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Global Existence and the Decay of Solutions to the Prandtl System with Small Analytic Data Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-04-12 Marius Paicu, Ping Zhang
In this paper, we prove the global existence and the large time decay estimate of solutions to Prandtl system with small initial data, which is analytical in the tangential variable. The key ingredient used in the proof is to derive a sufficiently fast decay-in-time estimate of some weighted analytic energy estimate to a quantity, which consists of a linear combination of the tangential velocity with
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A Posteriori Error Estimates for Numerical Solutions to Hyperbolic Conservation Laws Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-04-11 Alberto Bressan, Maria Teresa Chiri, Wen Shen
This paper is concerned with a posteriori error bounds for a wide class of numerical schemes, for \(n\times n\) hyperbolic conservation laws in one space dimension. These estimates are achieved by a “post-processing algorithm”, checking that the numerical solution retains small total variation, and computing its oscillation on suitable subdomains. The results apply, in particular, to solutions obtained
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Small Knudsen Rate of Convergence to Rarefaction Wave for the Landau Equation Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-04-09 Renjun Duan, Dongcheng Yang, Hongjun Yu
In this paper, we are concerned with the hydrodynamic limit to rarefaction waves of the compressible Euler system for the Landau equation with Coulomb potentials as the Knudsen number \(\varepsilon >0\) is vanishing. Precisely, whenever \(\varepsilon >0\) is small, for the Cauchy problem on the Landau equation with suitable initial data involving a scaling parameter \(a\in [\frac{2}{3},1]\), we construct
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Self-similar solutions of the compressible Navier–Stokes equations Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-04-03 Pierre Germain, Tsukasa Iwabuchi
We construct forward self-similar solutions (expanders) for the compressible Navier–Stokes equations. Some of these self-similar solutions are smooth, while others exhibit a singularity due to cavitation at the origin.
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The Relativistic Quantum Boltzmann Equation Near Equilibrium Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-04-02 Gi-Chan Bae, Jin Woo Jang, Seok-Bae Yun
The relativistic quantum Boltzmann equation (or the relativistic Uehling–Uhlenbeck equation) describes the dynamics of single-species fast-moving quantum particles. With the recent development of relativistic quantum mechanics, the relativistic quantum Boltzmann equation has been widely used in physics and engineering, for example in the quantum collision experiments and the simulations of electrons
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On the Problem of Maximal $$L^q$$ L q -regularity for Viscous Hamilton–Jacobi Equations Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-03-30 Marco Cirant, Alessandro Goffi
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 \(q > d \frac{\gamma -1}{\gamma }\).
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Stability of Traveling Wave Solutions of Nonlinear Dispersive Equations of NLS Type Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-03-28 Katelyn Plaisier Leisman, Jared C. Bronski, Mathew A. Johnson, Robert Marangell
We present a rigorous modulational stability theory for periodic traveling wave solutions to equations of nonlinear Schrödinger type. For Hamiltonian dispersive equations with a non-singular symplectic form and d conserved quantities (in addition to the Hamiltonian), one expects that generically \({{\mathcal {L}}}\), the linearization around a periodic traveling wave, will have a particular Jordan
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Vanishing Diffusion Limits and Long Time Behaviour of a Class of Forced Active Scalar Equations Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-03-25 Susan Friedlander, Anthony Suen
We investigate the properties of an abstract family of advection diffusion equations in the context of the fractional Laplacian. Two independent diffusion parameters enter the system, one via the constitutive law for the drift velocity and one as the prefactor of the fractional Laplacian. We obtain existence and convergence results in certain parameter regimes and limits. We study the long time behaviour
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Singularities and unsteady separation for the inviscid two-dimensional Prandtl system Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-03-23 Charles Collot, Tej-Eddine Ghoul, Nader Masmoudi
We consider the inviscid unsteady Prandtl system in two dimensions, motivated by the fact that it should model to leading order separation and singularity formation for the original viscous system. We give a sharp expression for the maximal time of existence of regular solutions, showing that singularities only happen at the boundary or on the set of zero vorticity, and that they correspond to boundary
