JournalTOCs

Archive for Rational Mechanics and Analysis

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ISSN (Print) 1432-0673 - ISSN (Online) 0003-9527
Published by Springer-Verlag

[2216 journals]

Isentropic Gas Flow for the Compressible Euler Equation in a Nozzle
- Abstract: Abstract We study the motion of isentropic gas in a nozzle. Nozzles are used to increase the thrust of engines or to accelerate a flow from subsonic to supersonic. Nozzles are essential parts for jet engines, rocket engines and supersonicwind tunnels. In the present paper, we consider unsteady flow, which is governed by the compressible Euler equation, and prove the existence of global solutions for the Cauchy problem. For this problem, the existence theorem has already been obtained for initial data away from the sonic state, (Liu in Commun Math Phys 68:141–172, 1979). Here, we are interested in the transonic flow, which is essential for engineering and physics. Although the transonic flow has recently been studied (Tsuge in J Math Kyoto Univ 46:457–524, 2006; Lu in Nonlinear Anal Real World Appl 12:2802–2810, 2011), these papers assume monotonicity of the cross section area. Here, we consider the transonic flow in a nozzle with a general cross section area. When we prove global existence, the most difficult point is obtaining a bounded estimate for approximate solutions. To overcome this, we employ a new invariant region that depends on the space variable. Moreover, we introduce a modified Godunov scheme. The corresponding approximate solutions consist of piecewise steady-state solutions of an auxiliary equation, which yield a desired bounded estimate. In order to prove their convergence, we use the compensated compactness framework.
  PubDate: 2013-08-01
Asymptotic Behaviour of a Pile-Up of Infinite Walls of Edge Dislocations
- Abstract: Abstract We consider a system of parallel straight edge dislocations and we analyse its asymptotic behaviour in the limit of many dislocations. The dislocations are represented by points in a plane, and they are arranged in vertical walls; each wall is free to move in the horizontal direction. The system is described by a discrete energy depending on the one-dimensional horizontal positions x i > 0 of the n walls; the energy contains contributions from repulsive pairwise interactions between all walls, a global shear stress forcing the walls to the left, and a pinned wall at x = 0 that prevents the walls from leaving through the left boundary. We study the behaviour of the energy as the number of walls, n, tends to infinity, and characterise this behaviour in terms of Γ-convergence. There are five different cases, depending on the asymptotic behaviour of the single dimensionless parameter β n , corresponding to ${\beta_n \ll 1/n, 1/n \ll \beta_n \ll 1}$ , and ${\beta_n \gg 1}$ , and the two critical regimes β n ~ 1/n and β n ~ 1. As a consequence we obtain characterisations of the limiting behaviour of stationary states in each of these five regimes. The results shed new light on the open problem of upscaling large numbers of dislocations. We show how various existing upscaled models arise as special cases of the theorems of this paper. The wide variety of behaviour suggests that upscaled models should incorporate more information than just dislocation densities. This additional information is encoded in the limit of the dimensionless parameter β n .
  PubDate: 2013-08-01
On the Stability in Weak Topology of the Set of Global Solutions to the Navier–Stokes Equations
- Abstract: Abstract Let X be a suitable function space and let ${\mathcal{G} \subset X}$ be the set of divergence free vector fields generating a global, smooth solution to the incompressible, homogeneous three-dimensional Navier–Stokes equations. We prove that a sequence of divergence free vector fields converging in the sense of distributions to an element of ${\mathcal{G}}$ belongs to ${\mathcal{G}}$ if n is large enough, provided the convergence holds “anisotropically” in frequency space. Typically, this excludes self-similar type convergence. Anisotropy appears as an important qualitative feature in the analysis of the Navier–Stokes equations; it is also shown that initial data which do not belong to ${\mathcal{G}}$ (hence which produce a solution blowing up in finite time) cannot have a strong anisotropy in their frequency support.
