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 Archive for Rational Mechanics and Analysis     [SJR: 3.43]   [H-I: 51]    [5 followers]  Follow        Hybrid journal (It can contain Open Access articles)    ISSN (Print) 1432-0673 - ISSN (Online) 0003-9527    Published by Springer-Verlag  [2210 journals]
• On Stability of the Spatially Inhomogeneous
Navier–Stokes–Boussinesq System with General Nonlinearity
• Abstract: Abstract This paper considers L 2-asymptotic stability of the spatially inhomogeneous Navier–Stokes–Boussinesq system with general nonlinearity including both power nonlinear terms and convective terms. We construct a local-in-time strong solution of the system by applying semigroup theory on Hilbert spaces and fractional powers of the Stokes–Laplace operator. It is also shown that under some assumptions on an energy inequality the system has a unique global-in-time strong solution when the initial datum is sufficiently small. Furthermore, we investigate the asymptotic stability of the global-in-time strong solution by using an energy inequality, maximal L p -in-time regularity for Hilbert space-valued functions, and fractional powers of linear operators in a solenoidal L 2-space. We introduce new methods for showing the asymptotic stability by applying an energy inequality and maximal L p -in-time regularity for Hilbert space-valued functions. Our approach in this paper can be applied to show the asymptotic stability of energy solutions for various incompressible viscous fluid systems and the stability of small stationary solutions whose structure is not clear.
PubDate: 2015-03-01

• A Double Bubble Assembly as a New Phase of a Ternary Inhibitory System
• Abstract: Abstract A ternary inhibitory system is a three component system characterized by two properties: growth and inhibition. A deviation from homogeneity has a strong positive feedback on its further increase. In the meantime a longer ranging confinement mechanism prevents unlimited spreading. Together they lead to a locally self-enhancing and self-organizing process. The model considered here is a planar nonlocal geometric problem derived from the triblock copolymer theory. An assembly of perturbed double bubbles is mathematically constructed as a stable stationary point of the free energy functional. Triple junction, a phenomenon in which the three components meet at a single point, is a key issue addressed in the construction. Coarsening, an undesirable scenario of excessive growth, is prevented by a lower bound on the long range interaction term in the free energy. The proof involves several ideas: perturbation of double bubbles in a restricted class; use of internal variables to remove nonlinear constraints, local minimization in a restricted class formulated as a nonlinear problem on a Hilbert space; and reduction to finite dimensional minimization. This existence theorem predicts a new morphological phase of a double bubble assembly.
PubDate: 2015-03-01

• Inverse Boundary Value Problem for the Stokes and the Navier–Stokes
Equations in the Plane
• Abstract: Abstract In this paper, we prove in two dimensions the global identifiability of the viscosity in an incompressible fluid by making boundary measurements. The main contribution of this work is to use more natural boundary measurements, the Cauchy forces, than the Dirichlet-to-Neumann map previously considered in Imanuvilov and Yamamoto (Global uniqueness in inverse boundary value problems for Navier–Stokes equations and Lamé ststem in two dimensions. arXiv:1309.1694, 2013) to prove the uniqueness of the viscosity for the Stokes equations and for the Navier–Stokes equations.
PubDate: 2015-03-01

• The Eshelby Theorem and its Variants for Piezoelectric Media
• Abstract: Abstract We prove the Eshelby theorem for an ellipsoidal piezoelectric inclusion in an infinite piezoelectric material. Explicit formulas for the link and polarization matrices are derived. Passing to the limits with respect to parameters in the corresponding equations, the result is extended to cases when either the inclusion or the surrounding material is purely elastic.
PubDate: 2015-03-01

• Quasi-Static Brittle Damage Evolution in Elastic Materials with Multiple
Damaged States
• Abstract: Abstract We present energetic and strain-threshold models for the quasi-static evolution of brutal brittle damage for geometrically-linear elastic materials. By allowing for anisotropic elastic moduli and multiple damaged states we present the issues for the first time in a truly elastic setting, and show that the threshold methods developed in (Garroni, A., Larsen, C. J., Threshold-based quasi-static brittle damage evolution, Archive for Rational Mechanics and Analysis 194 (2), 585–609, 2009) extend naturally to elastic materials with non-interacting damage. We show the existence of solutions and that energetic evolutions are also threshold evolutions.
PubDate: 2015-03-01

