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Showing papers in "Archive for Rational Mechanics and Analysis in 1991"



Journal ArticleDOI
TL;DR: In this article, the compatibility of the field equations and jump conditions of the one-dimensional theory of elastic bars with two additional constitutive requirements: a kinetic relation controlling the rate at which the phase transition takes place and a nucleation criterion for the initiation of phase transition.
Abstract: This paper treats the dynamics of phase transformations in elastic bars The specific issue studied is the compatibility of the field equations and jump conditions of the one-dimensional theory of such bars with two additional constitutive requirements: a kinetic relation controlling the rate at which the phase transition takes place and a nucleation criterion for the initiation of the phase transition A special elastic material with a piecewise-linear, non-monotonic stress-strain relation is considered, and the Riemann problem for this material is analyzed For a large class of initial data, it is found that the kinetic relation and the nucleation criterion together single out a unique solution to this problem from among the infinitely many solutions that satisfy the entropy jump condition at all strain discontinuities

390 citations



Journal ArticleDOI
TL;DR: In this paper, it was shown that any weakly harmonic mapping from a two-dimensional surface into a sphere is smooth, except possibly for a closed singular set of (n 2 ) dimensional Hausdorff measure zero.
Abstract: In an interesting recent paper [12], F. HI~LEIN has shown that any weakly harmonic mapping from a two-dimensional surface into a sphere is smooth. I present here a kind of generalization to higher dimensions, asserting in effect that a stationary harmonic mapping from an open subset of Nn(n __> 3) into a sphere is smooth, except possibly for a closed singular set of (n 2 ) dimensional Hausdorff measure zero. My proof crucially depends upon several of H~L~IN's observations (as streamlined by P.-L. LIoNs). To state the result precisely let us suppose that m, n => 2, U is a smooth open subset of Nn, and S m-1 denotes the unit sphere in Nm. A function u in the Sobolev space HI (U;Rm) , u = (u 1 . . . . ,urn), belongs to H I ( U ; S ml ) provided t u[ = 1 a.e. in U

286 citations


Journal ArticleDOI
TL;DR: In this article, the authors studied the problem of minimizing the functional complexity of the (1.1) problem on a smooth bounded domain, where 2N 2 < p < 2 < 2 = 2, 2E a + U { 0 }. N.
Abstract: In this paper we are concerned with the following problem: ~ u + 2u = u p ' in g?, u > 0 in 2 , (1.1) u ----0 on 0 Q where ~ Q R ~, N ~ 3, is a smooth bounded domain, and 2N 2 < p < 2 \" = ~ 2 , 2 E a + U { 0 } . N It is well known that problem (1.1) has at least one solution for every p E (2, 2*) and for every 2 E (--22, + co) and that this solution can be found by minimizing the functional e~(u) = f (IVul 2 + ,~u ~) dx D on the manifold

276 citations


Journal ArticleDOI
TL;DR: In this paper, the convergence of the homogenization of the Stokes or Navier-Stokes equations to a Dirichlet boundary condition was studied in a domain containing many tiny solid obstacles, periodically distributed in each direction of the axes.
Abstract: This paper treats the homogenization of the Stokes or Navier-Stokes equations with a Dirichlet boundary condition in a domain containing many tiny solid obstacles, periodically distributed in each direction of the axes. (For example, in the three-dimensional case, the obstacles have a size of e3 and are located at the nodes of a regular mesh of size e.) A suitable extension of the pressure is used to prove the convergence of the homogenization process to a Brinkman-type law (in which a linear zero-order term for the velocity is added to a Stokes or Navier-Stokes equation).

262 citations



Journal ArticleDOI
TL;DR: In this article, it was shown that for small obstacles, the limit problem reduces to the Stokes or Navier-Stokes equations, and for larger obstacles, to Darcy's law.
Abstract: This paper is devoted to the homogenization of the Stokes or Navier-Stokes equations with a Dirichlet boundary condition in a domain containing many tiny solid obstacles, periodically distributed in each direction of the axes. For obstacles of critical size it was established in Part I that the limit problem is described by a law of Brinkman type. Here we prove that for smaller obstacles, the limit problem reduces to the Stokes or Navier-Stokes equations, and for larger obstacles, to Darcy's law. We also apply the abstract framework of Part I to the case of a domain containing tiny obstacles, periodically distributed on a surface. (For example, in three dimensions, consider obstacles of size e2, located at the nodes of a regular plane mesh of period e.) This provides a mathematical model for fluid flows through mixing grids, based on a special form of the Brinkman law in which the additional term is concentrated on the plane of the grid.

169 citations






Journal ArticleDOI
TL;DR: In this article, positive radial solutions of a semilinear elliptic equation were studied in balls with zero Dirichlet boundary condition, and the uniqueness of the solution was established under quite general assumptions on g(r) and h(r).
Abstract: Positive radial solutions of a semilinear elliptic equation △u+g(r)u+h(r)u p =0, where r=|x|, xeR n , and p>1, are studied in balls with zero Dirichlet boundary condition. By means of a generalized Pohožaev identity which includes a real parameter, the uniqueness of the solution is established under quite general assumptions on g(r) and h(r). This result applies to Matukuma's equation and the scalar field equation and is shown to be sharp for these equations.


