Singular Limit Research Articles

Singular limit for the magnetohydrodynamics of the damped wave type in the critical Fourier–Sobolev space

We study the Cauchy problem of the incompressible damped wave type magnetohydrodynamic system in RN(N≥2). The purpose of this paper is to show the global well-posedness and a singular limit of the problem in Fourier–Sobolev spaces. For the proof of the results, we use the Lp-Lq type estimates for the fundamental solutions of the damped wave equation and end-point maximal regularity for the inhomogeneous heat equation in that space with a detailed estimate of difference between the symbol of the heat kernel and fundamental solution of the damped wave equation.

The non-ideal theory of the Aharonov\u2013Bohm effect

Elay Shech and John Earman have recently argued that the common topological interpretation of the Aharonov–Bohm (AB) effect is unsatisfactory because it fails to justify idealizations that it presupposes. In particular, they argue that an adequate account of the AB effect must address the role of boundary conditions in certain ideal cases of the effect. In this paper I defend the topological interpretation against their criticisms. I consider three types of idealization that might arise in treatments of the effect. First, Shech takes the AB effect to involve an idealization in the form of a singular limit, analogous to the thermodynamic limit in statistical mechanics. But, I argue, the AB effect itself features no singular limits, so it doesn’t involve idealizations in this sense. Second, I argue that Shech and Earman’s emphasis on the role of boundary conditions in the AB effect is misplaced. The idealizations that are useful in connecting the theoretical description of the AB effect to experiment do interact with facts about boundary conditions, but none of these idealizations are presupposed by the topological interpretation of the effect. Indeed, the boundary conditions for which Shech and demands justification are incompatible with some instances of the AB effect, so the topological interpretation ought not justify them. Finally, I address the role of the non-relativistic approximation usually presumed in discussions of the AB effect. This approximation is essential if—as the topological interpretation supposes—the AB effect constrains and justifies a relativistic theory of the electromagnetic interaction. In this case the ends justify the means. So the topological view presupposes no unjustified idealizations.

High friction limit for Euler–Korteweg and Navier–Stokes–Korteweg models via relative entropy approach

The aim of this paper is to investigate the singular relaxation limits for the Euler–Korteweg and the Navier–Stokes–Korteweg system in the high friction regime. We shall prove that the viscosity term is present only in higher orders in the proposed scaling and therefore it does not affect the limiting dynamics, and the two models share the same equilibrium equation. The analysis of the limit is carried out using the relative entropy techniques in the framework of weak, finite energy solutions of the relaxation models converging toward smooth solutions of the equilibrium. The results proved here take advantage of the enlarged formulation of the models in terms of the drift velocity introduced in [6], generalizing in this way the ones proved in [15] for the Euler–Korteweg model.

Singular limit of the nonisentropic compressible ideal MHD equations in a domain with boundary

ABSTRACT In this paper, we study the singular limit of three-dimensional nonisentropic compressible ideal MHD equations with horizontal boundary conditions. It is rigorously justified that, when the Mach number and Alfvén number are same and tend to zero, the solution of original equations converges to that of two-dimensional incompressible flow.

On a Singular Limit for the Compressible Rotating Euler System

The work addresses a singular limit for a rotating compressible Euler system in the low Mach number and low Rossby number regime. Based on the concept of dissipative measure-valued solution, the quasi-geostrophic system is identified as the limit problem in the case of ill-prepared initial data. The ill-prepared initial data will cause rapidly oscillating acoustic waves. Using dispersive estimates of Strichartz type, the effect of the acoustic waves in the asymptotic limit is eliminated.