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Uniqueness of Plane Stationary Navier–Stokes Flow Past an Obstacle Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-03-23 Mikhail Korobkov, Xiao Ren
We study the exterior problem for stationary Navier–Stokes equations in two dimensions describing a viscous incompressible fluid flowing past an obstacle. It is shown that, at small Reynolds numbers, the classical solutions constructed by Finn and Smith are unique in the class of D-solutions (that is, solutions with finite Dirichlet integral). No additional symmetry or decay assumptions are required
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Publisher Correction to: On the Boundary Layer Equations with Phase Transition in the Kinetic Theory of Gases Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-03-22 Niclas Bernhoff, François Golse
A Correction to this paper has been published: https://doi.org/10.1007/s00205-021-01608-9
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On the Serrin-Type Condition on One Velocity Component for the Navier–Stokes Equations Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-03-18 D. Chae, J. Wolf
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
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The Lieb–Thirring Inequality for Interacting Systems in Strong-Coupling Limit Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-03-16 Kevin Kögler, Phan Thành Nam
We consider an analogue of the Lieb–Thirring inequality for quantum systems with homogeneous repulsive interaction potentials, but without the antisymmetry assumption on the wave functions. We show that in the strong-coupling limit, the Lieb–Thirring constant converges to the optimal constant of the one-body Gagliardo–Nirenberg interpolation inequality without interaction.
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Nonlocal-Interaction Equation on Graphs: Gradient Flow Structure and Continuum Limit Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-03-15 Antonio Esposito, Francesco S. Patacchini, André Schlichting, Dejan Slepčev
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
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Particle Approximation of the BGK Equation Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-03-13 Paolo Buttà, Maxime Hauray, Mario Pulvirenti
In this paper we prove the convergence of a suitable particle system towards the BGK model. More precisely, we consider an interacting stochastic particle system in which each particle can instantaneously thermalize locally. We show that, under a suitable scaling limit, propagation of chaos does hold and the one-particle distribution function converges to the solution of the BGK equation.
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Emergence of Rigid Polycrystals from Atomistic Systems with Heitmann–Radin Sticky Disk Energy Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-03-12 Manuel Friedrich, Leonard Kreutz, Bernd Schmidt
We investigate the emergence of rigid polycrystalline structures from atomistic particle systems. The atomic interaction is governed by a suitably normalized pair interaction energy, where the ‘sticky disk’ interaction potential models the atoms as hard spheres that interact when they are tangential. The discrete energy is frame invariant and no underlying reference lattice on the atomistic configurations
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Crystallization to the Square Lattice for a Two-Body Potential Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-03-11 Laurent Bétermin, Lucia De Luca, Mircea Petrache
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.
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Limit Shapes of Local Minimizers for the Alt–Caffarelli Energy Functional in Inhomogeneous Media Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-03-11 William M. Feldman
This paper considers the Alt–Caffarelli free boundary problem in a periodic medium. This is a convenient model for several interesting phenomena appearing in the study of contact lines on rough surfaces, pinning, hysteresis and the formation of facets. We show the existence of an interval of effective pinned slopes at each direction \(e \in S^{d-1}\). In \(d=2\) we characterize the regularity properties
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Nernst–Planck–Navier–Stokes Systems far from Equilibrium Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-03-09 Peter Constantin, Mihaela Ignatova, Fizay-Noah Lee
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
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On the Uniqueness of Co-circular Four Body Central Configurations Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-03-08 Manuele Santoprete
We study central configurations lying on a common circle in the Newtonian four-body problem. Using a topological argument we prove that there is at most one co-circular central configuration for each cyclic ordering of the masses on the circle.
