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  Subjects -> PHYSICS (Total: 738 journals)
    - ELECTRICITY AND MAGNETISM (7 journals)
    - MECHANICS (19 journals)
    - NUCLEAR PHYSICS (44 journals)
    - OPTICS (84 journals)
    - PHYSICS (538 journals)
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    - THERMODYNAMICS (29 journals)

PHYSICS (538 journals)                  1 2 3 4 5 6 | Last

Acta Acustica united with Acustica     Full-text available via subscription   (Followers: 7)
Acta Mechanica     Hybrid Journal   (Followers: 15)
Acta Physica Slovaca     Open Access   (Followers: 3)
Advanced Composite Materials     Hybrid Journal   (Followers: 11)
Advanced Functional Materials     Hybrid Journal   (Followers: 34)
Advanced Materials     Hybrid Journal   (Followers: 317)
Advances in Condensed Matter Physics     Open Access   (Followers: 6)
Advances in Exploration Geophysics     Full-text available via subscription   (Followers: 3)
Advances in Geophysics     Full-text available via subscription   (Followers: 4)
Advances in High Energy Physics     Open Access   (Followers: 12)
Advances in Imaging and Electron Physics     Full-text available via subscription   (Followers: 1)
Advances in Materials Physics and Chemistry     Open Access   (Followers: 13)
Advances in Natural Sciences: Nanoscience and Nanotechnology     Open Access   (Followers: 14)
Advances in OptoElectronics     Open Access   (Followers: 3)
Advances In Physics     Hybrid Journal   (Followers: 7)
Advances in Physics Theories and Applications     Open Access   (Followers: 4)
Advances in Remote Sensing     Open Access   (Followers: 8)
Advances in Synchrotron Radiation     Hybrid Journal   (Followers: 1)
AIP Advances     Open Access   (Followers: 4)
AIP Conference Proceedings     Full-text available via subscription   (Followers: 1)
American Journal of Applied Sciences     Open Access   (Followers: 28)
American Journal of Condensed Matter Physics     Open Access   (Followers: 2)
Analysis and Mathematical Physics     Hybrid Journal   (Followers: 1)
Annalen der Physik     Hybrid Journal   (Followers: 2)
Annales Geophysicae (ANGEO)     Open Access   (Followers: 3)
Annales Henri PoincarĂ©     Hybrid Journal   (Followers: 2)
Annales UMCS, Physica     Open Access  
Annals of Nuclear Medicine     Hybrid Journal   (Followers: 3)
Annals of Physics     Hybrid Journal   (Followers: 2)
Annual Reports on NMR Spectroscopy     Full-text available via subscription   (Followers: 1)
Annual Review of Analytical Chemistry     Full-text available via subscription   (Followers: 9)
Annual Review of Condensed Matter Physics     Full-text available via subscription   (Followers: 1)
Annual Review of Materials Research     Full-text available via subscription   (Followers: 4)
APL Materials     Open Access   (Followers: 4)
Applied Composite Materials     Hybrid Journal   (Followers: 8)
Applied Physics A     Hybrid Journal   (Followers: 9)
Applied Physics Frontier     Open Access   (Followers: 1)
Applied Physics Letters     Hybrid Journal   (Followers: 25)
Applied Physics Research     Open Access   (Followers: 6)
Applied Physics Reviews     Hybrid Journal   (Followers: 7)
Applied Radiation and Isotopes     Hybrid Journal   (Followers: 5)
Applied Remote Sensing Journal     Open Access   (Followers: 9)
Applied Spectroscopy     Full-text available via subscription   (Followers: 12)
Applied Spectroscopy Reviews     Hybrid Journal   (Followers: 2)
Archive for Rational Mechanics and Analysis     Hybrid Journal   (Followers: 3)
Astronomy & Geophysics     Hybrid Journal   (Followers: 1)
Astrophysical Journal Letters     Full-text available via subscription   (Followers: 3)
Astrophysical Journal Supplement Series     Full-text available via subscription   (Followers: 3)
Atoms     Open Access  
Attention, Perception & Psychophysics     Full-text available via subscription   (Followers: 5)
Autonomous Mental Development, IEEE Transactions on     Hybrid Journal   (Followers: 5)
Axioms     Open Access  
Bangladesh Journal of Medical Physics     Open Access  
Bauphysik     Hybrid Journal   (Followers: 1)
Biomaterials     Hybrid Journal   (Followers: 27)
Biomedical Engineering, IEEE Reviews in     Full-text available via subscription   (Followers: 14)
Biomedical Engineering, IEEE Transactions on     Hybrid Journal   (Followers: 12)
Biomedical Imaging and Intervention Journal     Open Access   (Followers: 5)
Biophysical Reviews     Hybrid Journal  
Biophysical Reviews and Letters     Hybrid Journal   (Followers: 3)
BMC Biophysics     Open Access   (Followers: 7)
BMC Nuclear Medicine     Open Access   (Followers: 5)
Brazilian Journal of Physics     Hybrid Journal  
Broadcasting, IEEE Transactions on     Hybrid Journal   (Followers: 5)
Bulletin of Materials Science     Open Access   (Followers: 38)
Bulletin of the Atomic Scientists     Full-text available via subscription   (Followers: 4)
Bulletin of the Lebedev Physics Institute     Hybrid Journal   (Followers: 1)
Bulletin of the Russian Academy of Sciences: Physics     Hybrid Journal  
Caderno Brasileiro de Ensino de FĂ­sica     Open Access  
Canadian Journal of Physics     Full-text available via subscription   (Followers: 1)
Cells     Open Access   (Followers: 1)
Central European Journal of Physics     Hybrid Journal   (Followers: 1)
Chinese Journal of Astronomy and Astrophysics     Full-text available via subscription   (Followers: 1)
Chinese Journal of Chemical Physics     Hybrid Journal   (Followers: 1)
Chinese Physics B     Full-text available via subscription  
Chinese Physics C     Full-text available via subscription  
Chinese Physics Letters     Full-text available via subscription  
Cohesion and Structure     Full-text available via subscription   (Followers: 2)
Colloid Journal     Hybrid Journal   (Followers: 2)
Communications in Mathematical Physics     Hybrid Journal   (Followers: 2)
Communications in Numerical Methods in Engineering     Hybrid Journal   (Followers: 3)
Communications in Theoretical Physics     Full-text available via subscription   (Followers: 1)
Composites Part A : Applied Science and Manufacturing     Hybrid Journal   (Followers: 29)
Composites Part B : Engineering     Hybrid Journal   (Followers: 26)
Computational Materials Science     Hybrid Journal   (Followers: 18)
Computational Mathematics and Mathematical Physics     Hybrid Journal   (Followers: 1)
Computational Particle Mechanics     Hybrid Journal  
Computational Science and Discovery     Full-text available via subscription  
Computer Physics Communications     Hybrid Journal  
Contemporary Concepts of Condensed Matter Science     Full-text available via subscription  
Contemporary Physics     Hybrid Journal   (Followers: 10)
Continuum Mechanics and Thermodynamics     Hybrid Journal   (Followers: 3)
Contributions to Plasma Physics     Hybrid Journal   (Followers: 2)
COSPAR Colloquia Series     Full-text available via subscription   (Followers: 1)
Cryogenics     Hybrid Journal   (Followers: 13)
Current Applied Physics     Full-text available via subscription   (Followers: 4)
Diamond and Related Materials     Hybrid Journal   (Followers: 12)
Differential Equations and Nonlinear Mechanics     Open Access   (Followers: 4)
Doklady Physics     Hybrid Journal   (Followers: 1)
Dynamical Properties of Solids     Full-text available via subscription  