  PubDate: 2013-08-01
Mappings of Least Dirichlet Energy and their Hopf Differentials
- Abstract: Abstract The paper is concerned with mappings ${h \colon \mathbb{X}}$ ${{\begin{array}{ll} {\rm onto} \\ \longrightarrow \end{array}}}$ ${\mathbb{Y}}$ between planar domains having least Dirichlet energy. The existence and uniqueness (up to a conformal change of variables in ${\mathbb{X}}$ ) of the energy-minimal mappings is established within the class ${\overline{\fancyscript{H}}_2(\mathbb{X}, \mathbb{Y})}$ of strong limits of homeomorphisms in the Sobolev space ${\fancyscript{W}^{1,2}(\mathbb{X}, \mathbb{Y})}$ , a result of considerable interest in the mathematical models of nonlinear elasticity. The inner variation of the independent variable in ${\mathbb{X}}$ leads to the Hopf differential ${h_{z} \overline{h_{\bar{z}}} {\rm d}z \otimes {\rm d}z}$ and its trajectories. For a pair of doubly connected domains, in which ${\mathbb{X}}$ has finite conformal modulus, we establish the following principle: A mapping ${h \in \overline{\fancyscript{H}}_{2} (\mathbb{X}, \mathbb{Y})}$ is energy-minimal if and only if its Hopf-differential is analytic in ${\mathbb{X}}$ and real along ${\partial \mathbb{X}}$ . In general, the energy-minimal mappings may not be injective, in which case one observes the occurrence of slits in ${\mathbb{X}}$ (cognate with cracks). Slits are triggered by points of concavity of ${\mathbb{Y}}$ . They originate from ${\partial \mathbb{X}}$ and advance along vertical trajectories of the Hopf differential toward ${\mathbb{X}}$ where they eventually terminate, so no crosscuts are created.
  PubDate: 2013-08-01
Global Solutions of Nonlinear Wave Equations in Time Dependent Inhomogeneous Media
- Abstract: Abstract We consider the problem of small data global existence for a class of semilinear wave equations with null condition on a Lorentzian background ${(\mathbb{R}^{3 + 1}, g)}$ with a time dependent metric g coinciding with the Minkowski metric outside the cylinder ${\{(t, x) x \leq R\}}$ . We show that the small data global existence result can be reduced to two integrated local energy estimates and demonstrate that these estimates work in the particular case when g is merely C 1 close to the Minkowski metric. One of the novel aspects of this work is that it applies to equations on backgrounds which do not settle to any particular stationary metric.
  PubDate: 2013-08-01
Global Well-posedness of Incompressible Inhomogeneous Fluid Systems with Bounded Density or Non-Lipschitz Velocity
- Abstract: Abstract In this paper, we first prove the global existence of weak solutions to the d-dimensional incompressible inhomogeneous Navier–Stokes equations with initial data ${a_0 \in L^\infty (\mathbb{R}^d), u_0 = (u_0^h, u_0^d) \in \dot{B}^{-1+\frac{d}{p}}_{p, r} (\mathbb{R}^d)}$ , which satisfy ${(\mu \ a_0 \ _{L^\infty} + \ u_0^h\ _{\dot{B}^{-1+\frac{d}{p}}_{p, r}}) {\rm exp}(C_r{\mu^{-2r}}\ u_0^d\ _{\dot{B}^{-1+\frac{d}{p}}_{p,r}}^{2r}) \leqq c_0\mu}$ for some positive constants c 0, C r and 1 < p < d, 1 < r < ∞. The regularity of the initial velocity is critical to the scaling of this system and is general enough to generate non-Lipschitz velocity fields. Furthermore, with additional regularity assumptions on the initial velocity or on the initial density, we can also prove the uniqueness of such a solution. We should mention that the classical maximal L p (L q ) regularity theorem for the heat kernel plays an essential role in this context.
  PubDate: 2013-08-01
Asymptotics of the Solutions of the Stochastic Lattice Wave Equation
- Abstract: Abstract We consider the long time limit for the solutions of a discrete wave equation with weak stochastic forcing. The multiplicative noise conserves energy, and in the unpinned case also conserves momentum. We obtain a time-inhomogeneous Ornstein-Uhlenbeck equation for the limit wave function that holds for both square integrable and statistically homogeneous initial data. The limit is understood in the point-wise sense in the former case, and in the weak sense in the latter. On the other hand, the weak limit for square integrable initial data is deterministic.
  PubDate: 2013-08-01
On the Spectrum of the Poincaré Variational Problem for Two Close-to-Touching Inclusions in 2D
- Abstract: Abstract We study the spectrum of the Poincaré variational problem for two close to touching inclusions in R 2. We derive the asymptotics of its eigenvalues as the distance between the inclusions tends to zero.
  PubDate: 2013-08-01
Wigner Measure Propagation and Conical Singularity for General Initial Data
- Abstract: Abstract We study the evolution of Wigner measures of a family of solutions of a Schrödinger equation with a scalar potential displaying a conical singularity. Under a genericity assumption, classical trajectories exist and are unique, thus the question of the propagation of Wigner measures along these trajectories becomes relevant. We prove the propagation for general initial data.
