Parabolic Degeneracy Research Articles

Sonic-supersonic solutions for the two-dimensional steady compressible multiphase flow equations

This paper is concerned with the sonic-supersonic structures for the two-dimensional steady compressible inviscid multiphase flow equations. We construct a local classical supersonic solution near a given smooth sonic curve. This problem is originated from the transonic channel multiphase flows, which are one kind of the most important problems in mathematical fluid dynamics. In order to overcome the difficulties caused by the parabolic degeneracy near sonic and the multivariable dependence of pressure, we adopt the mixed variables of the pressure and angle functions and derive the characteristic decompositions of these quantities. In terms of the angle coordinate system, the multiphase flow equations can be transformed into a new degenerate hyperbolic system with an explicit singularity-regularity structure. We verify the convergence of the iterative sequence generated by the new system and then return the solution to the original physical variables. As a by-product, we obtain the existence of sonic-supersonic solutions for the steady full Euler equations with non-polytropic gases.

Degenerate Cauchy–Goursat problem for 2‐D steady isentropic Euler system with van der Waals gas

AbstractThis study concerns with the existence–uniqueness of local classical sonic‐supersonic solution to a degenerate Cauchy–Goursat problem that arises in transonic phenomena. The flow is governed by 2‐D steady isentropic Euler system with a polytropic van der Waals gas. The idea of characteristic decomposition has been used to convert the Euler system into a new system involving the angle variables . To overcome the parabolic degeneracy caused at the sonic curve, the partial hodograph transformation and a variety of dependent–independent variables have been introduced to transform the nonlinear system into a linear one with explicit singularity–regularity structure. The uniform convergence of the sequences has been discussed by employing the mathematical induction. Eventually, the inversion of the solution from partial hodograph plane to the original plane has been established.

The degenerate hyperbolic problem for the two‐dimensional nonlinear wave system

This paper constructs a local classical solution of degenerate hyperbolic problem for the two-dimensional nonlinear wave system. To deal with the parabolic degeneracy, we introduce a new set of coordinates to transform the nonlinear wave system to a new system that has explicitly singularity-regularity structures. By constructing a weighted metric space, we establish the existence of solution for the new system. Returning the solution to the original variables, we obtain the existence of the classical solution for the nonlinear wave system with degenerate boundary value problem.

The Nonlocal Problem for a Hyperbolic Equation with a Parabolic Degeneracy

The Nonlocal Problem for a Hyperbolic Equation with a Parabolic Degeneracy

Very high-order asymptotic-preserving schemes for hyperbolic systems of conservation laws with parabolic degeneracy on unstructured meshes

In this paper, we consider the numerical approximation of hyperbolic systems of conservation laws with stiff source terms and parabolic degeneracy in the asymptotic limit. We are more precisely interested in the design of high-order asymptotic-preserving schemes on unstructured meshes. Our approach is based on a very simple modification of the numerical flux associated with the usual HLL scheme and boils down to a sharp control of the underlying numerical diffusion. The strategy allows to capture the correct asymptotic parabolic behaviour and to preserve the high-order accuracy also in the asymptotic limit. Numerical experiments are proposed to illustrate these properties.

Sonic-Supersonic Solutions for the Two-Dimensional Steady Full Euler Equations

This paper focuses on the structure of classical sonic-supersonic solutions near sonic curves for the two-dimensional full Euler equations in gas dynamics. In order to deal with the parabolic degeneracy near the sonic curve, a novel set of dependent and independent variables are introduced to transform the Euler equations into a new system of governing equations which displays a clear regularity-singularity structure. With the help of technical characteristic decompositions, the existence of a local smooth solution for the new system is first established in a weighted metric space by using the iteration method and then expressed in terms of the original physical variables. This is the first time to construct a classical sonic-supersonic solution near a sonic curve for the full Euler equations.

On a degenerate boundary value problem to the two-dimensional self-similar nonlinear wave system

This paper focuses on a degenerate boundary value problem arising from the study of the two-dimensional Riemann problem to the nonlinear wave system. In order to deal with the parabolic degeneracy, we introduce a partial hodograph transformation to transform the nonlinear wave system into a new system, which displays a clear regularity–singularity structure. The local existence of classical solutions for the new system is established in a weighted metric space. Returning the solution to the original variables, we obtain the existence of classical solutions to the degenerate boundary value problem for the nonlinear wave system.