        1 2 3 4 5 6 | Last

Journal Cover Archive for Rational Mechanics and Analysis
   Journal TOC RSS feeds Export to Zotero [5 followers]  Follow    
   Hybrid Journal Hybrid journal (It can contain Open Access articles)
     ISSN (Print) 1432-0673 - ISSN (Online) 0003-9527
     Published by Springer-Verlag Homepage  [2209 journals]   [SJR: 3.43]   [H-I: 51]
  • Existence and Uniqueness for a Coupled Parabolic-Elliptic Model with
           Applications to Magnetic Relaxation
    • Abstract: Abstract We prove the existence, uniqueness and regularity of weak solutions of a coupled parabolic-elliptic model in 2D, and the existence of weak solutions in 3D; we consider the standard equations of magnetohydrodynamics with the advective terms removed from the velocity equation. Despite the apparent simplicity of the model, the proof in 2D requires results that are at the limit of what is available, including elliptic regularity in L 1 and a strengthened form of the Ladyzhenskaya inequality $$\ f \ _{L^{4}} \leqq c \ f \ _{L^{2,\infty}}^{1/2} \ \nabla f\ _{L^{2}}^{1/2},$$ which we derive using the theory of interpolation. The model potentially has applications to the method of magnetic relaxation introduced by Moffatt (J Fluid Mech 159:359–378, 1985) to construct stationary Euler flows with non-trivial topology.
      PubDate: 2014-11-01
       