The ergodic limit for weak solutions of elliptic equations with Neumann boundary condition

We consider the so-called it ergodic problem for weak solutions of elliptic equations in divergence form, complemented with Neumann boundary conditions. The simplest example reads as the following boundary value problem in a bounded domain of $\mathbb{R}^N$: $ \left\{ \begin{array}{l} - {\rm{div}}(A(x)\nabla u) + \lambda = H(x,\nabla u)\;\;\;\;\;{\rm{in}}\;\Omega ,\\ A(x)\nabla u \cdot \vec n = 0\;\;\;\;\;\;\;{\rm{on}}\;\partial \Omega , \end{array} \right. $ where A(x) is a coercive matrix with bounded coefficients, and $H(x, \nabla u)$ has Lipschitz growth in the gradient and measurable $x$-dependence with suitable growth in some Lebesgue space (typically, $|H(x, \nabla u)|\leq b(x) |\nabla u|+ f(x)$ for functions b(x)∈ LN(Ω) and f (x) ∈ Lm(Ω), $m\geq 1$). We prove that there exists a unique real value $\lambda$ for which the problem is solvable in Sobolev spaces and the solution is unique up to addition of a constant. We also characterize the ergodic limit, say the singular limit obtained by adding a vanishing zero order term in the equation. Our results extend to weak solutions and to data in Lebesgue spaces LN(Ω) (or in the dual space (H1(Ω))'), previous results which were proved in the literature for bounded solutions and possibly classical or viscosity formulations.

On contact point motion in the vibration analysis of elastic rods

We present a systematic method for analyzing the vibrations of elastic rods whose effective length is variable, with particular emphasis on rods in unilateral contact with rigid surfaces. Problems of this type abound in engineering applications at all length scales, from the laying of submarine pipelines to the stiction of cantilevers in microelectromechanical systems (MEMS). By a careful treatment of boundary conditions, we elucidate the circumstances under which a rod of variable length can be treated as one of fixed length for the sake of analyzing small-amplitude vibrations. In applying our method to a simple free vibration problem, we encounter an unusual singular limit and observe a close connection between vibration, stability, and existence.

The singular limits of solutions to the Riemann problem for the liquid–gas two-phase isentropic flow model

All the possible Riemann solutions are constructed in fully explicit forms for the one-dimensional inviscid liquid–gas two-phase isentropic flow model of drift-flux type. The concentration and cavitation phenomena are observed and investigated in the Riemann solutions when the pressure in the mixture momentum equation tends to zero. It is proved rigorously that the Riemann solution consisting of a 1-shock wave, 2-contact discontinuity, and 3-shock wave converges to a delta shock wave solution. By comparison, the Riemann solution consisting of a 1-rarefaction wave, 2-contact discontinuity, and 3-rarefaction wave converges to a solution consisting of 1-contact discontinuity, the vacuum state, and 3-contact discontinuity.

Fast reaction limit of reaction-diffusion systems

<p style='text-indent:20px;'>Singular limit problems of reaction-diffusion systems have been studied in cases where the effects of the reaction terms are very large compared with those of the other terms. Such problems appear in literature in various fields such as chemistry, ecology, biology, geology and approximation theory. In this paper, we deal with the singular limit of a general reaction-diffusion system including many problems in the literature. We formulate the problem, derive the limit equation and establish a rigorous mathematical theory.

Pressureless Euler with nonlocal interactions as a singular limit of degenerate Navier-Stokes system

We show that weak solutions of degenerate Navier-Stokes equations converge to the strong solutions of the pressureless Euler system with linear drag term, Newtonian repulsion and quadratic confinement. The proof is based on the relative entropy method using the artificial velocity formulation for the one-dimensional Navier-Stokes system.

THE TREE PROPERTY AT THE TWO IMMEDIATE SUCCESSORS OF A SINGULAR CARDINAL

AbstractWe present an alternative proof that from large cardinals, we can force the tree property at $\kappa ^+$ and $\kappa ^{++}$ simultaneously for a singular strong limit cardinal $\kappa $ . The advantage of our method is that the proof of the tree property at the double successor is simpler than in the existing literature. This new approach also works to establish the result for $\kappa =\aleph _{\omega ^2}$ .