Pointwise stabilization of the Poisson integral for the diffusion type equations with inertia

In this paper we consider the pointwise stabilization of the Poisson integral for the diffusion type equations with inertia in the case of finite number of parabolic degeneracy groups. We establish necessary and sufficient conditions of this stabilization for a class of bounded measurable initial functions.

Tricomi type equations with terms of lower order

In this paper, we study Tricomi type equations with terms of lower order in the plane. Solutions are required to satisfy conditions on one part of a characteristic in the hyperbolic region of the considered domain and on some parts of the line of parabolic degeneracy. We prove the uniqueness of solutions in suitable spaces of generalised solutions by application of extremum principle. The proof of the existence is based on the theory of singular integral equations, Wiener-Hopf equations and Fredholm integral equations.

Gas-cushioned droplet impacts with a thin layer of porous media

The pre-impact gas cushioning behaviour of a droplet approaching touchdown onto a thin layer of porous substrate is investigated. Although the model is applicable to droplet impacts with any porous substrate of limited height, a thin layer of porous medium is used as an idealized approximation of a regular array of pillars, which are frequently used to produced superhydrophobic- and superhydrophilic-textured surfaces. Bubble entrainment is predicted across a range of permeabilities and substrate heights, as a result of a gas pressure build-up in the viscous-gas squeeze film decelerating the droplet free-surface immediately below the centre of the droplet. For a droplet of water of radius 1 mm and impact approach speed 0.5 m s $$^{-1}$$ , the change from a flat rigid impermeable plate to a porous substrate of height $$5~\upmu $$ m and permeability $$2.5~\upmu $$ m $$^2$$ reduces the initial horizontal extent of the trapped air pocket by $$48~\%$$ , as the porous substrate provides additional pathways through which the gas can escape. Further increases in either the substrate permeability or substrate height can entirely eliminate the formation of a trapped gas pocket in the initial touchdown phase, with the droplet then initially hitting the top surface of the porous media at a single point. Droplet impacts with a porous substrate are qualitatively compared to droplet impacts with a rough impermeable surface, which provides a second approximation for a textured surface. This indicates that only small pillars can be successfully modelled by the porous media approximation. The effect of surface tension on gas-cushioned droplet impacts with porous substrates is also investigated. In contrast to the numerical predictions of a droplet free-surface above flat plate, when a porous substrate is included, the droplet free-surface touches down in finite time. Mathematically, this is due to the regularization of the parabolic degeneracy associated with the small gas-film-height limit the gas squeeze film equation, by non-zero substrate permeability and height, and physically suggests that the level of surface roughness is a critical parameter in determining the initial touchdown characteristics.

A problem with conditions given on inner characteristics and on the line of degeneracy for a mixed-type equation with singular coefficients

In an unbounded domain, we consider a problem with conditions given on inner characteristics in a hyperbolic part of the considered domain and on some parts of the line of parabolic degeneracy. We prove the unique solvability of the mentioned problem with the help of the extremum principle. The proof of solvability is based on the theory of singular integral equations, Wiener-Hopf equations and Fredholm integral equations.

О задаче с обобщeнными операторами дробного дифференцирования для уравнения смешанного типа с двумя линиями вырождения

The nonlocal problem for mixed-type equation with perpendicular lines of degeneracy is investigated for the case when the Dirichlet condition is given on the elliptic boundary, and the generalized derivatives of the solution values on the characteristics are pointwise related to the solution and its normal derivatives values on the lines of a parabolic degeneracy in its hyperbolic parts.

Degenerate Goursat-type boundary value problems arising from the study of two-dimensional isothermal Euler equations

As a ladder step to study transonic problems, we investigate two families of degenerate Goursat-type boundary value problems arising from the two-dimensional pseudo-steady isothermal Euler equations. The first family is about the genuinely two-dimensional full expansion of gas into a vacuum with a wedge; the other is a semi-hyperbolic patch that starts on sonic curves and ends at transonic shocks. Both the vacuum and the sonic sets cause parabolic degeneracy that results in substantial difficulties such as singularities of solutions and uniform a priori estimates. Main ingredients in this study are various characteristic decompositions for the pseudo-steady Euler equations in order to obtain necessary a priori estimates. Furthermore, we are able to verify the uniform Holder continuity of solutions with exponent 1/2 for the gas expansion problem and up to 2/7 for the semi-hyperbolic problem.

