Showing papers in "Archive for Rational Mechanics and Analysis in 2021"

Lipschitz Bounds and Nonautonomous Integrals

Abstract: We provide a general approach to Lipschitz regularity of solutions for a large class of vector-valued, nonautonomous variational problems exhibiting nonuniform ellipticity. The functionals considered here range from those with unbalanced polynomial growth conditions to those with fast, exponential type growth. The results obtained are sharp with respect to all the data considered and also yield new, optimal regularity criteria in the classical uniformly elliptic case. We give a classification of different types of nonuniform ellipticity, accordingly identifying suitable conditions to get regularity theorems.

Bounded Solutions of Ideal MHD with Compact Support in Space-Time

Abstract: We show that in 3-dimensional ideal magnetohydrodynamics there exist infinitely many bounded solutions that are compactly supported in space-time and have non-trivial velocity and magnetic fields. The solutions violate conservation of total energy and cross helicity, but preserve magnetic helicity. For the 2-dimensional case we show that, in contrast, no nontrivial compactly supported solutions exist in the energy space.

Positive Solutions of Transport Equations and Classical Nonuniqueness of Characteristic curves

Abstract: The seminal work of DiPerna and Lions (Invent Math 98(3):511–547, 1989) guarantees the existence and uniqueness of regular Lagrangian flows for Sobolev vector fields. The latter is a suitable selection of trajectories of the related ODE satisfying additional compressibility/semigroup properties. A long-standing open question is whether the uniqueness of the regular Lagrangian flow is a corollary of the uniqueness of the trajectory of the ODE for a.e. initial datum. Using Ambrosio’s superposition principle, we relate the latter to the uniqueness of positive solutions of the continuity equation and we then provide a negative answer using tools introduced by Modena and Szekelyhidi in the recent groundbreaking work (Modena and Szekelyhidi in Ann PDE 4(2):38, 2018). On the opposite side, we introduce a new class of asymmetric Lusin–Lipschitz inequalities and use them to prove the uniqueness of positive solutions of the continuity equation in an integrability range which goes beyond the DiPerna–Lions theory.

Nonlocal-to-Local Convergence of Cahn-Hilliard Equations: Neumann Boundary Conditions and Viscosity Terms.

Abstract: We consider a class of nonlocal viscous Cahn–Hilliard equations with Neumann boundary conditions for the chemical potential. The double-well potential is allowed to be singular (e.g. of logarithmic type), while the singularity of the convolution kernel does not fall in any available existence theory under Neumann boundary conditions. We prove well-posedness for the nonlocal equation in a suitable variational sense. Secondly, we show that the solutions to the nonlocal equation converge to the corresponding solutions to the local equation, as the convolution kernels approximate a Dirac delta. The asymptotic behaviour is analyzed by means of monotone analysis and Gamma convergence results, both when the limiting local Cahn–Hilliard equation is of viscous type and of pure type.

Anticipation Breeds Alignment

Abstract: We study the large-time behavior of systems driven by radial potentials, which react to anticipated positions, $$\mathbf{x}^\tau (t)=\mathbf{x}(t)+\tau \mathbf{v}(t)$$ , with anticipation increment $$\tau >0$$ . As a special case, such systems yield the celebrated Cucker–Smale model for alignment, coupled with pairwise interactions. Viewed from this perspective, such anticipation-driven systems are expected to emerge into flocking due to alignment of velocities, and spatial concentration due to confining potentials. We treat both the discrete dynamics and large crowd hydrodynamics, proving the decisive role of anticipation in driving such systems with attractive potentials into velocity alignment and spatial concentration. We also study the concentration effect near equilibrium for anticipated-based dynamics of pair of agents governed by attractive–repulsive potentials.

