19 results match your criteria Archive For Rational Mechanics And Analysis[Journal]

  • Page 1 of 1

Characterization of Minimizers of Aviles-Giga Functionals in Special Domains.

Elio Marconi

Arch Ration Mech Anal 2021 23;242(2):1289-1316. Epub 2021 Aug 23.

EPFL B, Station 8, 1015 Lausanne, Switzerland.

We consider the singularly perturbed problem on bounded domains . Under appropriate boundary conditions, we prove that if is an ellipse, then the minimizers of converge to the viscosity solution of the eikonal equation as . Read More

View Article and Full-Text PDF

Energy Minimisers with Prescribed Jacobian.

Arch Ration Mech Anal 2021 14;242(2):1059-1090. Epub 2021 Oct 14.

Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, 00014 Helsingin Yliopisto, Finland.

We consider the class of planar maps with Jacobian prescribed to be a fixed radially symmetric function and which, moreover, fixes the boundary of a ball; we then study maps which minimise the 2-Dirichlet energy in this class. We find a quantity which controls the symmetry, uniqueness and regularity of minimisers: if then minimisers are symmetric and unique; if is large but finite then there may be uncountably many minimisers, none of which is symmetric, although all of them have optimal regularity; if is infinite then generically minimisers have lower regularity. In particular, this result gives a negative answer to a question of Hélein (Ann. Read More

View Article and Full-Text PDF
October 2021

Echo Chains as a Linear Mechanism: Norm Inflation, Modified Exponents and Asymptotics.

Arch Ration Mech Anal 2021 30;242(1):643-700. Epub 2021 Jul 30.

BCAM - Basque Center for Applied Mathematics, Mazarredo 14, E48009 Bilbao, Basque Country Spain.

In this article we show that the Euler equations, when linearized around a low frequency perturbation to Couette flow, exhibit norm inflation in Gevrey-type spaces as time tends to infinity. Thus, echo chains are shown to be a (secondary) linear instability mechanism. Furthermore, we develop a more precise analysis of cancellations in the resonance mechanism, which yields a modified exponent in the high frequency regime. Read More

View Article and Full-Text PDF

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

Arch Ration Mech Anal 2021 22;242(1):1-147. Epub 2021 Jul 22.

Department for Mathematical Sciences, Chalmers University of Technology, Gothenburg, Sweden.

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 is optimal. Read More

View Article and Full-Text PDF

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

Arch Ration Mech Anal 2021 1;241(3):1529-1573. Epub 2021 Jun 1.

Department of Mathematics, Yonsei University, 50 Yonsei-Ro, Seodaemun-Gu, Seoul, 03722 Republic of Korea.

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. Read More

View Article and Full-Text PDF

A New Approach to the Rayleigh-Taylor Instability.

Arch Ration Mech Anal 2021 12;241(3):1243-1280. Epub 2021 Jun 12.

Mathematisches Institut, Universität Leipzig, Augustusplatz 10, 04109 Leipzig, Germany.

In this article we consider the inhomogeneous incompressible Euler equations describing two fluids with different constant densities under the influence of gravity as a differential inclusion. By considering the relaxation of the constitutive laws we formulate a general criterion for the existence of infinitely many weak solutions which reflect the turbulent mixing of the two fluids. Our criterion can be verified in the case that initially the fluids are at rest and separated by a flat interface with the heavier one being above the lighter one-the classical configuration giving rise to the Rayleigh-Taylor instability. Read More

View Article and Full-Text PDF

Free Boundary Regularity for Almost Every Solution to the Signorini Problem.

Arch Ration Mech Anal 2021 11;240(1):419-466. Epub 2021 Feb 11.

ICREA, Pg. Lluís Companys 23, 08010 Barcelona, Spain.

We investigate the regularity of the free boundary for the Signorini problem in . It is known that regular points are -dimensional and . However, even for obstacles , the set of non-regular (or degenerate) points could be very large-e. Read More

View Article and Full-Text PDF
February 2021

Derivation of the Landau-Pekar Equations in a Many-Body Mean-Field Limit.

Arch Ration Mech Anal 2021 26;240(1):383-417. Epub 2021 Feb 26.

Present Address: Institute of Science and Technology Austria (IST Austria), Am Campus 1, 3400 Klosterneuburg, Austria.

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 by the condensate, i. Read More

View Article and Full-Text PDF
February 2021

From Steklov to Neumann via homogenisation.

Arch Ration Mech Anal 2021 20;239(2):981-1023. Epub 2020 Nov 20.

Department of Mathematics, University College London, Gower Street, London, WC1E 6BT UK.

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. Read More

View Article and Full-Text PDF
November 2020

Fluctuations Around a Homogenised Semilinear Random PDE.

Arch Ration Mech Anal 2021 6;239(1):151-217. Epub 2020 Oct 6.

Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France.

We consider a semilinear parabolic partial differential equation in , where 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. Read More

View Article and Full-Text PDF
October 2020

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

Arch Ration Mech Anal 2021 4;239(1):117-149. Epub 2020 Oct 4.

Institut für Mathematik, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.

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. Read More

View Article and Full-Text PDF
October 2020

A Unified Model for Stress-Driven Rearrangement Instabilities.

Arch Ration Mech Anal 2020 20;238(1):415-488. Epub 2020 Jun 20.

Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern Platz 1, 1090 Wien, Austria.