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Positive Solutions of Transport Equations and Classical Nonuniqueness of Characteristic curves Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-03-08 Elia Brué, Maria Colombo, Camillo De Lellis
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
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On the Free Surface Motion of Highly Subsonic Heat-Conducting Inviscid Flows Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-03-06 Tao Luo, Huihui Zeng
For the free surface problem of highly subsonic heat-conducting inviscid flow in 2-D and 3-D, a priori estimates for geometric quantities of free surfaces, such as the second fundamental form and the injectivity radius of the normal exponential map, and the Sobolev norms of fluid variables, are proved by investigating the coupling of the boundary geometry and the interior solutions. An interesting
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On the Korteweg–de Vries Limit for the Fermi–Pasta–Ulam System Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-03-05 Younghun Hong, Chulkwang Kwak, Changhun Yang
In this paper, we develop dispersive PDE techniques for the Fermi–Pasta–Ulam (FPU) system with infinitely many oscillators, and we show that general solutions to the infinite FPU system can be approximated by counter-propagating waves governed by the Korteweg–de Vries (KdV) equation as the lattice spacing approaches zero. Our result not only simplifies the hypotheses but also reduces the regularity
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On the Boundary Layer Equations with Phase Transition in the Kinetic Theory of Gases Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-03-04 Niclas Bernhoff, François Golse
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
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Stationary Non-equilibrium Solutions for Coagulation Systems Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-03-02 Marina A. Ferreira, Jani Lukkarinen, Alessia Nota, Juan J. L. Velázquez
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
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Strong Convergence of the Vorticity for the 2D Euler Equations in the Inviscid Limit Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-03-01 Gennaro Ciampa, Gianluca Crippa, Stefano Spirito
In this paper we prove the uniform-in-time \(L^p\) convergence in the inviscid limit of a family \(\omega ^\nu \) 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 \(\omega ^{\nu }\) to \(\omega \) in
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Derivation of the Landau–Pekar Equations in a Many-Body Mean-Field Limit Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-02-26 Nikolai Leopold, David Mitrouskas, Robert Seiringer
We consider the Fröhlich Hamiltonian in a mean-field limit where many bosonic particles weakly couple to the quantized phonon field. For large particle numbers and a suitably small coupling, we show that the dynamics of the system is approximately described by the Landau–Pekar equations. These describe a Bose–Einstein condensate interacting with a classical polarization field, whose dynamics is effected
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Traveling Quasi-periodic Water Waves with Constant Vorticity Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-02-25 M. Berti, L. Franzoi, A. Maspero
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
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Correction to: The Landau Equation with the Specular Reflection Boundary Condition Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-02-23 Yan Guo, Hyung Ju Hwang, Jin Woo Jang, Zhimeng Ouyang
In the paper [5, Section 6], we quoted a
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3-D Axisymmetric Transonic Shock Solutions of the Full Euler System in Divergent Nozzles Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-02-21 Yong Park
We establish the stability of 3-D axisymmetric transonic shock solutions of the steady full Euler system in divergent nozzles under small perturbations of an incoming radial supersonic flow and a constant pressure at the exit of the nozzles. To study 3-D axisymmetric transonic shock solutions of the full Euler system, we use a stream function formulation of the full Euler system for a 3-D axisymmetric
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Regularity for $$C^{1,\alpha }$$ C 1 , α Interface Transmission Problems Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-02-19 Luis A. Caffarelli, María Soria-Carro, Pablo Raúl Stinga
We study the existence, uniqueness, and optimal regularity of solutions to transmission problems for harmonic functions with \(C^{1,\alpha }\) interfaces. For this, we develop a novel geometric stability argument based on the mean value property.