Curvature-Driven Wrinkling of Thin Elastic Shells

Abstract: How much energy does it take to stamp a thin elastic shell flat? Motivated by recent experiments on the wrinkling patterns of floating shells, we develop a rigorous method via $$\Gamma $$ -convergence for answering this question to leading order in the shell’s thickness and other small parameters. The observed patterns involve “ordered” regions of well-defined wrinkles alongside “disordered” regions whose local features are less robust; as little to no tension is applied, the preference for order is not a priori clear. Rescaling by the energy of a typical pattern, we derive a limiting variational problem for the effective displacement of the shell. It asks, in a linearized way, to cover up a maximum area with a length-shortening map to the plane. Convex analysis yields a boundary value problem characterizing the accompanying patterns via their defect measures. Partial uniqueness and regularity theorems follow from the method of characteristics on the ordered part of the shell. In this way, we can deduce from the principle of minimum energy the leading order features of stamped elastic shells.

Strong Convergence of the Vorticity for the 2D Euler Equations in the Inviscid Limit

Abstract: In this paper we prove the uniform-in-time $$L^p$$ convergence in the inviscid limit of a family $$\omega ^ u $$ of solutions of the 2D Navier–Stokes equations towards a renormalized/Lagrangian solution $$\omega $$ of the Euler equations. We also prove that, in the class of solutions with bounded vorticity, it is possible to obtain a rate for the convergence of $$\omega ^{ u }$$ to $$\omega $$ in $$L^p$$ . Finally, we show that solutions of the Euler equations with $$L^p$$ vorticity, obtained in the vanishing viscosity limit, conserve the kinetic energy. The proofs are given by using both a (stochastic) Lagrangian approach and an Eulerian approach.

Traveling quasi-periodic water waves with constant vorticity

Abstract: We prove the first bifurcation result of time quasi-periodic traveling wave solutions for space periodic water waves with vorticity. In particular, we prove the existence of small amplitude time quasi-periodic solutions of the gravity-capillary water waves equations with constant vorticity, for a bidimensional fluid over a flat bottom delimited by a space-periodic free interface. These quasi-periodic solutions exist for all the values of depth, gravity and vorticity, and restrict the surface tension to a Borel set of asymptotically full Lebesgue measure.

Gamma Calculus Beyond Villani and Explicit Convergence Estimates for Langevin Dynamics with Singular Potentials

Abstract: We apply Gamma calculus to the hypoelliptic and non-symmetric setting of Langevin dynamics under general conditions on the potential. This extension allows us to provide explicit estimates on the convergence rate (which is exponential) to equilibrium for the dynamics in a weighted $$H^1(\mu )$$ sense, $$\mu $$ denoting the unique invariant probability measure of the system. The general result holds for singular potentials, such as the well-known Lennard–Jones interaction and confining well, and it is applied in such a case to estimate the rate of convergence when the number of particles N in the system is large.

Continued Gravitational Collapse for Newtonian Stars

Abstract: The classical model of an isolated selfgravitating gaseous star is given by the Euler–Poisson system with a polytropic pressure law $$P(\rho )=\rho ^\gamma $$ , $$\gamma >1$$ . For any $$1<\gamma <\frac{4}{3}$$ , we construct an infinite-dimensional family of collapsing solutions to the Euler–Poisson system whose density is in general space inhomogeneous and undergoes gravitational blowup along a prescribed space-time surface, with continuous mass absorption at the origin. The leading order singular behavior is described by an explicit collapsing solution of the pressureless Euler–Poisson system.

Crystallization to the Square Lattice for a Two-Body Potential

Abstract: We consider two-dimensional zero-temperature systems of N particles to which we associate an energy of the form $$\begin{aligned} \mathcal {E}[V](X):=\sum _{1\leqq i\sqrt{2}$$ , in which case $${\overline{\mathcal E}_{\mathrm {sq}}[V]}=-4$$ . To the best of our knowledge, this is the first proof of crystallization to the square lattice for a two-body interaction energy.

Weak Solutions to the Muskat Problem with Surface Tension Via Optimal Transport

Abstract: Inspired by recent works on the threshold dynamics scheme for multi-phase mean curvature flow (by Esedoḡlu–Otto and Laux–Otto), we introduce a novel framework to approximate solutions of the Muskat problem with surface tension. Our approach is based on interpreting the Muskat problem as a gradient flow in a product Wasserstein space. This perspective allows us to construct weak solutions via a minimizing movements scheme. Rather than working directly with the singular surface tension force, we instead relax the perimeter functional with the heat content energy approximation of Esedoḡlu–Otto. The heat content energy allows us to show the convergence of the associated minimizing movement scheme in the Wasserstein space, and makes the scheme far more tractable for numerical simulations. Under a typical energy convergence assumption, we show that our scheme converges to weak solutions of the Muskat problem with surface tension. We then conclude the paper with a discussion on some numerical experiments and on equilibrium configurations.

