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 Archive for Rational Mechanics and Analysis    [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]   [SJR: 3.43]   [H-I: 51]
• On the Energy Dissipation Rate of Solutions to the Compressible Isentropic
Euler System
• Abstract: Abstract In this paper we extend and complement the results in Chiodaroli et al. (Global ill-posedness of the isentropic system of gas dynamics, 2014) on the well-posedness issue for weak solutions of the compressible isentropic Euler system in 2 space dimensions with pressure law p(ρ) = ρ γ , γ ≥ 1. First we show that every Riemann problem whose one-dimensional self-similar solution consists of two shocks admits also infinitely many two-dimensional admissible bounded weak solutions (not containing vacuum) generated by the method of De Lellis and Székelyhidi (Ann Math 170:1417–1436, 2009), (Arch Ration Mech Anal 195:225–260, 2010). Moreover we prove that for some of these Riemann problems and for 1 ≤ γ < 3 such solutions have a greater energy dissipation rate than the self-similar solution emanating from the same Riemann data. We therefore show that the maximal dissipation criterion proposed by Dafermos in (J Diff Equ 14:202–212, 1973) does not favour the classical self-similar solutions.
PubDate: 2014-12-01

• Well-Posedness and Singular Limit of a Semilinear Hyperbolic Relaxation
System with a Two-Scale Discontinuous Relaxation Rate
• Abstract: Abstract Nonlinear hyperbolic systems with relaxations may encounter different scales of relaxation time, which is a prototype multiscale phenomenon that arises in many applications. In such a problem the relaxation time is of O(1) in part of the domain and very small in the remaining domain in which the solution can be approximated by the zero relaxation limit which can be solved numerically much more efficiently. For the Jin–Xin relaxation system in such a two-scale setting, we establish its wellposedness and singular limit as the (smaller) relaxation time goes to zero. The limit is a multiscale coupling problem which couples the original Jin–Xin system on the domain when the relaxation time is O(1) with its relaxation limit in the other domain through interface conditions which can be derived by matched interface layer analysis.As a result, we also establish the well-posedness and regularity (such as boundedness in sup norm with bounded total variation and L 1-contraction) of the coupling problem, thus providing a rigorous mathematical foundation, in the general nonlinear setting, to the multiscale domain decomposition method for this two-scale problem originally proposed in Jin et al. in Math. Comp. 82, 749–779, 2013.
PubDate: 2014-12-01

• Hyperelastic Internal Balance by Multiplicative Decomposition of the
• Abstract: Abstract The multiplicative decomposition of the deformation gradient $${{\bf F} = {{\hat{\bf F}}}{\bf F}^*}$$ is often used in finite deformation continuum mechanics as a basis for treating mechanical effects including plasticity, biological growth, material swelling, and notions of material morphogenesis. Evolution rules for the particular effect from this list are then posed for F*. The tensor $${{{\hat{\bf F}}}}$$ is then invoked to describe a subsequent elastic accommodation, and a hyperelastic framework is put in place for its determination using an elastic energy density function, say $${W({\hat{\bf F}})}$$ , as a constitutive specification. Here we explore the theory that emerges if both F* and $${{\hat{\bf F}}}$$ are governed by hyperelastic criteria; thus we consider energy densities $${W({{\hat{\bf F}}}, {\bf F}^*)}$$ . The decomposition of F is itself determined by energy minimization, and the variation associated with the multiplicative decomposition gives a tensor relation that is interpreted as an internal balance requirement. Our initial development purposefully proceeds with minimal presumptions on the kinematic interpretation of the factors in the deformation gradient decomposition. Connections are then made to treatments that ascribe particular kinematic properties to the decomposition factors—the theory of structured deformations is especially significant in this regard. Such theories have broad utility in describing certain substructural reconfigurations in solids. To demonstrate in the context of the present variational treatment we consider a boundary value problem that involves an imposed twist. If the twist is small then the minimizer is classically smooth. At larger values of twist the energy minimizer exhibits a non-smooth deformation that localizes slip at a singular surface.
PubDate: 2014-12-01

