrow infty ).

We study minimisers of the p-conformal energy functionals,

defined for self mappings (f:{mathbb {D}}rightarrow {mathbb {D}}) with finite distortion and prescribed boundary values (f_0). Here

is the pointwise distortion functional and (mu _f(z)) is the Beltrami coefficient of f. We show that for quasisymmetric boundary data the limiting regimes (prightarrow infty ) recover the classical Teichmüller theory of extremal quasiconformal mappings (in part a result of Ahlfors), and for (prightarrow 1) recovers the harmonic mapping theory. Critical points of (textsf{E}_p) always satisfy the inner-variational distributional equation

We establish the existence of minimisers in the a priori regularity class (W^{1,frac{2p}{p+1}}({mathbb {D}})) and show these minimisers have a pseudo-inverse - a continuous (W^{1,2}({mathbb {D}})) surjection of ({mathbb {D}}) with ((hcirc f)(z)=z) almost everywhere. We then give a sufficient condition to ensure (C^{infty }({mathbb {D}})) smoothness of solutions to the distributional equation. For instance ({mathbb {K}}(z,f)in L^{p+1}_{loc}({mathbb {D}})) is enough to imply the solutions to the distributional equation are local diffeomorphisms. Further ({mathbb {K}}(w,h)in L^1({mathbb {D}})) will imply h is a homeomorphism, and together these results yield a diffeomorphic minimiser. We show such higher regularity assumptions to be necessary for critical points of the inner variational equation.

Moeckel (Math Z 205:499–517, 1990), Moeckel and Simó (SIAM J Math Anal 26:978–998, 1995) proved that, while continuously changing the masses, a 946-body planar central configuration bifurcates into a spatial central configuration. We show that this kind of bifurcation does not occur with 5 bodies. Question 17 in the list (Albouy et al. in Celest Mech Dyn Astr 113:369–375, 2012) is thus answered negatively.