• On Asymptotic Behaviour and W 2, p Regularity of Potentials in Optimal
Transportation
• Abstract: Abstract In this paper we study local properties of cost and potential functions in optimal transportation. We prove that in a proper normalization process, the cost function is uniformly smooth and converges locally smoothly to a quadratic cost x · y, while the potential function converges to a quadratic function. As applications we obtain the interior W 2, p estimates and sharp C 1, α estimates for the potentials, which satisfy a Monge–Ampère type equation. The W 2, p estimate was previously proved by Caffarelli for the quadratic transport cost and the associated standard Monge–Ampère equation.
PubDate: 2015-03-01

• From Microscopic Theory to Macroscopic Theory: a Systematic Study on
Modeling for Liquid Crystals
• Abstract: Abstract In this paper, we propose a systematic way of liquid crystal modeling to build connections between microscopic theory and macroscopic theory. In the first part, we propose a new Q-tensor model based on Onsager’s molecular theory for liquid crystals. The Oseen–Frank theory can be recovered from the derived Q-tensor theory by making a uniaxial assumption, and the coefficients in the Oseen–Frank model can be examined. In addition, the smectic-A phase can be characterized by the derived macroscopic model. In the second part, we derive a new dynamic Q-tensor model from Doi’s kinetic theory by the Bingham closure, which obeys the energy dissipation law. Moreover, the Ericksen–Leslie system can also be derived from new Q-tensor system by making an expansion near the local equilibrium.
PubDate: 2015-03-01

• A Rigorous Derivation of the Equations for the Clamped Biot-Kirchhoff-Love
Poroelastic Plate
• Abstract: Abstract In this paper we investigate the limit behavior of the solution to quasi-static Biot’s equations in thin poroelastic plates as the thickness tends to zero. We choose Terzaghi’s time corresponding to the plate thickness and obtain the strong convergence of the three-dimensional solid displacement, fluid pressure and total poroelastic stress to the solution of the new class of plate equations. In the new equations the in-plane stretching is described by the two dimensional Navier’s linear elasticity equations, with elastic moduli depending on Gassmann’s and Biot’s coefficients. The bending equation is coupled with the pressure equation and it contains the bending moment due to the variation in pore pressure across the plate thickness. The pressure equation is parabolic only in the vertical direction. As additional terms it contains the time derivative of the in-plane Laplacian of the vertical deflection of the plate and of the elastic in-plane compression term.
PubDate: 2015-03-01

• Minimizers of Anisotropic Surface Tensions Under Gravity: Higher
Dimensions via Symmetrization
• Abstract: Abstract We consider a variational model describing the shape of liquid drops and crystals under the influence of gravity, resting on a horizontal surface. Making use of anisotropic symmetrization techniques, we establish the existence, convexity and symmetry of minimizers for a class of surface tensions admissible to the symmetrization procedure. In the case of smooth surface tensions, we obtain the uniqueness of minimizers via an ODE characterization.
PubDate: 2015-02-01

• Thermal Creep of a Rarefied Gas on the Basis of Non-linear Korteweg-Theory
• Abstract: Abstract The study of thermal transpiration, more commonly called thermal creep, is accomplished by use of Korteweg’s theory of capillarity. Incorporation of this theory into the balance laws of continuum mechanics allows resolution of boundary value problems via solutions to systems of ordinary differential equations. The problem was originally considered by Maxwell in his classic 1879 paper Maxwell (Phil Trans Roy Soc (London) 170:231–256, 1879). In that paper Maxwell derived what is now called the Burnett higher order contribution to the Cauchy stress, but was not able to solve his newly derived system of partial differential equations. In this paper the authors note that a more appropriate higher order contribution to the Cauchy stress follows from Korteweg’s 1901 theory Korteweg (Arch Neerl Sci Exactes Nat Ser II 6:1–24, 1901). The appropriateness of Korteweg’s theory is based on the exact summation of the Chapman–Enskog expansion given by Gorban and Karlin. The resulting balance laws are solved exactly, qualitatively, and numerically and the results are qualitatively similar to the numerical and exact results given by Aoki et al., Loyalka et al., and Struchtrup et al.
PubDate: 2015-02-01