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An Index Theory for Collision, Parabolic and Hyperbolic Solutions of the Newtonian n -body Problem Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-02-18 Xijun Hu, Yuwei Ou, Guowei Yu
In the Newtonian n-body problem for solutions with arbitrary energy, which start and end either at a total collision or a parabolic/hyperbolic infinity, we prove some basic results about their Morse and Maslov indices. Moreover for homothetic solutions with arbitrary energy, we give a simple and precise formula that relates the Morse indices of these homothetic solutions to the spectra of the normalized
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Hard Spheres Dynamics: Weak Vs Strong Collisions Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-02-16 Denis Serre
We consider the motion of a finite though large number N of hard spheres in the whole space \({\mathbb R}^n\). Particles move freely until they experience elastic collisions. We use our recent theory of Compensated Integrability in order to estimate how much the particles are deviated by collisions. Our result, which is expressed in terms of hodographs, tells us that only \(O(N^2)\) collisions are
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Small Data Global Well-Posedness for a Boltzmann Equation via Bilinear Spacetime Estimates Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-02-16 Thomas Chen, Ryan Denlinger, Nataša Pavlović
We provide a new analysis of the Boltzmann equation with a constant collision kernel in two space dimensions. The scaling-critical Lebesgue space is \(L^2_{x,v}\); we prove the global well-posedness and a version of scattering, assuming that the data \(f_0\) is sufficiently smooth and localized, and the \(L^2_{x,v}\) norm of \(f_0\) is sufficiently small. The proof relies upon a new scaling-critical
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Hyperbolic Solutions to Bernoulli’s Free Boundary Problem Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-02-12 Antoine Henrot, Michiaki Onodera
Bernoulli’s free boundary problem is an overdetermined problem in which one seeks an annular domain such that the capacitary potential satisfies an extra boundary condition. There exist two different types of solutions called elliptic and hyperbolic solutions. Elliptic solutions are “stable” solutions and tractable by the super and subsolution method, variational methods and the implicit function theorem
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Anticipation Breeds Alignment Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-02-12 Ruiwen Shu, Eitan Tadmor
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
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Free Boundary Regularity for Almost Every Solution to the Signorini Problem Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-02-11 Xavier Fernández-Real, Xavier Ros-Oton
We investigate the regularity of the free boundary for the Signorini problem in \({\mathbb {R}}^{n+1}\). It is known that regular points are \((n-1)\)-dimensional and \(C^\infty \). However, even for \(C^\infty \) obstacles \(\varphi \), the set of non-regular (or degenerate) points could be very large—e.g. with infinite \({\mathcal {H}}^{n-1}\) measure. The only two assumptions under which a nice
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On $$\varGamma $$ Γ -Convergence of a Variational Model for Lithium-Ion Batteries Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-02-05 Kerrek Stinson
A singularly perturbed phase field model used to model lithium-ion batteries including chemical and elastic effects is considered. The underlying energy is given by $$\begin{aligned} I_\varepsilon [u,c ] := \int _{\varOmega }\left( \frac{1}{\varepsilon }f(c)+\varepsilon \Vert \nabla c\Vert ^2+\frac{1}{\varepsilon }{\mathbb {C}}(e(u)-ce_0):(e(u)-ce_0)\right) \, \mathrm{d}x, \end{aligned}$$ where f is
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$$\Gamma $$ Γ -Limit for Two-Dimensional Charged Magnetic Zigzag Domain Walls Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-02-04 Hans Knüpfer, Wenhui Shi
Charged domain walls are a type of domain wall in thin ferromagnetic films which appear due to global topological constraints. The non-dimensionalized micromagnetic energy for a uniaxial thin ferromagnetic film with in-plane magnetization \(m \in {\mathbb {S}}^1\) is given by $$\begin{aligned} E_\varepsilon [m] \ = \ \varepsilon \Vert \nabla m\Vert _{L^2}^2 + \frac{1}{\varepsilon } \Vert m \cdot e_2\Vert
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Symmetrization for Fractional Elliptic Problems: A Direct Approach Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-01-28 Vincenzo Ferone, Bruno Volzone
We provide new direct methods to establish symmetrization results in the form of a mass concentration (that is, integral) comparison for fractional elliptic equations of the type \((-\Delta )^{s}u=f \, (0
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Curvature-Driven Wrinkling of Thin Elastic Shells Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-01-18 Ian Tobasco
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
