Shock Formation for 2D Isentropic Euler Equations with Self-similar Variables

The author studies the 2D isentropic Euler equations with the ideal gas law.He exhibits a set of smooth initial data that give rise to shock formation at a single point near the planar symmetry.These solutions to the 2D isentropic Euler equations are associated with non-zero vorticity at the shock and have uniform-in-time-1/3-Holder bound.Moreover,these point shocks are of self-similar type and share the same profile,which is a solution to the 2D self-similar Burgers equation.The proof of the solutions,following the 3D construction of Buckmaster,Shkoller and Vicol(in 2023),is based on the stable 2D self-similar Burgers profile and the modulation method.

THE REGULAR SOLUTIONS OF THE ISENTROPIC EULER EQUATIONS WITH DEGENERATE LINEAR DAMPING

The regular solutions of the isentropic Euler equations with degenerate linear damping for a perfect gas are studied in this paper. And a critical degenerate linear damping coefficient is found, such that if the degenerate linear damping coefficient is larger than it and the gas lies in a compact domain initially, then the regular solution will blow up in finite time; if the degenerate linear damping coefficient is less than it, then under some hvpotheses on the initial data. the regular solution exists globally.

All Speed Scheme for the Low Mach Number Limit of the Isentropic Euler Equations

An all speed scheme for the Isentropic Euler equations is presented in thispaper. When the Mach number tends to zero, the compressible Euler equations converge to their incompressible counterpart, in which the density becomes a constant. Increasing approximation errors and severe stability constraints are the main difficultyin the low Mach regime. The key idea of our all speed scheme is the special semiimplicit time discretization, in which the low Mach number stiff term is divided intotwo parts, one being treated explicitly and the other one implicitly. Moreover, the fluxof the density equation is also treated implicitly and an elliptic type equation is derivedto obtain the density. In this way, the correct limit can be captured without requesting the mesh size and time step to be smaller than the Mach number. Compared withprevious semi-implicit methods [11,13,29], firstly, nonphysical oscillations can be suppressed by choosing proper parameter, besides, only a linear elliptic equation needs tobe solved implicitly which reduces much computational cost. We develop this semiimplicit time discretization in the framework of a first order Local Lax-Friedrichs (orRusanov) scheme and numerical tests are displayed to demonstrate its performances.

An All-Speed Asymptotic-Preserving Method for the Isentropic Euler and Navier-Stokes Equations

The computation of compressible flows becomesmore challengingwhen the Mach number has different orders of magnitude.When the Mach number is of order one,modern shock capturing methods are able to capture shocks and other complex structures with high numerical resolutions.However,if theMach number is small,the acoustic waves lead to stiffness in time and excessively large numerical viscosity,thus demanding much smaller time step and mesh size than normally needed for incompressible flow simulation.In this paper,we develop an all-speed asymptotic preserving(AP)numerical scheme for the compressible isentropic Euler and Navier-Stokes equations that is uniformly stable and accurate for all Mach numbers.Our idea is to split the system into two parts:one involves a slow,nonlinear and conservative hyperbolic system adequate for the use of modern shock capturing methods and the other a linear hyperbolic system which contains the stiff acoustic dynamics,to be solved implicitly.This implicit part is reformulated into a standard pressure Poisson projection system and thus possesses sufficient structure for efficient fast Fourier transform solution techniques.In the zero Mach number limit,the scheme automatically becomes a projection method-like incompressible solver.We present numerical results in one and two dimensions in both compressible and incompressible regimes.

Local Smooth Solutions to the 3-Dimensional Isentropic Relativistic Euler equations

The authors consider the local smooth solutions to the isentropic relativistic Euler equations in(3+1)-dimensional space-time for both non-vacuum and vacuum cases.The local existence is proved by symmetrizing the system and applying the FriedrichsLax-Kato theory of symmetric hyperbolic systems.For the non-vacuum case,according to Godunov,firstly a strictly convex entropy function is solved out,then a suitable symmetrizer to symmetrize the system is constructed.For the vacuum case,since the coefficient matrix blows-up near the vacuum,the authors use another symmetrization which is based on the generalized Riemann invariants and the normalized velocity.

Vanishing viscosity of isentropic Navier-Stokes equations for interacting shocks

We study the vanishing viscosity of the Navier-Stokes equations for interacting shocks. Given an entropy solution to p-system which consists of two different families of shocks interacting at some positive time,we show that such entropy solution is the vanishing viscosity limit of a family of global smooth solutions to the isentropic Navier-Stokes equations. The key point of the proofs is to derive the estimates separately before and after the interaction time and connect the incoming and outgoing viscous shock profiles.

High order semi-implicit weighted compact nonlinear scheme for the all-Mach isentropic Euler system

The computation of compressible flows at all Mach numbers is a very challenging problem.An efficient numerical method for solving this problem needs to have shock-capturing capability in the high Mach number regime,while it can deal with stiffness and accuracy in the low Mach number regime.This paper designs a high order semi-implicit weighted compact nonlinear scheme(WCNS)for the all-Mach isentropic Euler system of compressible gas dynamics.To avoid severe Courant-Friedrichs-Levy(CFL)restrictions for low Mach flows,the nonlinear fluxes in the Euler equations are split into stiff and non-stiff components.A third-order implicit-explicit(IMEX)method is used for the time discretization of the split components and a fifth-order WCNS is used for the spatial discretization of flux derivatives.The high order IMEX method is asymptotic preserving and asymptotically accurate in the zero Mach number limit.One-and two-dimensional numerical examples in both compressible and incompressible regimes are given to demonstrate the advantages of the designed IMEX WCNS.

Delta-shocks and vacuums in zero-pressure gas dynamics by the flux approximation

In this paper,firstly,by solving the Riemann problem of the zero-pressure flow in gas dynamics with a flux approximation,we construct parameterized delta-shock and constant density solutions,then we show that,as the flux perturbation vanishes,they converge to the delta-shock and vacuum state solutions of the zero-pressure flow,respectively.Secondly,we solve the Riemann problem of the Euler equations of isentropic gas dynamics with a double parameter flux approximation including pressure.Furthermore,we rigorously prove that,as the two-parameter flux perturbation vanishes,any Riemann solution containing two shock waves tends to a delta-shock solution to the zero-pressure flow;any Riemann solution containing two rarefaction waves tends to a two-contact-discontinuity solution to the zero-pressure flow and the nonvacuum intermediate state in between tends to a vacuum state.Finally,numerical results are given to present the formation processes of delta shock waves and vacuum states.