  • Bilinear Dispersive Estimates Via Space Time Resonances, Dimensions Two
           and Three
    • Abstract: Abstract Consider a bilinear interaction between two linear dispersive waves with a generic resonant structure (roughly speaking, space and time resonant sets intersect transversally). We derive an asymptotic equivalent of the solution for data in the Schwartz class, and bilinear dispersive estimates for data in weighted Lebesgue spaces. An application to water waves with infinite depth, gravity and surface tension is also presented.
      PubDate: 2014-11-01
       
  • The Jacobiator of Nonholonomic Systems and the Geometry of Reduced
           Nonholonomic Brackets
    • Abstract: Abstract In this paper, we consider the Hamiltonian formulation of nonholonomic systems with symmetries and study several aspects of the geometry of their reduced almost Poisson brackets, including the integrability of their characteristic distributions. Our starting point is establishing global formulas for the nonholonomic Jacobiators, before and after reduction, which are used to clarify the relationship between reduced nonholonomic brackets and twisted Poisson structures. For certain types of symmetries (generalizing the Chaplygin case), we obtain genuine Poisson structures on the reduced spaces and analyze situations in which the reduced nonholonomic brackets arise by applying a gauge transformation to these Poisson structures. We illustrate our results with mechanical examples, and in particular show how to recover several well-known facts in the special case of Chaplygin symmetries.
      PubDate: 2014-11-01
       
  • Global Weak Solutions to the Equations of Compressible Flow of Nematic
           Liquid Crystals in Two Dimensions
    • Abstract: Abstract We consider weak solutions to a simplified Ericksen–Leslie system of two-dimensional compressible flow of nematic liquid crystals. An initial-boundary value problem is first studied in a bounded domain. By developing new techniques and estimates to overcome the difficulties induced by the supercritical nonlinearity \({ \nabla\mathbf{d} ^2\mathbf{d}}\) in the equations of angular momentum on the direction field, and adapting the standard three-level approximation scheme and the weak convergence arguments for the compressible Navier–Stokes equations, we establish the global existence of weak solutions under a restriction imposed on the initial energy including the case of small initial energy. Then the Cauchy problem with large initial data is investigated, and we prove the global existence of large weak solutions by using the domain expansion technique and the rigidity theorem, provided that the second component of initial data of the direction field satisfies some geometric angle condition.
      PubDate: 2014-11-01
       
  • Gradient Damage Models Coupled with Plasticity and Nucleation of Cohesive
           Cracks
    • Abstract: Abstract In the framework of rate-independent systems, a family of elastic-plastic-damage models is proposed through a variational formulation. Since the goal is to account for softening behaviors until the total failure, the dissipated energy contains a gradient damage term in order to limit localization effects. The resulting model owns a great flexibility in the possible coupled responses, depending on the constitutive parameters. Moreover, considering the one-dimensional quasi-static problem of a bar under simple traction and constructing solutions with localization of damage, it turns out that in general a cohesive crack appears at the center of the damage zone before the rupture. The associated cohesive law is obtained in a closed form in terms of the parameters of the model.
      PubDate: 2014-11-01
       
  • A Quantitative Modulus of Continuity for the Two-Phase Stefan Problem
    • Abstract: Abstract We derive the quantitative modulus of continuity $$\omega(r)=\left[ p+\ln \left( \frac{r_0}{r}\right)\right]^{-\alpha (n, p)},$$ which we conjecture to be optimal for solutions of the p-degenerate two-phase Stefan problem. Even in the classical case p = 2, this represents a twofold improvement with respect to the early 1980’s state-of-the-art results by Caffarelli– Evans (Arch Rational Mech Anal 81(3):199–220, 1983) and DiBenedetto (Ann Mat Pura Appl 103(4):131–176, 1982), in the sense that we discard one logarithm iteration and obtain an explicit value for the exponent α(n, p).
      PubDate: 2014-11-01
       