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Partially Regular Weak Solutions of the Navier–Stokes Equations in $$\mathbb {R}^4 \times [0,\infty [$$ R 4 × [ 0 , ∞ [ Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-01-12 Bian Wu
We show that for any given solenoidal initial data in \(L^2\) and any solenoidal external force in \(L_{\text {loc}}^q \bigcap L^{3/2}\) with \(q>3\), there exist partially regular weak solutions of the Navier–Stokes equations in \(\mathbb {R}^4 \times [0,\infty [\) which satisfy certain local energy inequalities and whose singular sets have a locally finite 2-dimensional parabolic Hausdorff measure
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Two-Speed Solutions to Non-convex Rate-Independent Systems Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-01-08 Filip Rindler, Sebastian Schwarzacher, Juan J. L. Velázquez
We consider evolutionary PDE inclusions of the form $$\begin{aligned} -\lambda {\dot{u}}_\lambda + \Delta u - \mathrm {D}W_0(u) + f \ni \partial \mathscr {R}_1({\dot{u}}) \quad \text {in}\,\,{ (0,T) \times \Omega ,} \end{aligned}$$ where \(\mathscr {R}_1\) is a positively 1-homogeneous rate-independent dissipation potential and \(W_0\) is a (generally) non-convex energy density. This work constructs
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Long-time Asymptotics of the One-dimensional Damped Nonlinear Klein–Gordon Equation Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-01-08 Raphaël Côte, Yvan Martel, Xu Yuan
For the one-dimensional nonlinear damped Klein–Gordon equation $$\begin{aligned} \partial _{t}^{2}u+2\alpha \partial _{t}u-\partial _{x}^{2}u+u-|u|^{p-1}u=0 \quad \text{ on } \mathbb {R}\times \mathbb {R}, \end{aligned}$$ with \(\alpha >0\) and \(p>2\), we prove that any global finite energy solution either converges to 0 or behaves asymptotically as \(t\rightarrow \infty \) as the sum of \(K\ge 1\)
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Global Well-Posedness and Exponential Stability of 3D Navier–Stokes Equations with Density-Dependent Viscosity and Vacuum in Unbounded Domains Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-01-05 Cheng He, Jing Li, Boqiang Lü
We consider the global existence and large-time asymptotic behavior of strong solutions to the Cauchy problem of the three-dimensional (3D) nonhomogeneous incompressible Navier–Stokes equations with density-dependent viscosity and vacuum. After establishing some key a priori exponential decay-in-time rates of the strong solutions, we obtain both the global existence and exponential stability of strong
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Renormalized Energy Between Vortices in Some Ginzburg–Landau Models on 2-Dimensional Riemannian Manifolds Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-01-04 R. Ignat, R. L. Jerrard
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
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Large KAM Tori for Quasi-linear Perturbations of KdV Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2021-01-04 Massimiliano Berti, Thomas Kappeler, Riccardo Montalto
In this paper we prove the persistence of space periodic multi-solitons of arbitrary size under any quasi-linear Hamiltonian perturbation, which is smooth and sufficiently small. This answers positively a longstanding question of whether KAM techniques can be further developed to prove the existence of quasi-periodic solutions of arbitrary size of strongly nonlinear perturbations of integrable PDEs
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Surface Energy and Boundary Layers for a Chain of Atoms at Low Temperature Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2020-12-21 Sabine Jansen, Wolfgang König, Bernd Schmidt, Florian Theil
We analyze the surface energy and boundary layers for a chain of atoms at low temperature for an interaction potential of Lennard–Jones type. The pressure (stress) is assumed to be small but positive and bounded away from zero, while the temperature \(\beta ^{-1}\) goes to zero. Our main results are: (1) As \(\beta \rightarrow \infty \) at fixed positive pressure \(p>0\), the Gibbs measures \(\mu _\beta
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Darcy’s Law with a Source Term Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2020-11-25 Matt Jacobs, Inwon Kim, Jiajun Tong
We introduce a novel variant of the JKO scheme to approximate Darcy’s law with a pressure dependent source term. By introducing a new variable that implicitly controls the source term, our scheme is still able to use the standard Wasserstein-2-metric even though the total mass changes over time. Leveraging the dual formulation of our scheme, we show that the discrete-in-time approximations satisfy
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Cohesive Fracture in 1D: Quasi-static Evolution and Derivation from Static Phase-Field Models Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2020-11-24 Marco Bonacini, Sergio Conti, Flaviana Iurlano
In this paper we propose a notion of irreversibility for the evolution of cracks in the presence of cohesive forces, which allows for different responses in the loading and unloading processes, motivated by a variational approximation with damage models, and we investigate its applicability to the construction of a quasi-static evolution in a simple one-dimensional model. The cohesive fracture model