• Dispersion Equation for Water Waves with Vorticity and Stokes Waves on
Flows with Counter-Currents
• Abstract: Abstract The two-dimensional free-boundary problem of steady periodic waves with vorticity is considered for water of finite depth. We investigate how flows with small-amplitude Stokes waves on the free surface bifurcate from a horizontal parallel shear flow in which counter-currents may be present. Two bifurcation mechanisms are described: one for waves with fixed Bernoulli’s constant, and the other for waves with fixed wavelength. In both cases the corresponding dispersion equations serve for defining wavelengths from which Stokes waves bifurcate. Necessary and sufficient conditions for the existence of roots of these equations are obtained. Two particular vorticity distributions are considered in order to illustrate the general results.
PubDate: 2014-12-01

• Rate-Independent Dynamics and Kramers-Type Phase Transitions in Nonlocal
Fokker–Planck Equations with Dynamical Control
• Abstract: Abstract The hysteretic behavior of many-particle systems with non-convex free energy can be modeled by nonlocal Fokker–Planck equations that involve two small parameters and are driven by a time-dependent constraint. In this paper we consider the fast reaction regime related to Kramers-type phase transitions and show that the dynamics in the small-parameter limit can be described by a rate-independent evolution equation with hysteresis. For the proof we first derive mass-dissipation estimates by means of Muckenhoupt constants, formulate conditional stability estimates, and characterize the mass flux between the different phases in terms of moment estimates that encode large deviation results. Afterwards we combine all these partial results and establish the dynamical stability of localized peaks as well as sufficiently strong compactness results for the basic macroscopic quantities.
PubDate: 2014-12-01

• Newtonian Limit of Maxwell Fluid Flows
• Abstract: Abstract In this paper, we revise Maxwell’s constitutive relation and formulate a system of first-order partial differential equations with two parameters for compressible viscoelastic fluid flows. The system is shown to possess a nice conservation–dissipation (relaxation) structure and therefore is symmetrizable hyperbolic. Moreover, for smooth flows we rigorously verify that the revised Maxwell’s constitutive relations are compatible with Newton’s law of viscosity.
PubDate: 2014-12-01

• Quantitative Stochastic Homogenization of Elliptic Equations in
Nondivergence Form
• Abstract: Abstract We introduce a new method for studying stochastic homogenization of elliptic equations in nondivergence form. The main application is an algebraic error estimate, asserting that deviations from the homogenized limit are at most proportional to a power of the microscopic length scale, assuming a finite range of dependence. The results are new even for linear equations. The arguments rely on a new geometric quantity which is controlled in part by adapting elements of the regularity theory for the Monge–Ampère equation.
PubDate: 2014-12-01

• Existence, Uniqueness and Lipschitz Dependence for
Patlak–Keller–Segel and Navier–Stokes in
$${\mathbb{R}^2}$$ 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: 2014-12-01

• Uniform Bound of the Highest Energy for the Three Dimensional
Incompressible Elastodynamics
• Abstract: Abstract This article concerns the time growth of Sobolev norms of classical solutions to the three dimensional incompressible isotropic elastodynamics with small initial displacements. Given initial data in $${H^k_\Lambda}$$ for a fixed big integer k, the global well-posedness of this Cauchy problem has been established by Sideris and Thomases in (Commun Pure Appl Math 58(6):750–788, 2005) and (J Hyperbolic Differ Equ 3(4):673–690, 2006, Commun Pure Appl Math 60(12):1707–1730, 2007), where the highest-order generalized energy E k (t) may have a certain growth in time. Alinhac conjectured that such a growth in time may be a true phenomenon, in (Geometric analysis of hyperbolic differential equations: an introduction, lecture note series: 374. Mathematical Society, London) he proved that E k (t) is still uniformly bounded in time only for the three dimensional scalar quasilinear wave equation under a null condition. In this paper, we show that the highest-order generalized energy E k (t) is still uniformly bounded for the three dimensional incompressible isotropic elastodynamics. The equations of incompressible elastodynamics can be viewed as nonlocal systems of wave type and are inherently linearly degenerate in the isotropic case. There are three ingredients in our proof: the first is that we still have a decay rate of $${t^{-\frac{3}{2}}}$$ when we do the highest energy estimate away from the light cone even though in this case the Lorentz invariance is not available. The second one is that the $${L^\infty}$$ norm of the good unknowns, in particular $${\nabla(v + G\omega)}$$ , is shown to have a decay rate of $${t^{-\frac{3}{2}}}$$ near the light cone. The third one is that the pressure is estimated in a novel way as a nonlocal nonlinear term with null structure, as has been recently observed in [16]. The proof employs the generalized energy method of Klainerman, enhanced by weighted L 2 estimates and the ghost weight introduced by Alinhac.
PubDate: 2014-11-20