  • Degenerate Hyperbolic Conservation Laws with Dissipation: Reduction to and
           Validity of a Class of Burgers-Type Equations
    • Abstract: Abstract A conservation law is said to be degenerate or critical if the Jacobian of the flux vector evaluated on a constant state has a zero eigenvalue. In this paper, it is proved that a degenerate conservation law with dissipation will generate dynamics on a long time scale that resembles Burger’s dynamics. The case of k-fold degeneracy is also treated, and it is shown that it leads to a reduction to a quadratically coupled k-fold system of Burgers-type equations. Validity of the reduction and existence for the reduced system is proved in the class of uniformly local spaces, thereby capturing both finite and infinite energy solutions. The theory is applied to some examples, from stratified shallow-water hydrodynamics, that model the birth of hydraulic jumps.
      PubDate: 2014-11-01
       
  • A Refinement of the Local Serrin-Type Regularity Criterion for a Suitable
           Weak Solution to the Navier–Stokes Equations
    • Abstract: Abstract We formulate a new criterion for regularity of a suitable weak solution v to the Navier–Stokes equations at the space-time point (x 0, t 0). The criterion imposes a Serrin-type integrability condition on v only in a backward neighbourhood of (x 0, t 0), intersected with the exterior of a certain space-time paraboloid with vertex at point (x 0, t 0). We make no special assumptions on the solution in the interior of the paraboloid.
      PubDate: 2014-11-01
       
  • Shapes of Epitaxially Grown Quantum Dots
    • Abstract: Abstract A variational model introduced by Spencer and Tersoff (Appl. Phys. Lett. 96:073114, 2010) to describe optimal faceted shapes of epitaxially deposited films is studied analytically in the case in which there are a non-vanishing crystallographic miscut and a lattice incompatibility between the film and the substrate. The existence of faceted minimizers for every volume of the deposited film is established. In particular, it is shown that there is no wetting effect for small volumes. Geometric properties including a faceted version of the zero contact angle are derived, and the explicit shapes of minimizers for small volumes are identified.
      PubDate: 2014-11-01
       
  • Front Quenching in the G-equation Model Induced by Straining of Cellular
           Flow
    • Abstract: Abstract We study homogenization of the G-equation with a flow straining term (or the strain G-equation) in two dimensional periodic cellular flow. The strain G-equation is a highly non-coercive and non-convex level set Hamilton–Jacobi equation. The main objective is to investigate how the flow induced straining (the nonconvex term) influences front propagation as the flow intensity A increases. Three distinct regimes are identified. When A is below the critical level, homogenization holds and the turbulent flame speed s T (effective Hamiltonian) is well-defined for any periodic flow with small divergence and is enhanced by the cellular flow as s T ≧ O(A/log A). In the second regime where A is slightly above the critical value, homogenization breaks down, and s T is not well-defined along any direction. Solutions become a mixture of a fast moving part and a stagnant part. When A is sufficiently large, the whole flame front ceases to propagate forward due to the flow induced straining. In particular, along directions p = (±1, 0) and (0, ±1), s T is well-defined again with a value of zero (trapping). A partial homogenization result is also proved. If we consider a similar but relatively simpler Hamiltonian, the trapping occurs along all directions. The analysis is based on the two-player differential game representation of solutions, selection of game strategies and trapping regions, and construction of connecting trajectories.
      PubDate: 2014-10-01
       
  • Metastability and Dynamics of Discrete Topological Singularities in Two
           Dimensions: A Γ-Convergence Approach
    • Abstract: Abstract This paper aims at building a variational approach to the dynamics of discrete topological singularities in two dimensions, based on Γ-convergence. We consider discrete systems, described by scalar functions defined on a square lattice and governed by periodic interaction potentials. Our main motivation comes from XY spin systems, described by the phase parameter, and screw dislocations, described by the displacement function. For these systems, we introduce a discrete notion of vorticity. As the lattice spacing tends to zero we derive the first order Γ-limit of the free energy which is referred to as renormalized energy and describes the interaction of vortices. As a byproduct of this analysis, we show that such systems exhibit increasingly many metastable configurations of singularities. Therefore, we propose a variational approach to the depinning and dynamics of discrete vortices, based on minimizing movements. We show that, letting first the lattice spacing and then the time step of the minimizing movements tend to zero, the vortices move according with the gradient flow of the renormalized energy, as in the continuous Ginzburg–Landau framework.
      PubDate: 2014-10-01
       