  PubDate: 2013-07-01
Boundary Regularity of Rotating Vortex Patches
- Abstract: Abstract We show that the boundary of a rotating vortex patch (or V-state, in the terminology of Deem and Zabusky) is C ∞, provided the patch is close to the bifurcation circle in the Lipschitz norm. The rotating patch is also convex if it is close to the bifurcation circle in the C 2 norm. Our proof is based on Burbea’s approach to V-states.
  PubDate: 2013-07-01
Multiple Blow-Up Phenomena for the Sinh-Poisson Equation
- Abstract: Abstract We consider the sinh-Poisson equation $$(P) _ \lambda - \Delta{u} = \lambda \, {\rm sinh} \, u \quad {\rm in} \, \Omega, \quad u = 0 \quad {\rm on} \, \partial\Omega$$ , where Ω is a smooth bounded domain in ${\mathbb{R}^2}$ and λ is a small positive parameter. If ${0 \in \Omega}$ and Ω is symmetric with respect to the origin, for any integer k if λ is small enough, we construct a family of solutions to (P) λ , which blows up at the origin, whose positive mass is 4πk(k−1) and negative mass is 4πk(k + 1). This gives a complete answer to an open problem formulated by Jost et al. (Calc Var PDE 31(2):263–276, 2008).
  PubDate: 2013-07-01
An Estimate for the Morse Index of a Stokes Wave
- Abstract: Abstract Stokes waves are steady periodic water waves on the free surface of an infinitely deep irrotational two-dimensional flow under gravity without surface tension. They can be described in terms of solutions of the Euler–Lagrange equation of a certain functional; this allows one to define the Morse index of a Stokes wave. It is well known that if the Morse indices of the elements of a set of non-singular Stokes waves are bounded, then none of them is close to a singular one. The paper presents a quantitative variant of this result.
  PubDate: 2013-07-01
Nonlocal Nonlinear Schrödinger Equations in R ³
- Abstract: Abstract This paper studies a class of nonlocal nonlinear Schrödinger equations in R 3, which occurs in the infinite ion acoustic speed limit of the Zakharov system with magnetic fields in a cold plasma. The magnetic fields induce some nonlocal effects in these nonlinear Schrödinger systems, and the main goal of this paper is to understand these effects. The key is to establish some a priori estimates on the nonlocal terms generated by the magnetic field, through which we obtain various conclusions including finite time blow-ups, sharp threshold of global existence and instability of standing waves for these equations.
  PubDate: 2013-07-01
Existence of Quasipattern Solutions of the Swift–Hohenberg Equation
- Abstract: Abstract We consider the steady Swift–Hohenberg partial differential equation, a one-parameter family of PDEs on the plane that models, for example, Rayleigh–Bénard convection. For values of the parameter near its critical value, we look for small solutions, quasiperiodic in all directions of the plane, and which are invariant under rotations of angle ${\pi/q, q \geqq 4}$ . We solve an unusual small divisor problem and prove the existence of solutions for small parameter values, then address their stability with respect to quasi-periodic perturbations.
  PubDate: 2013-07-01
The Two-Dimensional Euler Equations on Singular Domains
- Abstract: Abstract We establish the existence of global weak solutions of the two-dimensional incompressible Euler equations for a large class of non-smooth open sets. Loosely, these open sets are the complements (in a simply connected domain) of a finite number of obstacles with positive Sobolev capacity. Existence of weak solutions with L p vorticity is deduced from a property of domain continuity for the Euler equations that relates to the so-called γ-convergence of open sets. Our results complete those obtained for convex domains in Taylor (Progress in Nonlinear Differential Equations and their Applications, Vol. 42, 2000), or for domains with asymptotically small holes (Iftimie et al. in Commun Partial Differ Equ 28(1–2), 349–379, 2003; Lopes Filho in SIAM J Math Anal 39(2), 422–436, 2007).