A variational model to simultaneously treat Stress-Driven Rearrangement Instabilities, such as boundary discontinuities, internal cracks, external filaments, edge delamination, wetting, and brittle fractures, is introduced. The model is characterized by an energy displaying both elastic and surface terms, and allows for a unified treatment of a wide range of settings, from epitaxially-strained thin films to crystalline cavities, and from capillarity problems to fracture models. The existence of minimizing configurations is established by adopting the direct method of the Calculus of Variations. Read More

View Article and Full-Text PDF

Higher-Order Linearization and Regularity in Nonlinear Homogenization.

Arch Ration Mech Anal 2020 9;237(2):631-741. Epub 2020 Apr 9.

2Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, (Gustaf Hällströmin katu 2), 00014 Helsinki, Finland.

We prove large-scale regularity for solutions of nonlinear elliptic equations with random coefficients, thereby obtaining a version of the statement of Hilbert's 19th problem in the context of homogenization. The analysis proceeds by iteratively improving three statements together: (i) the regularity of the homogenized Lagrangian  , (ii) the commutation of higher-order linearization and homogenization, and (iii) large-scale -type regularity for higher-order linearization errors. We consequently obtain a quantitative estimate on the scaling of linearization errors, a Liouville-type theorem describing the polynomially-growing solutions of the system of higher-order linearized equations, and an explicit (heterogenous analogue of the) Taylor series for an arbitrary solution of the nonlinear equations-with the remainder term optimally controlled. Read More

View Article and Full-Text PDF

Optimal Regularity and Structure of the Free Boundary for Minimizers in Cohesive Zone Models.

Arch Ration Mech Anal 2020 10;237(1):299-345. Epub 2020 Apr 10.

3Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zurich, Switzerland.

We study optimal regularity and free boundary for minimizers of an energy functional arising in cohesive zone models for fracture mechanics. Under smoothness assumptions on the boundary conditions and on the fracture energy density, we show that minimizers are , and that near non-degenerate points the fracture set is , for some . Read More

View Article and Full-Text PDF

Exponential Time Decay of Solutions to Reaction-Cross-Diffusion Systems of Maxwell-Stefan Type.

Arch Ration Mech Anal 2020 1;235(2):1059-1104. Epub 2019 Aug 1.

2Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010 Graz, Austria.

The large-time asymptotics of weak solutions to Maxwell-Stefan diffusion systems for chemically reacting fluids with different molar masses and reversible reactions are investigated. The diffusion matrix of the system is generally neither symmetric nor positive definite, but the equations admit a formal gradient-flow structure which provides entropy (free energy) estimates. The main result is the exponential decay to the unique equilibrium with a rate that is constructive up to a finite-dimensional inequality. Read More

View Article and Full-Text PDF

Global Stability of Minkowski Space for the Einstein-Vlasov System in the Harmonic Gauge.

Arch Ration Mech Anal 2020 24;235(1):517-633. Epub 2019 Jul 24.

2Department of Mathematics, Imperial College London, South Kensington Campus, London, SW7 2AZ UK.

Minkowski space is shown to be globally stable as a solution to the massive Einstein-Vlasov system. The proof is based on a harmonic gauge in which the equations reduce to a system of quasilinear wave equations for the metric, satisfying the weak null condition, coupled to a transport equation for the Vlasov particle distribution function. Central to the proof is a collection of vector fields used to control the particle distribution function, a function of both spacetime and momentum variables. Read More

View Article and Full-Text PDF

On a multiphase multicomponent model of biofilm growth.

Arch Ration Mech Anal 2014 Jan;211(1):257-300

Mathematical Biosciences Institute, and Department of Mathematics, Ohio State University, Columbus, Ohio 43210.

Biofilms are formed when free-floating bacteria attach to a surface and secrete polysaccharide to form an extracellular polymeric matrix (EPS). A general model of biofilm growth needs to include the bacteria, the EPS, and the solvent within the biofilm region (), and the solvent in the surrounding region (). The interface between the two regions, (), is a free boundary. Read More

View Article and Full-Text PDF
January 2014

Statistical foundations of liquid-crystal theory: I. Discrete systems of rod-like molecules.

Arch Ration Mech Anal 2012 Dec;206(3):1039-1072

Department of Mathematics and Statistics, 805 Sherbrooke Street West, Montreal, QC H3A 2K6, Tel.: 514-398-2998, ,

We develop a mechanical theory for systems of rod-like particles. Central to our approach is the assumption that the external power expenditure for any subsystem of rods is independent of the underlying frame of reference. This assumption is used to derive the basic balance laws for forces and torques. Read More

View Article and Full-Text PDF
December 2012

Statistical foundations of liquid-crystal theory: II: Macroscopic balance laws.

Arch Ration Mech Anal 2013 Jan;207(1):1-37

Department of Mathematics and Statistics, 805 Sherbrooke Street West, Montreal, QC H3A 2K6, Tel.: 514-398-2998, ,

Working on a state space determined by considering a discrete system of rigid rods, we use nonequilibrium statistical mechanics to derive macroscopic balance laws for liquid crystals. A probability function that satisfies the Liouville equation serves as the starting point for deriving each macroscopic balance. The terms appearing in the derived balances are interpreted as expected values and explicit formulas for these terms are obtained. Read More

View Article and Full-Text PDF
January 2013
  • Page 1 of 1