Archive for Rational Mechanics and Analysis最新文献

We establish the global-in-time existence of solutions of the Cauchy problem for the full Navier–Stokes equations for compressible heat-conducting flow in multidimensions with initial data that are large, discontinuous, spherically symmetric, and away from the vacuum. The solutions obtained here are of global finite total relative-energy including the origin, while cavitation may occur as balls centred at the origin of symmetry for which the interfaces between the fluid and the vacuum must be upper semi-continuous in space-time in the Eulerian coordinates. On any region strictly away from the possible vacuum, the velocity and specific internal energy are Hölder continuous, and the density has a uniform upper bound. To achieve this, our main strategy is to regard the Cauchy problem as the limit of a series of carefully designed initial-boundary value problems that are formulated in finite annular regions. For such approximation problems, we can derive uniform a priori estimates that are independent of both the inner and outer radii of the annuli considered in the spherically symmetric Lagrangian coordinates. The entropy inequality is recovered after taking the limit of the outer radius to infinity by using Mazur’s lemma and the convexity of the entropy function, which is required for the limit of the inner radius tending to zero. Then the global weak solutions of the original problem are attained via careful compactness arguments applied to the approximate solutions in the Eulerian coordinates.

We consider a gas of bosons interacting through a hard-sphere potential with radius (mathfrak {a}) in the thermodynamic limit. We derive an upper bound for the ground state energy per particle at low density. Our bound captures the leading term (4pi rho mathfrak {a}) and shows that corrections are smaller than (C rho mathfrak {a} (rho {{mathfrak {a}}}^3)^{1/2}), for a sufficiently large constant (C > 0). In combination with a known lower bound, our result implies that the first sub-leading term to the ground state energy of a dilute gas of hard spheres is, in fact, of the order (rho mathfrak {a}(rho {{mathfrak {a}}}^3)^{1/2}), in agreement with the Lee–Huang–Yang prediction.

Following a celebrated paper by Jordan, Kinderleherer and Otto, it is possible to discretize in time the Fokker–Planck equation (partial _tvarrho =Delta varrho +nabla cdot (varrho nabla V)) by solving a sequence of iterated variational problems in the Wasserstein space, and the sequence of piecewise constant curves obtained from the scheme is known to converge to the solution of the continuous PDE. This convergence is uniform in time valued in the Wasserstein space and also strong in (L^1) in space-time. We prove in this paper, under some assumptions on the domain (a bounded and smooth convex domain) and on the initial datum (which is supposed to be bounded away from zero and infinity and belong to (W^{1,p}) for an exponent p larger than the dimension), that the convergence is actually strong in (L^2_tH^2_x), hence strongly improving open the previously known results in terms of the order of derivation in space. The technique is based on some inequalities, obtained with optimal transport techniques, that can be proven on the discrete sequence of approximate solutions, and that mimic the corresponding continuous computations.

We show that small perturbations of the spatially homogeneous equilibrium of a thermally driven compressible viscous fluid are globally stable. Specifically, any weak solution of the evolutionary Navier–Stokes–Fourier system driven by thermal convection converges to an equilibrium as time goes to infinity. The main difficulty to overcome is the fact the problem does not admit any obvious Lyapunov function. The result applies, in particular, to the Rayleigh–Bénard convection problem.

We obtain uniform in time (L^infty )-bounds for the solutions to a class of thermo-diffusive systems. The nonlinearity is assumed to be at most sub-exponentially growing at infinity and have a linear behavior near zero.

Local existence and uniqueness for the two-dimensional hydrostatic Euler equations in Sobolev spaces has been established by Masmoudi and Wong (Arch Rational Mech Anal 204:231–271, 2012) under the local Rayleigh condition. Under certain assumptions, we show that such solution will either develop singularities or produce the collapse of the local Rayleigh condition. In addition, we find necessary conditions for global solvability in Sobolev spaces. Finally, for certain class of initial data, we establish the finite time blow-up of solutions of the semi-Lagrangian equations introduced by Brenier (Nonlinearity 12:495–512, 1999). Our proof relies on new monotonicity identities for the solution of the hydrostatic Euler equations under the local Rayleigh condition.

This work is devoted to the large-scale rheology of suspensions of non-Brownian inertialess rigid particles, possibly self-propelling, suspended in a Stokes flow. Starting from a hydrodynamic model, we derive a semi-dilute mean-field description in form of a Doi-type model, which is given by a ‘macroscopic’ effective Stokes equation coupled with a ‘microscopic’ Vlasov equation for the statistical distribution of particle positions and orientations. This accounts for some non-Newtonian effects since the viscosity in the effective Stokes equation depends on the local distribution of particle orientations via Einstein’s formula. The main difficulty is the detailed analysis of multibody hydrodynamic interactions between the particles, which we perform by means of a cluster expansion combined with a multipole expansion in a suitable dilute regime.

We classify regularity for Lagrangian mean curvature type equations, which include the potential equation for prescribed Lagrangian mean curvature and those for Lagrangian mean curvature flow self-shrinkers and expanders, translating solitons, and rotating solitons. Convex solutions of the second boundary value problem for certain such equations were constructed by Brendle-Warren (J Differ Geom 84(2):267-287, 2010), Huang (J Funct Anal 269(4):1095-1114, 2015), and Wang-Huang-Bao (Calc Var Partial Differ Equ 62(3):74 2023). We first show that convex viscosity solutions are regular provided the Lagrangian angle or phase is (C^2) and convex in the gradient variable. We next show that for merely Hölder continuous phases, convex solutions are regular if they are (C^{1,beta }) for sufficiently large (beta ). Singular solutions are given to show that each condition is optimal and that the Hölder exponent is sharp. Along the way, we generalize the constant rank theorem of Bian and Guan to include arbitrary dependence on the Legendre transform.

We provide a concise PDE-based proof of anomalous diffusion in the Kraichan model—a stochastic, white-in-time model of passive scalar turbulence; that is, we show an exponential rate of (L^2) decay in expectation of a passive scalar advected by a certain white-in-time, correlated-in-space, divergence-free Gaussian field, uniform in the initial data and the diffusivity of the passive scalar. Additionally, we provide examples of correlated-in-time versions of the Kraichnan model which fail to exhibit anomalous diffusion despite their (formal) white-in-time limits exhibiting anomalous diffusion. As part of this analysis, we prove that anomalous diffusion of a scalar advected by some flow implies non-uniqueness of the ODE trajectories of that flow.

We present a new proof of linear stability of the Schwarzschild solution to gravitational perturbations. Our approach employs the system of linearised gravity in the new geometric gauge of Benomio (A new gauge for gravitational perturbations of Kerr spacetimes I: the linearised theory, 2022, https://arxiv.org/abs/2211.00602), specialised to the (|a|=0) case. The proof fundamentally relies on the novel structure of the transport equations in the system. Indeed, while exploiting the well-known decoupling of two gauge invariant linearised quantities into spin (pm 2) Teukolsky equations, we make enhanced use of the red-shifted transport equations and their stabilising properties to control the gauge dependent part of the system. As a result, an initial-data gauge normalisation suffices to establish both orbital and asymptotic stability for all the linearised quantities in the system. The absence of future gauge normalisations is a novel element in the linear stability analysis of black hole spacetimes in geometric gauges governed by transport equations. In particular, our approach simplifies the proof of Dafermos et al. (Acta Math 222:1–214, 2019), which requires a future normalised (double-null) gauge to establish asymptotic stability for the full system.