Low stratification of the complete Euler system

The present paper takes advantage of the concept of dissipative measure-valued solutions to show the rigorous derivation of the Euler–Boussinesq (EB) system that has been successfully used in various meteorological models. In particular, we show that EB system can be obtained as a singular limit of the complete Euler system. We provide two types of result—firstly, we treat the case of well-prepared initial data for any sufficiently regular bounded domain. Secondly, we use the dispersive estimates for acoustic equation to tackle the case of the ill-prepared initial data on an unbounded exterior domain.

On singular limit for the compressible Navier–Stokes system with non-monotone pressure law

We consider a singular limit for the compressible Navier–Stokes system with general non-monotone pressure law in the asymptotic regime of low Mach number and large Reynolds numbers. We show that any dissipative weak solution approaches the solution of incompressible Euler equation both for well-prepared initial data and ill-prepared initial data.

Singular limits of reaction diffusion equations and geometric flows with discontinuous velocity

We consider the singular limit of a bistable reaction diffusion equation in the case when the velocity of the traveling wave solution depends on the space variable and converges to a discontinuous function. We show that the family of solutions converges to the stable equilibria off a front propagating with a discontinuous velocity. The convergence is global in time by applying the weak geometric flow uniquely defined through the theory of viscosity solutions and the level-set equation.

A Multi-scale Limit of a Randomly Forced Rotating 3-D Compressible Fluid

We study a singular limit of a scaled compressible Navier–Stokes–Coriolis system driven by both a deterministic and stochastic forcing terms in three dimensions. If the Mach number is comparable to the Froude number with both proportional to say \(\varepsilon \ll 1\), whereas the Rossby number scales like \(\varepsilon ^m\) for \(m>1\) large, then we show that any family of weak martingale solution to the 3-D randomly forced rotating compressible equation (under the influence of a deterministic centrifugal force) converges in probability, as \(\varepsilon \rightarrow 0\), to the 2-D incompressible Navier–Stokes system with a corresponding random forcing term.

The Cauchy problem for the Moore-Gibson-Thompson equation in the dissipative case

In this paper, we study the Cauchy problem for the linear and semilinear Moore-Gibson-Thompson (MGT) equation in the dissipative case. Concerning the linear MGT model, by utilizing WKB analysis associated with Fourier analysis, we derive some $L^2$ estimates of solutions, which improve those in the previous research [46]. Furthermore, asymptotic profiles of the solution and an approximate relation in a framework of the weighted $L^1$ space are derived. Next, with the aid of the classical energy method and Hardy's inequality, we get singular limit results for an energy and the solution itself. Concerning the semilinear MGT model, basing on these sharp $L^2$ estimates and constructing time-weighted Sobolev spaces, we investigate global (in time) existence of Sobolev solutions with different regularities. Finally, under a sign assumption on initial data, nonexistence of global (in time) weak solutions is proved by applying a test function method.

A detailed examination of anisotropy and timescales in three-dimensional incompressible magnetohydrodynamic turbulence

When magnetohydrodynamic turbulence evolves in the presence of a large-scale mean magnetic field, an anisotropy develops relative to that preferred direction. The well-known tendency is to develop stronger gradients perpendicular to the magnetic field, relative to the direction along the field. This anisotropy of the spectrum is deeply connected with the anisotropy of estimated timescales for dynamical processes and requires reconsideration of basic issues such as scale locality and spectral transfer. Here, analysis of high-resolution three-dimensional simulations of unforced magnetohydrodynamic turbulence permits quantitative assessment of the behavior of theoretically relevant timescales in Fourier wavevector space. We discuss the distribution of nonlinear times, Alfvén times, and estimated spectral transfer rates. Attention is called to the potential significance of special regions of the spectrum, such as the two-dimensional limit and the “critical balance” region. A formulation of estimated spectral transfer in terms of a suppression factor supports a conclusion that the quasi-two-dimensional fluctuations (characterized by strong nonlinearities) are not a singular limit, but may be in general expected to make important contributions.