  • A Polyconvex Integrand; Euler–Lagrange Equations and Uniqueness of
           Equilibrium
    • Abstract: Abstract In this manuscript we are interested in stored energy functionals W defined on the set of d × d matrices, which not only fail to be convex but satisfy \({{\rm lim}_{\det \xi \rightarrow 0^+} W(\xi)=\infty.}\) We initiate a study which we hope will lead to a theory for the existence and uniqueness of minimizers of functionals of the form \({E(\mathbf{u})=\int_\Omega (W(\nabla \mathbf{u}) -\mathbf{F} \cdot \mathbf{u}) {\rm d}x}\) , as well as their Euler–Lagrange equations. The techniques developed here can be applied to a class of functionals larger than those considered in this manuscript, although we keep our focus on polyconvex stored energy functionals of the form \({W(\xi)=f(\xi) +h( {\rm det} \xi)}\) – such that \({{\rm lim}_{t \rightarrow 0^+} h(t)=\infty}\) – which appear in the study of Ogden material. We present a collection of perturbed and relaxed problems for which we prove uniqueness results. Then, we characterize these minimizers by their Euler–Lagrange equations.
      PubDate: 2014-10-01
       
  • Decaying Turbulence in the Generalised Burgers Equation
    • Abstract: Abstract We consider the generalised Burgers equation $$\frac{\partial u}{\partial t} + f'(u)\frac{\partial u}{\partial x} - \nu \frac{\partial^2 u}{\partial x^2} = 0, \,\, t \geqq 0, \,\, x \in S^1,$$ where f is strongly convex and ν is small and positive. We obtain sharp estimates for Sobolev norms of u (upper and lower bounds differ only by a multiplicative constant). Then, we obtain sharp estimates for the dissipation length scale and the small-scale quantities which characterise the decaying Burgers turbulence, i.e., the structure functions and the energy spectrum. The proof uses a quantitative version of an argument by Aurell et al. (J Fluid Mech 238:467–486, 1992). Note that we are dealing with decaying, as opposed to stationary turbulence. Thus, our estimates are not uniform in time. However, they hold on a time interval [T 1, T 2], where T 1 and T 2 depend only on f and the initial condition, and do not depend on the viscosity. These results allow us to obtain a rigorous theory of the one-dimensional Burgers turbulence in the spirit of Kolmogorov’s 1941 theory. In particular, we obtain two results which hold in the inertial range. On one hand, we explain the bifractal behaviour of the moments of increments, or structure functions. On the other hand, we obtain an energy spectrum of the form k −2. These results remain valid in the inviscid limit.
      PubDate: 2014-10-01
       
  • Local and Global Well-Posedness of Strong Solutions to the 3D Primitive
           Equations with Vertical Eddy Diffusivity
    • Abstract: Abstract In this paper, we consider the initial-boundary value problem of the viscous 3D primitive equations for oceanic and atmospheric dynamics with only vertical diffusion in the temperature equation. Local and global well-posedness of strong solutions are established for this system with H 2 initial data.
      PubDate: 2014-10-01
       
  • Quasilinear and Hessian Type Equations with Exponential Reaction and
           Measure Data
    • Abstract: Abstract We prove existence results concerning equations of the type \({-\Delta_pu=P(u)+\mu}\) for p > 1 and F k [−u] = P(u) + μ with \({1 \leqq k < \frac{N}{2}}\) in a bounded domain Ω or the whole \({\mathbb{R}^N}\) , where μ is a positive Radon measure and \({P(u)\sim e^{au^\beta}}\) with a > 0 and \({\beta \geqq 1}\) . Sufficient conditions for existence are expressed in terms of the fractional maximal potential of μ. Two-sided estimates on the solutions are obtained in terms of some precise Wolff potentials of μ. Necessary conditions are obtained in terms of Orlicz capacities. We also establish existence results for a general Wolff potential equation under the form \({u={\bf W}_{\alpha, p}^R[P(u)]+f}\) in \({\mathbb{R}^N}\) , where \({0 < R \leqq \infty}\) and f is a positive integrable function.
      PubDate: 2014-10-01
       