  PubDate: 2013-07-01
Compatibility Equations of Nonlinear Elasticity for Non-Simply-Connected Bodies
- Abstract: Abstract Compatibility equations of elasticity are almost 150 years old. Interestingly, they do not seem to have been rigorously studied, to date, for non-simply-connected bodies. In this paper we derive necessary and sufficient compatibility equations of nonlinear elasticity for arbitrary non-simply-connected bodies when the ambient space is Euclidean. For a non-simply-connected body, a measure of strain may not be compatible, even if the standard compatibility equations (“bulk” compatibility equations) are satisfied. It turns out that there may be topological obstructions to compatibility; this paper aims to understand them for both deformation gradient F and the right Cauchy-Green strain C = F T F. We show that the necessary and sufficient conditions for compatibility of deformation gradient F are the vanishing of its exterior derivative and all its periods, that is, its integral over generators of the first homology group of the material manifold. We will show that not every non-null-homotopic path requires supplementary compatibility equations for F and linearized strain e. We then find both necessary and sufficient compatibility conditions for the right Cauchy-Green strain tensor C for arbitrary non-simply-connected bodies when the material and ambient space manifolds have the same dimensions. We discuss the well-known necessary compatibility equations in the linearized setting and the Cesàro-Volterra path integral. We then obtain the sufficient conditions of compatibility for the linearized strain when the body is not simply-connected. To summarize, the question of compatibility reduces to two issues: i) an integrability condition, which is d(F dX) = 0 for the deformation gradient and a curvature vanishing condition for C, and ii) a topological condition. For F dx this is a homological condition because the equation one is trying to solve takes the form dφ = F dX. For C, however, parallel transport is involved, which means that one needs to solve an equation of the form dR/ ds = RK, where R takes values in the orthogonal group. This is, therefore, a question about an orthogonal representation of the fundamental group, which, as the orthogonal group is not commutative, cannot, in general, be reduced to a homological question.
  PubDate: 2013-07-01
Classical Limit for a System of Non-Linear Random Schrödinger Equations
- Abstract: Abstract This work is concerned with the semi-classical analysis of mixed state solutions to a Schrödinger–Position equation perturbed by a random potential with weak amplitude and fast oscillations in time and space. We show that the Wigner transform of the density matrix converges weakly and in probability to solutions of a Vlasov–Poisson–Boltzmann equation with a linear collision kernel.Aconsequence of this result is that a smooth non-linearity such as the Poisson potential (repulsive or attractive) does not change the statistical stability property of the Wigner transform observed in linear problems.We obtain, in addition, that the local density and current are self-averaging, which is of importance for some imaging problems in random media. The proof brings together the martingale method for stochastic equations with compactness techniques for non-linear PDEs in a semi-classical regime. It relies partly on the derivation of an energy estimate that is straightforward in a deterministic setting but requires the use of a martingale formulation and well-chosen perturbed test functions in the random context.
  PubDate: 2013-07-01
Hadamard Variational Formula for the Green’s Function of the Boundary Value Problem on the Stokes Equations
- Abstract: Abstract For every ${\varepsilon > 0}$ , we consider the Green’s matrix ${G_{\varepsilon}(x, y)}$ of the Stokes equations describing the motion of incompressible fluids in a bounded domain ${\Omega_{\varepsilon} \subset \mathbb{R}^d}$ , which is a family of perturbation of domains from ${\Omega\equiv \Omega_0}$ with the smooth boundary ${\partial\Omega}$ . Assuming the volume preserving property, that is, ${\mbox{vol.}\Omega_{\varepsilon} = \mbox{vol.}\Omega}$ for all ${\varepsilon > 0}$ , we give an explicit representation formula for ${\delta G(x, y) \equiv \lim_{\varepsilon\to +0}\varepsilon^{-1}(G_{\varepsilon}(x, y) - G_0(x, y))}$ in terms of the boundary integral on ${\partial \Omega}$ of ${G_0(x, y)}$ . Our result may be regarded as a classical Hadamard variational formula for the Green’s functions of the elliptic boundary value problems.
  PubDate: 2013-06-01
Nondispersive solutions to the L ²-critical Half-Wave Equation
- Abstract: Abstract We consider the focusing L 2-critical half-wave equation in one space dimension, $$i \partial_t u = D u - u ^2 u$$ , where D denotes the first-order fractional derivative. Standard arguments show that there is a critical threshold ${M_{*} > 0}$ such that all H 1/2 solutions with ${\ u\ _{L^2} < M_*}$ extend globally in time, while solutions with ${\ u\ _{L^2} \geq M_*}$ may develop singularities in finite time. In this paper, we first prove the existence of a family of traveling waves with subcritical arbitrarily small mass. We then give a second example of nondispersive dynamics and show the existence of finite-time blowup solutions with minimal mass ${\ u_0\ _{L^2} = M_*}$ . More precisely, we construct a family of minimal mass blowup solutions that are parametrized by the energy E 0 > 0 and the linear momentum ${P_0 \in \mathbb{R}}$ . In particular, our main result (and its proof) can be seen as a model scenario of minimal mass blowup for L 2-critical nonlinear PDEs with nonlocal dispersion.
  PubDate: 2013-05-03
Breakdown of Smoothness for the Muskat Problem
- Abstract: Abstract In this paper we show that there exists analytic initial data in the stable regime for the Muskat problem such that the solution turns to the unstable regime and later breaks down, that is, no longer belongs to C 4.
  PubDate: 2013-04-06