• Existence and Symmetry of Ground States to the Boussinesq abcd Systems
• Abstract: We consider a four-parameter family of Boussinesq systems derived by Bona et al. (J Nonlinear Sci 12:283–318, 2002). We establish the existence of the ground states which are solitary waves minimizing the action functional of the systems. We further show that in the presence of large surface tension the ground states are even up to translation.
PubDate: 2014-11-18

• Generalized Harnack Inequality for Nonhomogeneous Elliptic Equations
• Abstract: Abstract This paper is concerned with nonlinear elliptic equations in nondivergence form $$F(D^{2}u, Du, x) = 0$$ where F has a drift term which is not Lipschitz continuous. Under this condition the equations are nonhomogeneous and nonnegative solutions do not satisfy the classical Harnack inequality. This paper presents a new generalization of the Harnack inequality for such equations. As a corollary we obtain the optimal Harnack type of inequality for p(x)-harmonic functions which quantifies the strong minimum principle.
PubDate: 2014-11-15

• A Kac Model for Fermions
• Abstract: Abstract We introduce a stochastic N-particle system and show that, as N → ∞, an effective description ruled by the homogeneous fermionic Uehling–Uhlenbeck equation is recovered. The particle model we consider is the same as the Kac model for the homogeneous Boltzmann equation with an additional exclusion constraint taking into account the Pauli Exclusion Principle.
PubDate: 2014-11-08

• Localized Concentration of Semi-Classical States for Nonlinear Dirac
Equations
• Abstract: Abstract The present paper studies concentration phenomena of the semiclassical approximation of a massive Dirac equation with general nonlinear self-coupling: $$\begin{array}{ll}-i \hbar \alpha \cdot \nabla w+a \beta w + V (x) w = g ( w ) w.\end{array}$$ Compared with some existing issues, the most interesting results obtained here are twofold: the solutions concentrating around local minima of the external potential; and the nonlinearities assumed to be either super-linear or asymptotically linear at the infinity. As a consequence one sees that, if there are k bounded domains $${\Lambda_j \subset \mathbb{R}^3}$$ such that $${-a < \min_{\Lambda_j} V=V(x_j) < \min_{\partial \Lambda_j}V}$$ , $${x_j\in\Lambda_j}$$ , then the k-families of solutions $${w_\hbar^j}$$ concentrate around x j as $${\hbar\to 0}$$ , respectively. The proof relies on variational arguments: the solutions are found as critical points of an energy functional. The Dirac operator has a continuous spectrum which is not bounded from below and above, hence the energy functional is strongly indefinite. A penalization technique is developed here to obtain the desired solutions.
PubDate: 2014-11-08

• Differential Complexes in Continuum Mechanics
• Abstract: Abstract We study some differential complexes in continuum mechanics that involve both symmetric and non-symmetric second-order tensors. In particular, we show that the tensorial analogue of the standard grad-curl-div complex can simultaneously describe the kinematics and the kinetics of motion of a continuum. The relation between this complex and the de Rham complex allows one to readily derive the necessary and sufficient conditions for the compatibility of displacement gradient and the existence of stress functions on non-contractible bodies.We also derive the local compatibility equations in terms of the Green deformation tensor for motions of 2D and 3D bodies, and shells in curved ambient spaces with constant curvatures.
PubDate: 2014-11-05

• 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

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