• Stability of the Melting Hedgehog in the Landau–de Gennes Theory of
Nematic Liquid Crystals
• Abstract: Abstract We investigate stability properties of the radially symmetric solution corresponding to the vortex defect (the so called “melting hedgehog”) in the framework of the Landau–de Gennes model of nematic liquid crystals. We prove local stability of the melting hedgehog under arbitrary Q-tensor valued perturbations in the temperature regime near the critical supercooling temperature. As a consequence of our method, we also rediscover the loss of stability of the vortex defect in the deep nematic regime.
PubDate: 2015-02-01

• Regularity for Double Phase Variational Problems
• Abstract: Abstract We prove sharp regularity theorems for minimisers of a class of variational integrals whose integrand switches between two different types of degenerate elliptic phases, according to the zero set of a modulating coefficient $${a(\cdot)}$$ . The model case is given by the functional $$w \mapsto \int ( Dw ^p + a(x) Dw ^q) \, {\rm d}x,$$ where q > p and $${a(\cdot) \geqq 0}$$ .
PubDate: 2015-02-01

• Domain Walls in the Coupled Gross–Pitaevskii Equations
• Abstract: Abstract A thorough study of domain wall solutions in coupled Gross–Pitaevskii equations on the real line is carried out including existence of these solutions; their spectral and nonlinear stability; their persistence and stability under a small localized potential. The proof of existence is variational and is presented in a general framework: we show that the domain wall solutions are energy minimizers within a class of vector-valued functions with nontrivial conditions at infinity. The admissible energy functionals include those corresponding to coupled Gross–Pitaevskii equations, arising in modeling of Bose–Einstein condensates. The results on spectral and nonlinear stability follow from properties of the linearized operator about the domain wall. The methods apply to many systems of interest and integrability is not germane to our analysis. Finally, sufficient conditions for persistence and stability of domain wall solutions are obtained to show that stable pinning occurs near maxima of the potential, thus giving rigorous justification to earlier results in the physics literature.
PubDate: 2015-02-01

• Collective Excitations of Bose Gases in the Mean-Field Regime
• Abstract: Abstract We study the spectrum of a large system of N identical bosons interacting via a two-body potential with strength 1/N. In this mean-field regime, Bogoliubov’s theory predicts that the spectrum of the N-particle Hamiltonian can be approximated by that of an effective quadratic Hamiltonian acting on Fock space, which describes the fluctuations around a condensed state. Recently, Bogoliubov’s theory has been justified rigorously in the case that the low-energy eigenvectors of the N-particle Hamiltonian display complete condensation in the unique minimizer of the corresponding Hartree functional. In this paper, we shall justify Bogoliubov’s theory for the high-energy part of the spectrum of the N-particle Hamiltonian corresponding to (non-linear) excited states of the Hartree functional. Moreover, we shall extend the existing results on the excitation spectrum to the case of non-uniqueness and/or degeneracy of the Hartree minimizer. In particular, the latter covers the case of rotating Bose gases, when the rotation speed is large enough to break the symmetry and to produce multiple quantized vortices in the Hartree minimizer.
PubDate: 2015-02-01

• Diffusion Limit of Kinetic Equations for Multiple Species Charged
Particles
• Abstract: Abstract In ionic solutions, there are multi-species charged particles (ions) with different properties like mass, charge etc. Macroscopic continuum models like the Poisson–Nernst–Planck (PNP) systems have been extensively used to describe the transport and distribution of ionic species in the solvent. Starting from the kinetic theory for the ion transport, we study a Vlasov–Poisson–Fokker–Planck (VPFP) system in a bounded domain with reflection boundary conditions for charge distributions and prove that the global renormalized solutions of the VPFP system converge to the global weak solutions of the PNP system, as the small parameter related to the scaled thermal velocity and mean free path tends to zero. Our results may justify the PNP system as a macroscopic model for the transport of multi-species ions in dilute solutions.
PubDate: 2015-02-01

