Singularities in complex interfaces

1990 ◽  
Vol 333 (1631) ◽  
pp. 379-389 ◽  
Keyword(s):  
Unit Circle ◽  
Blow Up ◽  
The Self ◽  
Conformal Map ◽  
Local Equation ◽  
Material Interface ◽  
Two Fluids ◽  
Non Local ◽  
Self Similar ◽  
Similar Solutions

We analyse an equation describing the motion of the material interface between two fluids in a pressure field. The interface can be expressed as the image of the unit circle under a certain time depending conformal map. This conformal transformation maps the exterior of the unit circle onto the region occupied by one of the fluids. The conformal map has singularities in the unit disc. As long as these singularities are close to the origin, the complicated non-local equation governing the evolution of the conformal map can be approximated by a somewhat simpler, local equation. We prove that there exist self-similar solutions of this equation, that they have singularities away from the origin, that these singularities hit in finite time the unit circle and that the self-similar blow up is stable to perturbations that respect the symmetry of the self-similar profile.

Critical diffusion exponents for self-similar blow-up solutions of a quasilinear parabolic equation with an exponential source

1996 ◽  
Vol 126 (2) ◽  
pp. 413-441 ◽  
Author(s):  
Chris J. Budd ◽  
Victor A. Galaktionov
Keyword(s):  
Parabolic Equation ◽  
Blow Up ◽  
Infinite Sequence ◽  
Quasilinear Parabolic Equation ◽  
Homoclinic Bifurcation ◽  
The Self ◽  
Self Similar ◽  
Image Position ◽  
Similar Solutions ◽  
Quasilinear Parabolic

We study the self-similar solutions of the quasilinear parabolic equationWe show that there is an exponentsuch that if σ> then the equation admits a countable set {uk(x, t)} of self-similar blow-up solutions. These solutions have the formwhere T> 0 is a finite blow-up time, θ(ξ) solves a nonlinear ODE and each function uk(x, t) is nonconstant in a neighbourhood of the origin and has exactly k maxima and minima for x ≧ 0. There is a further critical exponent σ = ф such that if σ > ф there is a second set of self-similar solutions which are constant (in x) in a neighbourhood of the origin. We conjecture (and provide formal arguments and numerical evidence for) the existence of an infinite sequence σk→σ∞ of critical values, such that σ1 = 0 and uk exists only in the range σ>σk (when σ> 0 the equation has no nontrivial self-similar solutions). The proof of existence when σ>σ∞(σ>ф) is obtained by a combination of comparison and dynamical systems arguments and relates the existence of the self-similar solutions to a homoclinic bifurcation in an appropriate phase-space.

On a self-similar solution for the decay of turbulent bursts

1992 ◽  
Vol 3 (4) ◽  
pp. 319-341 ◽  
Author(s):  
S. P. Hastings ◽  
L. A. Peletier
Keyword(s):  
Asymptotic Behaviour ◽  
Physical Parameter ◽  
Dimensional Analysis ◽  
Existence And Uniqueness ◽  
The Self ◽  
Similar Solution ◽  
Self Similar Solution ◽  
Eigenvalue Parameter ◽  
Self Similar ◽  
Similar Solutions

We discuss the self-similar solutions of the second kind associated with the propagation of turbulent bursts in a fluid at rest. Such solutions involve an eigenvalue parameter μ, which cannot be determined from dimensional analysis. Existence and uniqueness are established and the dependence of μ on a physical parameter λ in the problem is studied: estimates are obtained and the asymptotic behaviour as λ → ∞ is established.