Physical topology of three-dimensional unsteady flows with spheroidal invariant surfaces.

Scope is the response of Lagrangian flow topologies of three-dimensional time-periodic flows consisting of spheroidal invariant surfaces to perturbation. Such invariant surfaces generically accommodate nonintegrable Hamiltonian dynamics and, in consequence, intrasurface topologies composed of islands and chaotic seas. Computational studies predict a response to arbitrary perturbation that is dramatically different from the classical case of toroidal invariant surfaces: said islands and chaotic seas evolve into tubes and shells, respectively, that merge into "tube-and-shell" structures consisting of two shells connected via (a) tube(s) by a mechanism termed "resonance-induced merger" (RIM). This paper provides conclusive experimental proof of RIM and advances the corresponding structures as the physical topology of realistic flows with spheroidal invariant surfaces; the underlying unperturbed state is a singular limit that exists only for ideal conditions and cannot be achieved in a physical experiment. This paper furthermore expands existing theory on certain instances of RIM to a comprehensive theory (supported by experiments) that explains all observed instances of this phenomenon. This theory reveals that RIM ensues from perturbed periodic lines via three possible scenarios: truncation of tubes by (i) manifolds of isolated periodic points emerging near elliptic lines or by either (ii) local or (iii) global segmentation of periodic lines into elliptic and hyperbolic parts. The RIM scenario for local segmentation includes a perturbation-induced change from elliptic to hyperbolic dynamics near degenerate points on entirely elliptic lines (denoted "virtual local segmentation"). This theory furthermore demonstrates that RIM indeed accomplishes tube-shell merger by exposing the existence of invariant surfaces that smoothly extend from the tubes into the chaotic shells. These phenomena set the response to perturbation-and physical topology-of flows with spheroidal invariant surfaces fundamentally apart from flows with toroidal invariant surfaces. Its entirely kinematic nature and reliance solely on continuity and solenoidality of the velocity field render the comprehensive theory and its findings universal and generically applicable for (arbitrary perturbation of) basically any incompressible flow-in fact any smooth solenoidal vector field-accommodating spheroidal invariant surfaces.

Hyperbolicity of first and second order extended thermodynamics theory of polyatomic rarefied gases

The balance laws of Rational Extended Thermodynamics describe well the evolution of rarefied gases in non-equilibrium. Usually, it is necessary to approximate the theory in a neighborhood of an equilibrium state and consequently, its hyperbolicity property remains valid only in a neighborhood of the equilibrium state of the field variables, called hyperbolicity region. The goal of this paper is first to determine the differential system with 14 fields for a rarefied polyatomic polytropic gas, approximated at the second-order in the non-equilibrium variables. Then, we investigate and compare the hyperbolicity property of the first-order and second-order systems. In particular, we analyze the role played by the dynamic pressure and the molecular degrees of freedom. Finally, we also show that in the monatomic singular limit the quadratic theory for a polyatomic gas converges to the corresponding quadratic theory for a monatomic gas.

The Regularized Visible Fold Revisited

The planar visible fold is a simple singularity in piecewise smooth systems. In this paper, we consider singularly perturbed systems that limit to this piecewise smooth bifurcation as the singular perturbation parameter $$\epsilon \rightarrow 0$$ . Alternatively, these singularly perturbed systems can be thought of as regularizations of their piecewise counterparts. The main contribution of the paper is to demonstrate the use of consecutive blowup transformations in this setting, allowing us to obtain detailed information about a transition map near the fold under very general assumptions. We apply this information to prove, for the first time, the existence of a locally unique saddle-node bifurcation in the case where a limit cycle, in the singular limit $$\epsilon \rightarrow 0$$ , grazes the discontinuity set. We apply this result to a mass-spring system on a moving belt described by a Stribeck-type friction law.