A quantitative description of skyrmions in ultrathin ferromagnetic films and rigidity of degree $\pm1$ harmonic maps from $\mathbb{R}^2$ to $\mathbb{S}^2$

Abstract: We characterize skyrmions in ultrathin ferromagnetic films as local minimizers of a reduced micromagnetic energy appropriate for quasi two-dimensional materials with perpendicular magnetic anisotropy and interfacial Dzyaloshinskii–Moriya interaction. The minimization is carried out in a suitable class of two-dimensional magnetization configurations that prevents the energy from going to negative infinity, while not imposing any restrictions on the spatial scale of the configuration. We first demonstrate the existence of minimizers for an explicit range of the model parameters when the energy is dominated by the exchange energy. We then investigate the conformal limit, in which only the exchange energy survives and identify the asymptotic profiles of the skyrmions as degree 1 harmonic maps from the plane to the sphere, together with their radii, angles and energies. A byproduct of our analysis is a quantitative rigidity result for degree $$\pm \,1$$ harmonic maps from the two-dimensional sphere to itself.

Renormalized Energy Between Vortices in Some Ginzburg–Landau Models on 2-Dimensional Riemannian Manifolds

Abstract: We study a variational Ginzburg–Landau type model depending on a small parameter $$\varepsilon >0$$ for (tangent) vector fields on a 2-dimensional Riemannian manifold S. As $$\varepsilon \rightarrow 0$$ , these vector fields tend to have unit length so they generate singular points, called vortices, of a (non-zero) index if the genus $${\mathfrak {g}}$$ of S is different than 1. Our first main result concerns the characterization of canonical harmonic unit vector fields with prescribed singular points and indices. The novelty of this classification involves flux integrals constrained to a particular vorticity-dependent lattice in the $$2{\mathfrak {g}}$$ -dimensional space of harmonic 1-forms on S if $${\mathfrak {g}}\geqq 1$$ . Our second main result determines the interaction energy (called renormalized energy) between vortex points as a $$\Gamma $$ -limit (at the second order) as $$\varepsilon \rightarrow 0$$ . The renormalized energy governing the optimal location of vortices depends on the Gauss curvature of S as well as on the quantized flux. The coupling between flux quantization constraints and vorticity, and its impact on the renormalized energy, are new phenomena in the theory of Ginzburg–Landau type models. We also extend this study to two other (extrinsic) models for embedded hypersurfaces $$S\subset {{\mathbb {R}}}^3$$ , in particular, to a physical model for non-tangent maps to S coming from micromagnetics.

Well-posedness of Free Boundary Problem in Non-relativistic and Relativistic Ideal Compressible Magnetohydrodynamics

Abstract: We consider the free boundary problem for non-relativistic and relativistic ideal compressible magnetohydrodynamics in two and three spatial dimensions with the total pressure vanishing on the plasma–vacuum interface. We establish the local-in-time existence and uniqueness of solutions to this nonlinear characteristic hyperbolic problem under the Rayleigh–Taylor sign condition on the total pressure. The proof is based on certain tame estimates in anisotropic Sobolev spaces for the linearized problem and a modification of the Nash–Moser iteration scheme. Our result is uniform in the speed of light and appears to be the first well-posedness result for the free boundary problem in ideal compressible magnetohydrodynamics with zero total pressure on the moving boundary.