Natural convection flow far from a horizontal plate

1999 ◽  
Vol 387 ◽  
pp. 227-254 ◽  
Author(s):  
VALOD NOSHADI ◽  
WILHELM SCHNEIDER
Keyword(s):  
Boundary Layer ◽  
Prandtl Number ◽  
Stokes Equations ◽  
Axisymmetric Flow ◽  
The Self ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Boundary Layer Equations ◽  
Self Similar ◽  
Similar Solutions

Plane and axisymmetric (radial), horizontal laminar jet flows, produced by natural convection on a horizontal finite plate acting as a heat dipole, are considered at large distances from the plate. It is shown that physically acceptable self-similar solutions of the boundary-layer equations, which include buoyancy effects, exist in certain Prandtl-number regimes, i.e. 0.5<Pr[les ]1.470588 for plane, and Pr>1 for axisymmetric flow. In the plane flow case, the eigenvalues of the self-similar solutions are independent of the Prandtl number and can be determined from a momentum balance, whereas in the axisymmetric case the eigenvalues depend on the Prandtl number and are to be determined as part of the solution of the eigenvalue problem. For Prandtl numbers equal to, or smaller than, the lower limiting values of 0.5 and 1 for plane and axisymmetric flow, respectively, the far flow field is a non-buoyant jet, for which self-similar solutions of the boundary-layer equations are also provided. Furthermore it is shown that self-similar solutions of the full Navier–Stokes equations for axisymmetric flow, with the velocity varying as 1/r, exist for arbitrary values of the Prandtl number.Comparisons with finite-element solutions of the full Navier–Stokes equations show that the self-similar boundary-layer solutions are asymptotically approached as the plate Grashof number tends to infinity, whereas the self-similar solution to the full Navier–Stokes equations is applicable, for a given value of the Prandtl number, only to one particular, finite value of the Grashof number.In the Appendices second-order boundary-layer solutions are given, and uniformly valid composite expansions are constructed; asymptotic expansions for large values of the lateral coordinate are performed to study the decay of the self-similar boundary-layer flows; and the stability of the jets is investigated using transient numerical solutions of the Navier–Stokes equations.

scholarly journals Analysis of the self-similar solutions of a generalized Burger's equation with nonlinear damping

2001 ◽  
Vol 7 (3) ◽  
pp. 253-282 ◽  
Author(s):  
Ch. Srinivasa Rao ◽  
P. L. Sachdev ◽  
Mythily Ramaswamy
Keyword(s):  
Initial Conditions ◽  
Nonlinear Damping ◽  
Nonlinear Ordinary Differential Equation ◽  
The Self ◽  
Similar Reduction ◽  
Generalized Burgers Equation ◽  
Different Types ◽  
Self Similar ◽  
Parameter Ranges ◽  
Similar Solutions

The nonlinear ordinary differential equation resulting from the self-similar reduction of a generalized Burgers equation with nonlinear damping is studied in some detail. Assuming initial conditions at the origin we observe a wide variety of solutions – (positive) single hump, unbounded or those with a finite zero. The existence and nonexistence of positive bounded solutions with different types of decay (exponential or algebraic) to zero at infinity for specific parameter ranges are proved.

scholarly journals Development of singularities of the Skyrme model

2020 ◽  
Vol 17 (01) ◽  
pp. 61-73
Author(s):  
Michael McNulty
Keyword(s):  
Sigma Model ◽  
Blow Up ◽  
Strong Field ◽  
Skyrme Model ◽  
Nonlinear Sigma Model ◽  
Field Limit ◽  
Wave Maps ◽  
Self Similar ◽  
Similar Solutions ◽  
Strong Field Limit

The Skyrme model is a geometric field theory and a quasilinear modification of the Nonlinear Sigma Model (Wave Maps). In this paper, we study the development of singularities for the equivariant Skyrme Model, in the strong-field limit, where the restoration of scale invariance allows us to look for self-similar blow-up behavior. After introducing the Skyrme Model and reviewing what’s known about formation of singularities in equivariant Wave Maps, we prove the existence of smooth self-similar solutions to the [Formula: see text]-dimensional Skyrme Model in the strong-field limit, and use that to conclude that the solution to the corresponding Cauchy problem blows-up in finite time, starting from a particular class of everywhere smooth initial data.

