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Register a new userPoisson Commuting Energies for a System of Infinitely Many Bosons
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We consider the cubic Gross-Pitaevskii (GP) hierarchy in one spatial dimension. We establish the existence of an infinite sequence of observables such that the corresponding trace functionals, which we call ``energies, commute with respect to the weak Lie-Poisson structure defined by the authors in arXiv:1908.03847. The Hamiltonian equation associated to the third energy functional is precisely the GP hierarchy. The equations of motion corresponding to the remaining energies generalize the well-known nonlinear Schrodinger hierarchy, the third element of which is the one-dimensional cubic nonlinear Schrodinger equation. This work provides substantial evidence for the GP hierarchy as a new integrable system.
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We prove the existence of ground states for the semi-relativistic Schrodinger-Poisson-Slater energy $$I^{alpha,beta}(rho)=inf_{substack{uin H^frac 12(R^3) int_{R^3}|u|^2 dx=rho}} frac{1}{2}|u|^2_{H^frac 12(R^3)} +alphaintint_{R^{3}timesR^{3}} frac{| u(x)|^{2}|u(y)|^2}{|x-y|}dxdy-betaint_{R^{3}}|u|^{frac{8}{3}}dx$$ $alpha,beta>0$ and $rho>0$ is small enough. The minimization problem is $L^2$ critical and in order to characterize of the values $alpha, beta>0$ such that $I^{alpha, beta}(rho)>-infty$ for every $rho>0$, we prove a new lower bound on the Coulomb energy involving the kinetic energy and the exchange energy. We prove the existence of a constant $S>0$ such that $$frac{1}{S}frac{|varphi|_{L^frac 83(R^3)}}{|varphi|_{dot H^frac 12(R^3)}^frac 12}leq left (intint_{R^3times R^3} frac{|varphi(x)|^2|varphi(y)|^2}{|x-y|}dxdyright)^frac 18 $$ for all $varphiin C^infty_0(R^3)$. Eventually we show that similar compactness property fails provided that in the energy above we replace the inhomogeneous Sobolev norm $|u|^2_{H^frac 12(R^3)}$ by the homogeneous one $|u|_{dot H^frac 12(R^3)}$.
We study the asymptotic behavior of ground state energy for Schrodinger-Poisson-Slater energy functional. We show that ground state energy restricted to radially symmetric functions is above the ground state energy when the number of particles is sufficiently large.
The compressible Navier-Stokes-Poisson system is concerned in the present paper, and the global existence and uniqueness of the strong solution is shown in the framework of hybrid Besov spaces in three and higher dimensions.
We study the statistics of backward clusters in a gas of hard spheres at low density. A backward cluster is defined as the group of particles involved directly or indirectly in the backwards-in-time dynamics of a given tagged sphere. We derive upper and lower bounds on the average size of clusters by using the theory of the homogeneous Boltzmann equation combined with suitable hierarchical expansions. These representations are known in the easier context of Maxwellian molecules (Wild sums). We test our results with a numerical experiment based on molecular dynamics simulations.
In the early 1980s, S. T. Yau conjectured that any compact Riemannian three-manifold admits an infinite number of closed immersed minimal surfaces. We use min-max theory for the area functional to prove this conjecture in the positive Ricci curvature setting. More precisely, we show that every compact Riemannian manifold with positive Ricci curvature and dimension at most seven contains infinitely many smooth, closed, embedded minimal hypersurfaces. In the last section we mention some open problems related with the geometry of these minimal hypersurfaces.
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