On the Serrin-Type Condition on One Velocity Component for the Navier-Stokes Equations

Abstract: In this paper we consider the regularity problem of the Navier–Stokes equations in $$ {\mathbb {R}}^{3} $$ . We show that the Serrin-type condition imposed on one component of the velocity $$ u_3\in L^p(0,T; L^q({\mathbb {R}}^{3} ))$$ with $$ \frac{2}{p}+ \frac{3}{q} <1$$ , $$ 3

Nonlocal-Interaction Equation on Graphs: Gradient Flow Structure and Continuum Limit

Abstract: We consider dynamics driven by interaction energies on graphs. We introduce graph analogues of the continuum nonlocal-interaction equation and interpret them as gradient flows with respect to a graph Wasserstein distance. The particular Wasserstein distance we consider arises from the graph analogue of the Benamou–Brenier formulation where the graph continuity equation uses an upwind interpolation to define the density along the edges. While this approach has both theoretical and computational advantages, the resulting distance is only a quasi-metric. We investigate this quasi-metric both on graphs and on more general structures where the set of “vertices” is an arbitrary positive measure. We call the resulting gradient flow of the nonlocal-interaction energy the nonlocal nonlocal-interaction equation (NL $$^2$$ IE). We develop the existence theory for the solutions of the NL $$^2$$ IE as curves of maximal slope with respect to the upwind Wasserstein quasi-metric. Furthermore, we show that the solutions of the NL $$^2$$ IE on graphs converge as the empirical measures of the set of vertices converge weakly, which establishes a valuable discrete-to-continuum convergence result.

Isodiametry, Variance, and Regular Simplices from Particle Interactions

Abstract: Consider a collection of particles interacting through an attractive-repulsive potential given as a difference of power laws and normalized so that its unique minimum occurs at unit separation. For a range of exponents corresponding to mild repulsion and strong attraction, we show that the minimum energy configuration is uniquely attained—apart from translations and rotations—by equidistributing the particles over the vertices of a regular top-dimensional simplex (i.e. an equilateral triangle in two dimensions and regular tetrahedron in three). If the attraction is not assumed to be strong, we show that these configurations are at least local energy minimizers in the relevant $$d_\infty $$ metric from optimal transportation, as are all of the other uncountably many unbalanced configurations with the same support. We infer the existence of phase transitions. The proof is based in part on a simple isodiametric variance bound which characterizes regular simplices; it shows that among probability measures on $${{\mathbf {R}}}^n$$ whose supports have at most unit diameter, the variance around the mean is maximized precisely by those measures which assign mass $$1/(n+1)$$ to each vertex of a (unit-diameter) regular simplex.

Stationary non-equilibrium solutions for coagulation systems

Abstract: We study coagulation equations under non-equilibrium conditions which are induced by the addition of a source term for small cluster sizes. We consider both discrete and continuous coagulation equations, and allow for a large class of coagulation rate kernels, with the main restriction being boundedness from above and below by certain weight functions. The weight functions depend on two power law parameters, and the assumptions cover, in particular, the commonly used free molecular and diffusion limited aggregation coagulation kernels. Our main result shows that the two weight function parameters already determine whether there exists a stationary solution under the presence of a source term. In particular, we find that the diffusive kernel allows for the existence of stationary solutions while there cannot be any such solutions for the free molecular kernel. The argument to prove the non-existence of solutions relies on a novel power law lower bound, valid in the appropriate parameter regime, for the decay of stationary solutions with a constant flux. We obtain optimal lower and upper estimates of the solutions for large cluster sizes, and prove that the solutions of the discrete model behave asymptotically as solutions of the continuous model.

Fluctuations Around a Homogenised Semilinear Random PDE

Abstract: We consider a semilinear parabolic partial differential equation in $$\mathbf{R}_+\times [0,1]^d$$ , where $$d=1, 2$$ or 3, with a highly oscillating random potential and either homogeneous Dirichlet or Neumann boundary condition. If the amplitude of the oscillations has the right size compared to its typical spatiotemporal scale, then the solution of our equation converges to the solution of a deterministic homogenised parabolic PDE, which is a form of law of large numbers. Our main interest is in the associated central limit theorem. Namely, we study the limit of a properly rescaled difference between the initial random solution and its LLN limit. In dimension $$d=1$$ , that rescaled difference converges as one might expect to a centred Ornstein–Uhlenbeck process. However, in dimension $$d=2$$ , the limit is a non-centred Gaussian process, while in dimension $$d=3$$ , before taking the CLT limit, we need to subtract at an intermediate scale the solution of a deterministic parabolic PDE, subject (in the case of Neumann boundary condition) to a non-homogeneous Neumann boundary condition. Our proofs make use of the theory of regularity structures, in particular of the very recently developed methodology allowing to treat parabolic PDEs with boundary conditions within that theory.

