Archive for Rational Mechanics and Analysis最新文献

In the limit of vanishing lattice spacing we provide a rigorous variational coarse-graining result for a next-to-nearest neighbor lattice model of a simple crystal. We show that the (Gamma )-limit of suitable scaled versions of the model leads to an energy describing a continuum mechanical model depending on partial dislocations and stacking faults. Our result highlights the necessary multiscale character of the energies setting the groundwork for more comprehensive models that can better explain and predict the mechanical behavior of materials with complex defect structures.

We rigorously construct the first steady traveling wave solutions of the 2D incompressible Euler equation that take the form of a contiguous vortex-patch dipole, which can be viewed as the vortex-patch counterpart of the well-known Lamb–Chaplygin dipole. Our construction is based on a novel fixed-point approach that determines the patch boundary as the fixed point of a certain nonlinear map. Smoothness and other properties of the patch boundary are also obtained.

We consider a family of isolated inhomogeneous steady states of the gravitational Vlasov–Poisson system with a point mass at the centre. These are parametrised by the polytropic index (k>1/2), so that the phase space density of the steady state is (C^1) at the vacuum boundary if and only if (k>1). We prove the following sharp dichotomy result: if (k>1), the linear perturbations Landau damp and if (1/2< kle 1) they do not. The above dichotomy is a new phenomenon and highlights the importance of steady state regularity at the vacuum boundary in the discussion of the long-time behaviour of the perturbations. Our proof of (nonquantitative) gravitational relaxation around steady states with (k>1) is the first such result for the gravitational Vlasov–Poisson system. The key novelty of this work is the proof that no embedded eigenvalues exist in the essential spectrum of the linearised system.

We study a dilute system of N interacting bosons coupled to an impurity particle via a pair potential in the Gross–Pitaevskii regime. We derive an expansion of the ground state energy up to order one in the boson number, and show that the difference of excited eigenvalues to the ground state is given by the eigenvalues of the renormalized Bogoliubov–Fröhlich Hamiltonian in the limit (Nrightarrow infty ).

We consider the Hookean dumbbell model, a system of nonlinear PDEs arising in the kinetic theory of homogeneous dilute polymeric fluids. It consists of the unsteady incompressible Navier–Stokes equations in a bounded Lipschitz domain, coupled to a Fokker–Planck-type parabolic equation with a centre-of-mass diffusion term, for the probability density function, modelling the evolution of the configuration of noninteracting polymer molecules in the solvent. The micro–macro interaction is reflected by the presence of a drag term in the Fokker–Planck equation and the divergence of a polymeric extra-stress tensor in the Navier–Stokes balance of momentum equation. We introduce the concept of generalised dissipative solution—a relaxation of the usual notion of weak solution, allowing for the presence of a, possibly nonzero, defect measure in the momentum equation. This defect measure accounts for the lack of compactness in the polymeric extra-stress tensor. We prove global existence of generalised dissipative solutions satisfying additionally an energy inequality for the macroscopic deformation tensor. Using this inequality, we establish a conditional regularity result: any generalised dissipative solution with a sufficiently regular velocity field is a weak solution to the Hookean dumbbell model. Additionally, in two space dimensions we provide a rigorous derivation of the macroscopic closure of the Hookean model and discuss its relationship with the Oldroyd-B model with stress diffusion. Finally, we improve a result by Barrett and Süli (Nonlinear Anal. Real World Appl. 39:362–395, 2018) by establishing the global existence of weak solutions for a larger class of initial data.

We establish the time-asymptotic stability of generic Riemann solutions to the one-dimensional compressible Navier–Stokes–Fourier equations. The Riemann solution under consideration is a generic combination of a shock, a contact discontinuity, and a rarefaction wave. We prove that the perturbed solution of Navier–Stokes–Fourier converges, uniformly in space as time goes to infinity, to an ansatz composed of viscous shock with time-dependent shift, a viscous contact wave and an inviscid rarefaction wave. This is a first resolution of the time-asymptotic stability of three waves of different kinds associated with the generic Riemann solutions. Our approach relies on the method of a-contraction with shifts and relative entropy, specifically applied to both the shock wave and the contact wave. It enables the application of a global energy method for the generic combination of three waves.

In this paper, we prove an abstract Birkhoff normal form theorem for some unbounded infinite dimensional Hamiltonian systems. Based on this result we obtain that the solution to Derivative Nonlinear Schrödinger equations under periodic boundary condition with typical small enough initial value remains small in the Sobolev norm ( H^{textbf{s}}(mathbb {T})) over a long time interval. The length of the time interval is equal to (e^{|ln R|^{1+gamma }}) with (0<gamma <1/5) as the initial value is smaller than (Rll 1).

We study space–time fluctuations of a hard sphere system at thermal equilibrium, and prove that the covariance converges to the solution of a linearized Boltzmann equation in the low density limit, globally in time. This result was obtained previously in Bodineau et al. (Commun Pure Appl Math 76:3852–3911, 2021) by using uniform bounds on the number of recollisions of dispersing systems of hard spheres [as provided for instance in Burago et al. (Ann Math (2), 147(3):695–708, 1998)]. We present a self-contained proof with substantial differences, which does not use this geometric result. This can be regarded as the first step of a program aiming of deriving the fluctuation theory of the rarefied gas for interaction potentials different from hard spheres.

We construct d–dimensional polyhedral chains such that the distribution of tangent planes is close to a prescribed measure on the Grassmannian and the chains are either cycles (if the barycenter of the prescribed measure, considered as a measure on (bigwedge ^d mathbb {R}^n), is 0), or their boundary is the boundary of a unit d–cube (if the barycenter of the prescribed measure is a simple d–vector). Such fillings were first proven to exist by Burago and Ivanov (Geom Funct Anal 14:469–490, 2004); our work gives an explicit construction, which is also flexible to generalizations. For instance, in the case that the measure on the Grassmannian is supported on the set of positively oriented d–planes, we can construct fillings that are Lipschitz multigraphs. We apply this construction to prove the surprising fact that, for anisotropic integrands, polyconvexity is equivalent to quasiconvexity of the associated Q-integrands (that is, ellipticity for Lipschitz multigraphs) and to show that strict polyconvexity is necessary for the atomic condition to hold.

We are concerned with the global existence of finite-energy entropy solutions of the one-dimensional compressible Euler equations with (possibly) damping, alignment forces, and the nonlocal interactions of Newtonian repulsion and quadratic confinement. Both the polytropic gas law and the general gas law are analyzed. This is achieved by constructing a sequence of solutions of the one-dimensional compressible Navier–Stokes-type equations with density-dependent viscosity on expanding intervals with the stress-free boundary condition and then taking the vanishing viscosity limit. The main difficulties in this paper arise from the appearance of the nonlocal terms. In particular, some uniform higher moment estimates of the solutions for the compressible Navier–Stokes equations on the expanding intervals with stress-free boundary condition are obtained by careful design of the approximate initial data.