Journal ArticleDOI
TL;DR: In this article, the bifurcation from the essential spectrum was studied and the exact solutions of the equations of Maxwell's equations were found to be the same as in this paper.
Abstract: Keywords: bifurcation from the essential spectrum ; exact solutions ; of Maxwell's equations Reference ANA-ARTICLE-1991-001doi:10.1007/BF00380816View record in Web of Science Record created on 2008-12-10, modified on 2016-08-08



Journal ArticleDOI
TL;DR: In this paper, the Stokes problem is treated in exterior Lipschitz continuous domains of ℝ2 andℝ3 using the weighted Sobolev spaces of Hanouzet and Giroire.
Abstract: This paper treats the Stokes problem in exterior Lipschitz-continuous domains of ℝ2 and ℝ3. Using the weighted Sobolev spaces of Hanouzet (in ℝ3) and Giroire (in ℝ2), we establish the inf-sup condition between the velocity and pressure spaces. This fundamental result shows that the variational Stokes problem is well-posed in those spaces. In the last paragraph, we obtain additional regularity of the solution when the data are smoother.


Journal ArticleDOI
TL;DR: In this paper, the global existence of smooth solutions to the equations of nonlinear thermoelasticity is shown for a one-dimensional homogeneous reference configuration, and the asymptotic behavior of the solutions as t→∞ is described.
Abstract: The global existence of smooth solutions to the equations of nonlinear thermoelasticity is shown for a one-dimensional homogeneous reference configuration. Dirichlet boundary conditions are studied and the asymptotic behaviour of the solutions as t→∞ is described.


Journal ArticleDOI
TL;DR: In this article, it was shown that for a large class of initial data, blow-up occurs at the boundary of the domain when the nonlinearity is no worse than quadratic.
Abstract: This paper treats a superlinear parabolic equation, degenerate in the time derivative. It is shown that the solution may blow up in finite time. Moreover, it is proved that for a large class of initial data, blow-up occurs at the boundary of the domain when the nonlinearity is no worse than quadratic. Various estimates are obtained which determine the asymptotic behaviour near the blow-up. The mathematical analysis is then extended to equations with other degeneracies.


Journal ArticleDOI
TL;DR: In this paper, the authors study wave patterns of hyperbolic systems in which rotational symmetry creates a specific kind of degeneracy, and they give a unified presentation of examples from continuum mechanics.
Abstract: The purpose of the paper is to study wave patterns of hyperbolic systems in which rotational symmetry creates a specific kind of degeneracy. In this situation hyperbolicity is necessarily non-strict, so that the elementary waves have interesting patterns. The discussion is centered around a theorem on existence and uniqueness of solutions of the Riemann problem. We give a unified presentation of examples from continuum mechanics


Journal ArticleDOI
David Hoff1
TL;DR: In this article, the authors prove local existence and study properties of discontinuous solutions of the Navier-Stokes equations for one-dimensional, compressible, nonisentropic flow.
Abstract: We prove local existence and study properties of discontinuous solutions of the Navier-Stokes equations for one-dimensional, compressible, nonisentropic flow. We assume that, modulo a step function, the initial data is in L2 and the initial velocity and density are in the space BV. We show that the velocity and the temperature become smoothed out in positive time, and that discontinuities in the density, pressure, and gradients of the velocity and temperature persist for all time. We also show that for stable gases these discontinuities decay exponentially in time, more rapidly for smaller viscosities.


Journal ArticleDOI
TL;DR: In this article, the degree of microscopic order is represented as a scalar-valued function, which vanishes where the fluid becomes isotropic, and is represented by a set of functions.
Abstract: In this paper we represent the degree of microscopic order through a scalar-valued function, the degree of orientation, which vanishes where the fluid becomes isotropic

Journal ArticleDOI
TL;DR: In this article, a nonlinear, nonlocal model of a shear-free, inextensible beam attached to a rotating rigid body is discussed, and it is shown that linearization of the equations of motion about certain relative equilibrium configurations leads to a partial differential equation.
Abstract: The dynamical effects of imposing constraints on the relative motions of component parts in a rotating mechanical system or structure are explored. It is noted that various simplifying assumptions in modeling the dynamics of elastic beams imply strain constraints, i.e., that the structure being modeled is rigid in certain directions. In a number of cases, such assumptions predict features in both the equilibrium and dynamic behavior which are qualitatively different from what is seen if the assumptions are relaxed. It is argued that many pitfalls may be avoided by adopting so-called geometrically exact models, and examples from the recent literature are cited to demonstrate the consequences of not doing this. These remarks are brought into focus by a detailed discussion of the nonlinear, nonlocal model of a shear-free, inextensible beam attached to a rotating rigid body. Here it is shown that linearization of the equations of motion about certain relative equilibrium configurations leads to a partial differential equation. Such spatially localized models are not obtained in general, however, and this leaves open questions regarding the way in which the geometry of a complex structure influences computational requirements and the possibility of exploiting parallelism in performing simulations. A general treatment of linearization about implicit solutions to equilibrium equations is presented and it is shown that this approach avoids unintended imposition of constraints on relative motions in the models. Finally, the example of a rotating kinematic chain shows how constraining the relative motions in a rotating mechanical system may destabilize uniformly rotating states.

Journal ArticleDOI
TL;DR: In this article, it was shown that the empirical processes of the positions and velocities respectively converge to solutions of the continuity equation and the Euler equation, in the limit as the particle number tends to infinity.
Abstract: We consider certain Hamiltonian systems with many particles interacting through a potential whose range is large in comparison with the typical distance between neighbouring particles. It is shown that the empirical processes of the positions and the velocities respectively converge to solutions of the continuity equation and the Euler equation, in the limit as the particle number tends to infinity.