Solving a Singular Limit Problem Arising With Euler–Korteweg Dispersive Waves

We consider the singular limit problem for the Cauchy problem of the (Patlak–) Keller–Segel system of parabolic-parabolic type. The problem is considered in the uniformly local Lebesgue spaces and the singular limit problem as the relaxation parameter tau goes to infinity, the solution to the Keller–Segel equation converges to a solution to the drift-diffusion system in the strong uniformly local topology. For the proof, we follow the former result due to Kurokiba–Ogawa [20–22] and we establish maximal regularity for the heat equation over the uniformly local Lebesgue and Morrey spaces which are non-UMD Banach spaces and apply it for the strong convergence of the singular limit problem in the scaling critical local spaces.

The non-monotonicity of the local speed of sound in dense matter at baryon number densities much higher than the nuclear saturation density ($n_0 \approx 0.16\,$fm$^{-3}$) suggests the possible existence of a non-convex thermodynamics which will lead to a non-convex dynamics. Here, we explore the rich and complex dynamics that an equation of state (EoS) with non-convex regions in the pressure-density plane may develop as a result of genuinely relativistic effects, without a classical counterpart. To this end, we have introduced a phenomenological EoS, whose parameters can be restricted heeding to causality and thermodynamic stability constraints. This EoS shall be regarded as a toy-model with which we may mimic realistic (and far more complex) EoS of practical use in the realm of Relativistic Hydrodynamics.

Solution of the Euler equations on Cartesian grids

A mathematical model with a small parameter, which describes the hardening process of the binary tin–lead alloy, is investigated on the basis of nonlinear asymptotic analysis. A singular limit problem, namely an extended Stefan problem in the case of short relaxation time in the phase transformation zone, is derived. We prove the existence of an asymptotic solution with any accuracy on the time interval where the solution to the singular limit problem exists. The phase-separation interface is determined uniquely by three leading approximations. We also show that the stability of the separation interface depends on the so-called dissipation condition obtained for the solutions of the interface problem. Nonsymmetry of the surface tension tensor leads to a situation where the limit values of concentration distributions are in dependence on the geometry of the interface. This provokes the dispersion of the interface problem solutions on the part of the interface that not is tangent to the main crystallographic axis.

We deal with the numerical approximation of the complex structure in special relativistic hydrodynamics (SRHD) when the system is closed with a non-convex equation of state (EOS). We consider a recently introduced phenomenological EOS (Ibanez et al. in MNRAS 476:1100, 2018) that mimics the loss of classical behavior when the fluid enters into a non-convex—thermodynamically—region in the relativistic regime. We introduce a flux formulation to approximate the solution of Riemann problems in SRHD such that the non-classical dynamics is detected and well resolved. We also design a strategy to recover primitive variables based on iterative procedures and present a detailed analysis providing a sufficient condition to ensure convergence. We propose a set of Riemann problems in one and two dimensions including blast waves, colliding slabs and expanding slabs, illustrating the strong complex dynamics arising in non-convex SRHD.

Thermodynamic and Gasdynamic Aspects of a Boiling Liquid Expanding Vapour Explosion

We consider a singular limit problem of the Cauchy problem for the Patlak–Keller–Segel equation in a scaling critical function space. It is shown that a solution to the Patlak–Keller–Segel system in a scaling critical function space involving the class of bounded mean oscillations converges to a solution to the drift-diffusion system of parabolic-elliptic type (simplified Keller–Segel model) strongly as the relaxation time parameter tau rightarrow infty . For the proof, we show generalized maximal regularity for the heat equation in the homogeneous Besov spaces and the class of bounded mean oscillations and we utilize them systematically as well as the continuous embeddings between the interpolation spaces dot{B}^s_{q,sigma }({mathbb {R}}^n) and dot{F}^s_{q,sigma }({mathbb {R}}^n) for the proof of the singular limit. In particular, end-point maximal regularity in BMO and space time modified class introduced by Koch–Tataru is utilized in our proof.