Existence of solutions to the Riemann problem for a class of hyperbolic conservation laws exhibiting a parabolic degeneracy

Existence of solutions to the Riemann problem for a class of hyperbolic conservation laws exhibiting a parabolic degeneracy

On the influence of damping in hyperbolic equations with parabolic degeneracy

This paper examines the effect of damping on a nonstrictly hyperbolic 2 × 2 2\times 2 system. It is shown that the growth of singularities is not restricted as in the strictly hyperbolic case where dissipation can be strong enough to preserve the smoothness of small solutions globally in time. Here, irrespective of the stabilizing properties of damping, solutions are found to break down in finite time on a line where two eigenvalues coincide in state space.

Oblique derivative problem for quasilinear mixed equations with parabolic degeneracy in multiply connected domains

This article deals with the oblique derivative problem for second order quasilinear equations of mixed type in multiply connected domains, which includes the Tricomi problem of the Chaplygin equation in gas dynamics as a special case. We first give the representation of solutions of the boundary value problem for the equations, and then prove the uniqueness and existence of solutions for the problem by a new method, using complex functions in the domain of ellipticity and hyperbolic complex functions in the domain where the equation is hyperbolic.

Interacting fermions on the honeycomb bilayer: From weak to strong coupling

Many-body instabilities of the half-filled honeycomb bilayer are studied using weak-coupling renormalization group (RG) as well as strong-coupling expansion. For spinless fermions and assuming parabolic degeneracy, there are four independent four-fermion contact couplings. While the dominant instability depends on the microscopic values of the couplings, the broken symmetry state is typically a gapped insulator with either broken inversion symmetry or broken time-reversal symmetry, with a quantized anomalous Hall effect. Under certain conditions, the dominant instability may appear in the particle-particle (pairing) channel. For some nongeneric fine-tuned initial conditions, weak-coupling RG trajectories flow into the noninteracting fixed point, although generally we find runaway flows which we associate with ordering tendencies. Additionally, a tight-binding model with nearest-neighbor hopping and nearest-neighbor repulsion is studied in weak and strong couplings and in each regime a gapped phase with inversion symmetry breaking is found. In the strong-coupling limit, the ground-state wave function is constructed for vanishing in-plane hopping but finite interplane hopping, which explicitly displays the broken inversion symmetry and a finite difference between the number of particles on the two layers. Finally, we discuss the spin-1/2 case and use Fierz identities to show that the number of independent four-fermion contact couplings is 9. The corresponding RG equations in the spin-1/2 case are also presented, and used to show that, just as in strong coupling, the most dominant weak-coupling instability of the repulsive Hubbard model (at half filling) is an antiferromagnet.

Oblique derivative problems for second order nonlinear mixed equations with degenerate line

The present article deals with oblique derivative problems for some nonlinear mixed equations with parabolic degeneracy, which include the Tricomi problem as a special case. First, the formulation of the problems for the equations is given; next, the representation and estimates of solutions for the above problems are obtained; finally, the existence of solutions for the problems is proved by the successive iteration and the compactness principle of solutions of the problems. In this article, the author uses the complex method, namely, the complex functions in the elliptic domain and the hyperbolic complex functions in hyperbolic domain are used.

The Tricomi problem for second order systems of mixed type with parabolic degeneracy

The present article deals with the Tricomi problem for systems of second order equations of mixed type with parabolic degeneracy. First the formulation of the problem for the systems is given, next the representations and estimates of solutions for the above problem are obtained, finally the existence of solutions for the problem is proved by the successive iteration and the method of parameter extension.

The mixed boundary‐value problem for second order elliptic equations with degenerate curve on the sides of an angle

AbstractIn [1, 2, 3, 4, 6], the authors posed and discussed some boundary‐value problems of second order elliptic equations with parabolic degeneracy. In [5], the authors posed and discussed some boundary‐value problems of second order mixed equations with degenerate curve on the sides of an angle, but the coefficients of the equations possess strong restrictions. The present paper deals with the mixed problem for elliptic equations with degenerate curve on the sides of an angle, where the coefficients satisfy general conditions. We first give the formulation of the problem and estimates of solutions of the problem for the equations, and then prove the existence of solutions for the above problem by the Leray–Schauder theorem. In this paper, we use the complex analytic method, namely we first introduce the new notation (2.1) below and reduce the degenerate elliptic equations of second order to the corresponding problem for degenerate elliptic complex equations of first order, afterwards the above problem of second order degenerate elliptic equations can be solved. The results in this paper will be used in a subsequent paper to handle the Tricomi problem of second order equations of mixed type with degenerate curve on the sides of an angle. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)