scholarly journals Examples of non connective C *-algebras

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Anna Gąsior ◽  
Andrzej Szczepański
Keyword(s):  
Fixed Point ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Blow Up ◽  
Fractional Diffusion ◽  
Uniqueness Of Solutions ◽  
C Algebras ◽  
Space Fractional Diffusion Equation ◽  
Self Similar ◽  
Similar Solutions

Abstract This paper investigates the problem of the existence and uniqueness of solutions under the generalized self-similar forms to the space-fractional diffusion equation. Therefore, through applying the properties of Schauder’s and Banach’s fixed point theorems; we establish several results on the global existence and blow-up of generalized self-similar solutions to this equation.

scholarly journals Theoretical interpretation of fetch limited wind-drivensea observations

2005 ◽  
Vol 12 (6) ◽  
pp. 1011-1020 ◽  
Author(s):  
V. E. Zakharov
Keyword(s):  
Gravity Waves ◽  
Wave Breaking ◽  
Field Studies ◽  
The Self ◽  
Theoretical Interpretation ◽  
Simple Analytical Model ◽  
Surface Gravity Waves ◽  
Interaction Terms ◽  
Self Similar ◽  
Similar Solutions

Abstract. We show that the results of major fetch limited field studies of wind-generated surface gravity waves on deep water can be explained in the framework of simple analytical model. The spectra measured in these experiments are described by self-similar solutions of ``conservative" Hasselmann equation that includes only advective and nonlinear interaction terms. Interaction with the wind and dissipation due to the wave breaking indirectly defines parameters of the self-similar solutions.

scholarly journals ACCRETION AND PLASMA OUTFLOW FROM DISSIPATIONLESS DISCS

2010 ◽  
Vol 19 (03) ◽  
pp. 339-365 ◽  
Author(s):  
S. V. BOGOVALOV ◽  
S. R. KELNER
Keyword(s):  
Angular Momentum ◽  
Similar Case ◽  
Gravitational Energy ◽  
Accretion Disc ◽  
The Self ◽  
Low Viscosity ◽  
Variable Polarity ◽  
Self Similar ◽  
Similar Solutions ◽  
High Electric Conductivity

We consider the specific case of disc accretion for negligibly low viscosity and infinitely high electric conductivity. The key component in this model is the outflowing magnetized wind from the accretion disc, since this wind effectively carries away angular momentum of the accreting matter. Assuming magnetic field has variable polarity in the disc (to avoid magnetic flux and energy accumulation at the gravitational center), this leads to radiatively inefficient accretion of the disc matter onto the gravitational center. In such a case, the wind forms an outflow, which carries away all the energy and angular momentum of the accreted matter. Interestingly, in this framework, the basic properties of the outflow (as well as angular momentum and energy flux per particle in the outflow) do not depend on the structure of accretion disc. The self-similar solutions obtained prove the existence of such an accreting regime. In the self-similar case, the disc accretion rate (Ṁ) depends on the distance to the gravitational center, r, as [Formula: see text], where λ is the dimensionless Alfvenic radius. Thus, the outflow predominantly occurs from the very central part of the disc provided that λ ≫ 1 (it follows from the conservation of matter). The accretion/outflow mechanism provides transformation of the gravitational energy from the accreted matter into the energy of the outflowing wind with efficiency close to 100%. The flow velocity can essentially exceed the Kepler velocity at the site of the wind launch.

scholarly journals Self-Similar Blow-Up Solutions of the KPZ Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-4 ◽  
Author(s):  
Alexander Gladkov
Keyword(s):  
Asymptotic Behavior ◽  
Blow Up ◽  
Kpz Equation ◽  
Self Similar ◽  
The Kpz Equation ◽  
Similar Solutions

Self-similar blow-up solutions for the generalized deterministic KPZ equationut=uxx+|ux|qwithq>2are considered. The asymptotic behavior of self-similar solutions is studied.

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