  • The Fokker–Planck Equation with Absorbing Boundary Conditions
    • Abstract: Abstract We study the initial-boundary value problem for the Fokker–Planck equation in an interval with absorbing boundary conditions. We develop a theory of well-posedness of classical solutions for the problem. We also prove that the resulting solutions decay exponentially for long times. To prove these results we obtain several crucial estimates, which include hypoellipticity away from the singular set for the Fokker–Planck equation with absorbing boundary conditions, as well as the Hölder continuity of the solutions up to the singular set.
      PubDate: 2014-10-01
       
  • Variational Proof of the Existence of the Super-Eight Orbit in the
           Four-Body Problem
    • Abstract: Abstract Using the variational method, Chenciner and Montgomery (Ann Math 152:881–901, 2000) proved the existence of an eight-shaped periodic solution of the planar three-body problem with equal masses. Just after the discovery, Gerver numerically found a similar periodic solution called “super-eight” in the planar four-body problem with equal mass. In this paper we prove the existence of the super-eight orbit by using the variational method. The difficulty of the proof is to eliminate the possibility of collisions. In order to solve it, we apply the scaling technique established by Tanaka (Ann Inst H Poincaré Anal Non Linéaire 10:215–238, 1993), (Proc Am Math Soc 122:275–284, 1994) and investigate the asymptotic behavior of a binary collision.
      PubDate: 2014-10-01
       
  • Intrinsic Geometry and Analysis of Diffusion Processes and        class="a-plus-plus">L        
    • Abstract: Abstract The aim of this paper is twofold. First, we obtain a better understanding of the intrinsic distance of diffusion processes. Precisely, (a) for all n ≧ 1, the diffusion matrix A is weak upper semicontinuous on Ω if and only if the intrinsic differential and the local intrinsic distance structures coincide; (b) if n = 1, or if n ≧ 2 and A is weak upper semicontinuous on Ω, the intrinsic distance and differential structures always coincide; (c) if n ≧ 2 and A fails to be weak upper semicontinuous on Ω, the (non-)coincidence of the intrinsic distance and differential structures depend on the geometry of the non-weak-upper-semicontinuity set of A. Second, for an arbitrary diffusion matrix A, we show that the intrinsic distance completely determines the absolute minimizer of the corresponding L ∞-variational problem, and then obtain the existence and uniqueness for given boundary data. We also give an example of a diffusion matrix A for which there is an absolute minimizer that is not of class C 1. When A is continuous, we also obtain the linear approximation property of the absolute minimizer.
      PubDate: 2014-10-01
       
  • Existence, Uniqueness and Lipschitz Dependence for
           Patlak–Keller–Segel and Navier–Stokes in        class="a-plus-plus inline-equation id-i-eq1">        class="a-plus-plus equation-source
           format-t-e-x">\({\mathbb{R}^2}\)
    with
           Measure-Valued Initial Data
    • Abstract: Abstract We establish a new local well-posedness result in the space of finite Borel measures for mild solutions of the parabolic–elliptic Patlak–Keller–Segel (PKS) model of chemotactic aggregation in two dimensions. Our result only requires that the initial measure satisfy the necessary assumption \({\max_{x \in \mathbb{R}^2} \mu (\{x\}) < 8 \pi}\) . This work improves the small-data results of Biler (Stud Math 114(2):181–192, 1995) and the existence results of Senba and Suzuki (J Funct Anal 191:17–51, 2002). Our work is based on that of Gallagher and Gallay (Math Ann 332:287–327, 2005), who prove the uniqueness and log-Lipschitz continuity of the solution map for the 2D Navier–Stokes equations (NSE) with measure-valued initial vorticity. We refine their techniques and present an alternative version of their proof which yields existence, uniqueness and Lipschitz continuity of the solution maps of both PKS and NSE. Many steps are more difficult for PKS than for NSE, particularly on the level of the linear estimates related to the self-similar spreading solutions.
      PubDate: 2014-09-20
       
  • The Sharp Corner Formation in 2D Euler Dynamics of Patches: Infinite
           Double Exponential Rate of Merging
    • Abstract: Abstract For the 2D Euler dynamics of patches, we investigate the convergence to the singular stationary solution in the presence of a regular strain. It is proved that the rate of merging can be double exponential infinitely in time and the estimates we obtain are sharp.
      PubDate: 2014-09-18
       
 
 
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