On the Problem of Maximal L q -regularity for Viscous Hamilton-Jacobi Equations

Abstract: In this paper we prove a conjecture by P.-L. Lions on maximal regularity of $$L^q$$ -type for periodic solutions to $$-\Delta u + |Du|^\gamma = f$$ in $$\mathbb {R}^d$$ , under the (sharp) assumption that $$q > d \frac{\gamma -1}{\gamma }$$ .

Nernst–Planck–Navier–Stokes Systems far from Equilibrium

Abstract: We consider ionic electrodiffusion in fluids, described by the Nernst–Planck–Navier–Stokes system. We prove that the system has global smooth solutions for arbitrary smooth data in bounded domains with a smooth boundary in three space dimensions, in the following situations. We consider: a arbitrary positive Dirichlet boundary conditions for the ionic concentrations, arbitrary Dirichlet boundary conditions for the potential, arbitrary positive initial concentrations, and arbitrary regular divergence-free initial velocities. Global regularity holds for any positive, possibly different diffusivities of the ions, in the case of two ionic species, coupled to Stokes equations for the fluid. The result also holds in the case of Navier–Stokes coupling, if the velocity is regular. The same global smoothness of solutions is proved to hold for arbitrarily many ionic species as well, but in that case we require all their diffusivities to be the same.

Mean-Field Limits: From Particle Descriptions to Macroscopic Equations.

Abstract: We rigorously derive pressureless Euler-type equations with nonlocal dissipative terms in velocity and aggregation equations with nonlocal velocity fields from Newton-type particle descriptions of swarming models with alignment interactions. Crucially, we make use of a discrete version of a modulated kinetic energy together with the bounded Lipschitz distance for measures in order to control terms in its time derivative due to the nonlocal interactions.

Almost Global Solutions to the Three-Dimensional Isentropic Inviscid Flows with Damping in a Physical Vacuum Around Barenlatt Solutions

Abstract: For the three-dimensional vacuum free boundary problem with physical singularity where the sound speed is $$C^{1/2}$$ -Holder continuous across the vacuum boundary of the compressible Euler equations with damping, without any symmetry assumptions, we prove the almost global existence of smooth solutions when the initial data are small perturbations of the Barenblatt self-similar solutions to the corresponding porous media equations simplified via Darcy’s law. It is proved that if the initial perturbation is of the size of $$\varepsilon $$ , then the existing time for smooth solutions is at least of the order of $$\exp (\varepsilon ^{-2/3})$$ . The key issue for the analysis is the slow sub-linear growth of vacuum boundaries of the order of $$t^{1/(3\gamma -1)}$$ , where $$\gamma >1$$ is the adiabatic exponent for the gas. This is in sharp contrast to the currently available global-in-time existence theory of expanding solutions to the vacuum free boundary problems with physical singularity of compressible Euler equations for which the expanding rate of vacuum boundaries is linear. The results obtained in this paper are closely related to the open question in multiple dimensions framed by T.-P. Liu’s construction of particular solutions in 1996.

Corrector Equations in Fluid Mechanics: Effective Viscosity of Colloidal Suspensions

Abstract: Consider a colloidal suspension of rigid particles in a steady Stokes flow. In a celebrated work, Einstein argued that in the regime of dilute particles the system behaves at leading order like a Stokes fluid with some explicit effective viscosity. In the present contribution, we rigorously define a notion of effective viscosity, regardless of the dilute regime assumption. More precisely, we establish a homogenization result for when particles are distributed according to a given stationary and ergodic random point process. The main novelty is the introduction and analysis of suitable corrector equations.