• Existence of Magnetic Compressible Fluid Stars
• Abstract: Abstract The existence of magnetic star solutions which are axi-symmetric stationary solutions for the Euler–Poisson system of compressible fluids coupled to a magnetic field is proved in this paper by a variational method. Our method of proof consists in deriving an elliptic equation for the magnetic potential in cylindrical coordinates in $${\mathbb{R}^3}$$ , and obtaining the estimates of the Green’s function for this elliptic equation by transforming it to 5-Laplacian.
PubDate: 2015-02-01

• The Nonlocal Porous Medium Equation: Barenblatt Profiles and Other Weak
Solutions
• Abstract: Abstract A degenerate nonlinear nonlocal evolution equation is considered; it can be understood as a porous medium equation whose pressure law is nonlinear and nonlocal. We show the existence of sign-changing weak solutions to the corresponding Cauchy problem. Moreover, we construct explicit compactly supported self-similar solutions which generalize Barenblatt profiles—the well-known solutions of the classical porous medium equation.
PubDate: 2015-02-01

• 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: 2015-02-01

• On Inviscid Limits for the Stochastic Navier–Stokes Equations and
Related Models
• Abstract: Abstract We study inviscid limits of invariant measures for the 2D stochastic Navier–Stokes equations. As shown by Kuksin (J Stat Phys 115(1–2):469–492, 2004), the noise scaling $${\sqrt{\nu}}$$ is the only one which leads to non-trivial limiting measures, which are invariant for the 2D Euler equations. Using a Moser-type iteration for stochastic drift-diffusion equations, we show that any limiting measure $${\mu_{0}}$$ is in fact supported on bounded vorticities. Relationships of $${\mu_{0}}$$ to the long term dynamics of 2D Euler in $${L^{\infty}}$$ with the weak* topology are discussed. We also obtain a drift-independent modulus of continuity for a stationary deterministic model problem, which leads us to conjecture that in fact $${\mu_0}$$ is supported on $${C^0}$$ . Moreover, in view of the Batchelor–Krainchnan 2D turbulence theory, we consider inviscid limits for a weakly damped stochastic Navier–Stokes equation. In this setting we show that only an order zero noise scaling (with respect to ν) leads to a nontrivial limiting measure in the inviscid limit.
PubDate: 2015-01-21

• Convex Entropy, Hopf Bifurcation, and Viscous and Inviscid Shock Stability
• Abstract: Abstract We consider by a combination of analytical and numerical techniques, some basic questions regarding the relations between inviscid and viscous stability and the existence of a convex entropy. Specifically, for a system possessing a convex entropy, in particular for the equations of gas dynamics with a convex equation of state, we ask: (1) can inviscid instability occur? (2) can viscous instability not detected by inviscid theory occur? (3) can there occur the—necessarily viscous—effect of Hopf bifurcation, or “galloping instability”? and, perhaps most important from a practical point of view, (4) as shock amplitude is increased from the (stable) weak-amplitude limit, can there occur a first transition from viscous stability to instability that is not detected by inviscid theory? We show that (1) does occur for strictly hyperbolic, genuinely nonlinear gas dynamics with certain convex equations of state, while (2) and (3) do occur for an artifically constructed system with convex viscosity-compatible entropy. We do not know of an example for which (4) occurs, leaving this as a key open question in viscous shock theory, related to the principal eigenvalue property of Sturm Liouville and related operators. In analogy with, and partly proceeding close to, the analysis of Smith on (non-)uniqueness of the Riemann problem, we obtain convenient criteria for shock (in)stability in the form of necessary and sufficient conditions on the equation of state.
PubDate: 2015-01-20

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