A relaxation-projection method for compressible flows. Part I: The numerical equation of state for the Euler equations

The parameter identification problem considered herewith consists of identifying the measurable, bounded and strictly positive position - dependent conductivity from interior potential measurements in one spatial dimension. At least one solution is assumed to exist. Since conductivity satisfies a first order ordinary differential equation (ODE), uniqueness follows from stating either a regular or a singular initial value problem. Regular problems arise when a Cauchy datum is available at a point where the coefficient of a <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</sub> does not vanish. Singular problems are met when the potential is stationary, according to the conditions originally given by Kitamura and Nakagiri [SIAM J. Control and Optimiz., 15, (1977), pp 785 - 802] or when the {potential; source term} pair satisfies a non - local property known as self-identifiability. Potentials shall be piecewise differentiable in all cases. When stability is examined, potentials are chosen in a subset of W <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2,1</sup> ; the starting point is an ODE for the conductivity difference, the defect equation. Two distinct integration procedures are applied to the latter, according to whether uniqueness is due to a regular or a singular problem. Standard stability theory applies to the former and leads to L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> -estimates, provided Gronwall - Bellman's inequality is extended to measurable functions. Singular problems yield at most L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> - stability estimates. Some examples and counterexamples are provided. The results help in interpreting the performance of some known distributed parameter identification methods.

Eulerian droplet model: Delta-shock waves and solution of the Riemann problem

A numerical procedure has been developed for the computation of the steady, inviscid supersonic flow over aircraft configurations. The technique accounts for major nonlinear effects (shock waves, blunt leading edges) at low to moderate supersonic speeds. A fully implicit marching technique for the full potential equation is utilized in a stereographically projected, conformally mapped, spherical coordinate frame. Cross-flow planes are efficiently solved by type-dependen t relaxation techniques. Results are presented for several delta wing configurations and bodies of revolution, and are compared with existing experimental data, Euler's equations solutions, and results from linearized theories (panel methods). I. Introduction T HE goal of the present investigation is the development of a highly efficient and general numerical procedure for the computation of the steady, inviscid supersonic flow over aircraft configurations. In particular, this paper will concentrate on the development and verification of the method through application to isolated delta wings and bodies of revolution. Emphasis will be given to the prediction of nonlinear effects at imbedded shock waves and blunt leading edges in the low to moderate supersonic Mach number range. Existing computational techniques in this regime may be broadly characterized as panel methods and finite-difference methods. Panel methods, which have been demonstrated to be widely versatile and quite efficient, have reached a very advanced stage of development and utility for subsonic flows. Efforts are continuing to improve their capabilities at supersonic speeds (i.e., Ehlers et al. *). These methods address the solution to the first-order linear small-disturbance equation with various approximations for the boundary conditions. Finite-difference techniques for wing planforms in the supersonic regime concentrate on solutions to Euler's equations with differing treatments of shock waves, shock capturing2 or shock fitting,3'5 and a variety of methods for grid generation (i.e., Moretti5) and geometry input. The major drawbacks of these methods are associated with their large computational times which are due primarily to the fine grids necessary to resolve thin wings and the accompanying strict stability requirements. Other problem areas for the Euler solutions in the Mach number range considered here are due to subsonic Mach numbers in the axial marching direction, initial conditions at the apex of the wing or body, and vortical singularities of the leeward surface. In a recent paper by Mason and daForno6 the opportunities for performance gains through nonlinear aerodynamic effects are discussed including the impact on nonlinear computational analysis and design. A convincing case is made for the necessity of the inclusion of nonlinear effects particularly for wing leading edges and imbedded shock waves for the aerodynamic design and analysis of wings.

A number of papers[1–9] have appeared in recent years reporting numerical solutions of the Euler equations for flows past sharp edge delta wings with leading edge vortices. A schematic diagram of the problem under consideration is shown in Figure 1. Solutions have been compared with experiment [1,8], with incompressible panel method calculations [6], with other Euler equation solutions [8,5], and with Navier-Stokes calculations [9]. Although one cannot conclude that these comparisons show complete agreement, it is reasonable to say that they establish that finite volume calculations of the Euler equations appear to give reasonable predictions of the primary vortex generated by this configuration. A point of some concern in these papers [1,3,4] relates to the mechanism by which the Euler equations cause the flow to separate from the body and form a strong vortex which has associated with it a significant total pressure loss.

A relaxation model for the non-isothermal Navier-Stokes-Korteweg equations in confined domains

The high Weissenberg number limit of the UCM model and the Euler equations

A CFD decompression model for CO2 mixture and the influence of non-equilibrium phase transition