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

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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

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http://dx.doi.org/10.1007/s00205-020-01519-1DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7217891PMC

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

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http://dx.doi.org/10.1007/s00205-020-01509-3DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7176608PMC

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

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http://dx.doi.org/10.1007/s00205-019-01439-9DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7021190PMC

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

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http://dx.doi.org/10.1007/s00205-019-01425-1DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7010697PMC

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

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http://dx.doi.org/10.1007/s00205-013-0665-1DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3979576PMC
January 2014
35 Reads

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

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http://dx.doi.org/10.1007/s00205-012-0550-3DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3679949PMC
December 2012
3 Reads

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

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http://dx.doi.org/10.1007/s00205-012-0551-2DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3611664PMC
January 2013
20 Reads
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