Nonlinear Hyperbolic Waves in Relativistic Gases of Massive Particles with Synge Energy

Abstract: In this article, we study some fundamental properties of nonlinear waves and the Riemann problem of Euler’s relativistic system when the constitutive equation for energy is that of Synge for a monatomic rarefied gas or its generalization for diatomic gas. These constitutive equations are the only ones compatible with the relativistic kinetic theory for massive particles in the whole range from the classical to the ultra-relativistic regime. They involve modified Bessel functions of the second kind and this makes Euler’s relativistic system rather complex. Based on delicate estimates of the Bessel functions, we prove: (i) a limit on the speed of sound of $$1{/}\sqrt{3}$$ times the speed of light (which a fortiori implies subluminality, that is causality), (ii) the genuine non-linearity of the acoustic waves, (iii) the compatibility of Rankine–Hugoniot relations with the second law of thermodynamics (entropy growth through all Lax shocks), and (iv) the unique resolvability of the initial value problem of Riemann (if we include the possibility of vacuum as in the non-relativistic context).

Asymptotic Stability of Minkowski Space-Time with Non-compactly Supported Massless Vlasov Matter.

Abstract: We prove the global asymptotic stability of the Minkowski space for the massless Einstein–Vlasov system in wave coordinates. In contrast with previous work on the subject, no compact support assumptions on the initial data of the Vlasov field in space or the momentum variables are required. In fact, the initial decay in v is optimal. The present proof is based on vector field and weighted vector field techniques for Vlasov fields, as developed in previous work of Fajman, Joudioux, and Smulevici, and heavily relies on several structural properties of the massless Vlasov equation, similar to the null and weak null conditions. To deal with the weak decay rate of the metric, we propagate well-chosen hierarchized weighted energy norms which reflect the strong decay properties satisfied by the particle density far from the light cone. A particular analytical difficulty arises at the top order, when we do not have access to improved pointwise decay estimates for certain metric components. This difficulty is resolved using a novel hierarchy in the massless Einstein–Vlasov system, which exploits the propagation of different growth rates for the energy norms of different metric components.

Bifurcation of Symmetric Domain Walls for the Bénard–Rayleigh Convection Problem

Abstract: We prove the existence of domain walls for the Benard-Rayleigh convection problem. Our approach relies upon a spatial dynamics formulation of the hydrodynamic problem, a center manifold reduction, and a normal form analysis of a reduced system. Domain walls are constructed as heteroclinic orbits of this reduced system.

On the Diffusive-Mean Field Limit for Weakly Interacting Diffusions Exhibiting Phase Transitions

Abstract: The objective of this article is to analyse the statistical behaviour of a large number of weakly interacting diffusion processes evolving under the influence of a periodic interaction potential. We focus our attention on the combined mean field and diffusive (homogenisation) limits. In particular, we show that these two limits do not commute if the mean field system constrained to the torus undergoes a phase transition, that is to say, if it admits more than one steady state. A typical example of such a system on the torus is given by the noisy Kuramoto model of mean field plane rotators. As a by-product of our main results, we also analyse the energetic consequences of the central limit theorem for fluctuations around the mean field limit and derive optimal rates of convergence in relative entropy of the Gibbs measure to the (unique) limit of the mean field energy below the critical temperature.

On the Boundary Layer Equations with Phase Transition in the Kinetic Theory of Gases

Abstract: Consider the steady Boltzmann equation with slab symmetry for a monatomic, hard sphere gas in a half space. At the boundary of the half space, it is assumed that the gas is in contact with its condensed phase. The present paper discusses the existence and uniqueness of a uniformly decaying boundary layer type solution of the Boltzmann equation in this situation, in the vicinity of the Maxwellian equilibrium with zero bulk velocity, with the same temperature as that of the condensed phase, and whose pressure is the saturating vapor pressure at the temperature of the interface. This problem has been extensively studied, first by Sone, Aoki and their collaborators, by means of careful numerical simulations. See section 2 of (Bardos et al. in J Stat Phys 124:275–300, 2006) for a very detailed presentation of these works. More recently, Liu and Yu (Arch Ration Mech Anal 209:869–997, 2013) proposed an extensive mathematical strategy to handle the problems studied numerically by Sone, Aoki and their group. The present paper offers an alternative, possibly simpler proof of one of the results discussed in Liu and Yu (2013).