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Corrector Equations in Fluid Mechanics: Effective Viscosity of Colloidal Suspensions Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2020-11-22 Mitia Duerinckx, Antoine Gloria
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
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From Steklov to Neumann via homogenisation Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2020-11-20 Alexandre Girouard, Antoine Henrot, Jean Lagacé
We study a new link between the Steklov and Neumann eigenvalues of domains in Euclidean space. This is obtained through an homogenisation limit of the Steklov problem on a periodically perforated domain, converging to a family of eigenvalue problems with dynamical boundary conditions. For this problem, the spectral parameter appears both in the interior of the domain and on its boundary. This intermediary
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Nonlinear Hyperbolic Waves in Relativistic Gases of Massive Particles with Synge Energy Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2020-11-17 Tommaso Ruggeri, Qinghua Xiao, Huijiang Zhao
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
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Well-posedness of Free Boundary Problem in Non-relativistic and Relativistic Ideal Compressible Magnetohydrodynamics Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2020-11-12 Yuri Trakhinin, Tao Wang
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
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Symmetry and Rigidity of Minimal Surfaces with Plateau-like Singularities Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2020-11-12 Jacob Bernstein, Francesco Maggi
By employing the method of moving planes in a novel way we extend some classical symmetry and rigidity results for smooth minimal surfaces to surfaces that have singularities of the sort typically observed in soap films.
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On the Isoperimetric Inequality and Surface Diffusion Flow for Multiply Winding Curves Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2020-11-10 Tatsuya Miura, Shinya Okabe
In this paper we establish a general form of the isoperimetric inequality for immersed closed curves (possibly non-convex) in the plane under rotational symmetry. As an application, we obtain a global existence result for the surface diffusion flow, providing that an initial curve is \(H^2\)-close to a multiply covered circle and is sufficiently rotationally symmetric.
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Piecewise Smooth Stationary Euler Flows with Compact Support Via Overdetermined Boundary Problems Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2020-11-09 Miguel Domínguez-Vázquez, Alberto Enciso, Daniel Peralta-Salas
We construct new stationary weak solutions of the 3D Euler equation with compact support. The solutions, which are piecewise smooth and discontinuous across a surface, are axisymmetric with swirl. The range of solutions we find is different from, and larger than, the family of smooth stationary solutions recently obtained by Gavrilov and Constantin–La–Vicol; in particular, these solutions are not localizable
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Structural Stability of Supersonic Solutions to the Euler–Poisson System Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2020-11-04 Myoungjean Bae, Ben Duan, Jingjing Xiao, Chunjing Xie
The structural stability for supersonic solutions of the Euler–Poisson system for hydrodynamical model in semiconductor devices and plasmas in two dimensional domain is established, under the perturbation of the flow velocity and the strength of electric field in the horizontal direction at the entrance of a channel. First, the Euler–Poisson system in the supersonic region is reformulated into a second
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Bifurcation of Symmetric Domain Walls for the Bénard–Rayleigh Convection Problem Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2020-11-03 Mariana Haragus, Gérard Iooss
We prove the existence of domain walls for the Bénard–Rayleigh convection problem. Our approach relies upon a spatial dynamics formulation of the hydrodynamic problem, a center manifold reduction, and a normal forms analysis of an eight-dimensional reduced system. Domain walls are constructed as heteroclinic solutions connecting suitably chosen periodic solutions of this reduced system.
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Torus-like Solutions for the Landau-de Gennes Model. Part I: The Lyuksyutov Regime Arch. Rational Mech. Anal. (IF 2.42) Pub Date : 2020-10-23 Federico Dipasquale, Vincent Millot, Adriano Pisante
We study global minimizers of a continuum Landau-de Gennes energy functional for nematic liquid crystals, in three-dimensional domains, under a Dirichlet boundary condition. In a relevant range of parameters (which we call the Lyuksyutov regime), the main result establishes the nontrivial topology of the biaxiality sets of minimizers for a large class of boundary conditions including the homeotropic