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AbstractThe phenomena of concentration and cavitation in the Riemann solutions of the relativistic Euler equations with non-ideal isothermal dusty gas are considered. The Riemann problem is solved first, and the solutions consisting of rarefaction waves and shock waves are obtained. Then, by the vanishing pressure approach, we investigate the limiting behaviors of Riemann solutions and observe the concentration and cavitation phenomena. It is rigorously proved that as the dusty gas pressure vanishes with double parameters, the solution containing two shock waves and two rarefaction waves respectively converges to the delta-shock wave and vacuum state of the pressureless relativistic Euler equations. [ABSTRACT FROM AUTHOR]

A novel approach for the stabilization of the Discontinuous Galerkin method based on the Dafermos entropy rate crition is presented. First, estimates for the maximal possible entropy dissipation rate of a weak solution are derived. Second, families of conservative Hilbert–Schmidt operators are identified to dissipate entropy. Steering these operators using the bounds on the entropy dissipation results in high-order accurate shock-capturing DG schemes for the one-dimensional Euler equations, satisfying the entropy rate criterion and an entropy inequality. Other testcases include the one-dimensional Buckley–Leverett equation. [ABSTRACT FROM AUTHOR]

In this paper we study the compressible magnetohydrodynamics equations in three dimensions, which offer a good model for plasmas. Formation of singularity for C 1 -solution in finite time is proved with axisymmetric initial data. The key observation is that the magnetic force term admits a good structure with axisymmetric assumption. [ABSTRACT FROM AUTHOR]

In this paper, the design and analysis of high order accurate IMEX finite volume schemes for the compressible Euler–Poisson (EP) equations in the quasineutral limit is presented. As the quasineutral limit is singular for the governing equations, the time discretisation is tantamount to achieving an accurate numerical method. To this end, the EP system is viewed as a differential algebraic equation system (DAEs) via static condensation. As a consequence of this vantage point, high order linearly semi-implicit (SI) time discretisation are realised by employing a novel combination of the direct approach used for implicit discretisation of DAEs and, two different classes of IMEX-RK schemes: the additive and the multiplicative. For both the time discretisation strategies, in order to account for rapid plasma oscillations in quasineutral regimes, the nonlinear Euler fluxes are split into two different combinations of stiff and non-stiff components. The high order scheme resulting from the additive approach is designated as a classical scheme while the one generated by the multiplicative approach possesses the asymptotic preserving (AP) property. Time discretisations for the classical and the AP schemes are performed by standard IMEX-RK and SI-IMEX-RK methods, respectively so that the stiff terms are treated implicitly and the non-stiff ones explicitly. In order to discretise in space a Rusanov-type central flux is used for the non-stiff part, and simple central differencing for the stiff part. Results of numerical experiments are presented, which confirm that the high order schemes based on the SI-IMEX-RK time discretisation achieve uniform second order convergence with respect to the Debye length and are AP in the quasineutral limit. [ABSTRACT FROM AUTHOR]

This work revisits a recent finding by the first author concerning the local convergence of a regularized scalar conservation law. We significantly improve the original statement by establishing a global convergence result within the Lebesgue spaces L loc ∞ (R + ; L p (R)) , for any p ∈ [ 1 , ∞) , as the regularization parameter ℓ approaches zero. Notably, we demonstrate that this stability result is accompanied by a quantifiable rate of convergence. A key insight in our proof lies in the observation that the fluctuations of the solutions remain under control in low regularity spaces, allowing for a potential quantification of their behavior in the limit as ℓ → 0 . This is achieved through a careful asymptotic analysis of the perturbative terms in the regularized equation, which, in our view, constitutes a pivotal contribution to the core findings of this paper. [ABSTRACT FROM AUTHOR]

This paper concerns smooth transonic flows with nonzero vorticity in De Laval nozzles for a quasi two dimensional steady Euler flow model which is a generalization of the classical quasi one dimensional model. First we examine the existence and uniqueness of smooth transonic flows to the quasi one-dimensional model, which start from a subsonic state at the entrance and accelerate to reach a sonic state at the throat and then become supersonic are proved by a reduction of degeneracy of the velocity near the sonic point and the implicit function theorem. These flows can have positive or zero acceleration at their sonic points and the degeneracy types near the sonic point are classified precisely. We then establish the structural stability of the smooth one dimensional transonic flow with positive acceleration at the sonic point for the quasi two dimensional steady Euler flow model under small perturbations of suitable boundary conditions, which yields the existence and uniqueness of a class of smooth transonic flows with nonzero vorticity and positive acceleration to the quasi two dimensional model. The positive acceleration of the one dimensional transonic solutions plays an important role in searching for an appropriate multiplier for the linearized second order mixed type equations. A deformation-curl decomposition for the quasi two dimensional model is utilized to deal with the transonic flows with nonzero vorticity. [ABSTRACT FROM AUTHOR]

A Riemann problem for the conservation law u t + [ ϕ (u) ] x = k H (x - v t) , where x, t, k, v and u = u (x , t) are real numbers, is studied with the goal of getting singular solutions in a convenient space of distributions that contains delta shock waves. Here ϕ stands for an entire function taking real values on the real axis and H represents the Heaviside function. When u is seen as a density of matter some surprises may appear such as the creation of matter from a vacuum state. In a particular case, as the time goes on, such a matter grows continuously, running away from any spatial bounded region, what can be viewed as a unidimensional model of universe. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Riemann problem and wave interactions for a 2 × 2 non-strictly hyperbolic system of conservation laws with non-convex flux. Riemann problem can not be solved always in the usual class of elementary waves and the solution develops δ -waves. We construct the exact solution to the Riemann problem for the governing system and discuss all possible cases of wave interactions involving δ -wave. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the Riemann problem and the perturbed Riemann problem for the Chaplygin pressure Aw–Rascle model with Coulomb-like friction, which can also be seen as the nonsymmetric Keyfitz–Kranzer system with Chaplygin pressure and Coulomb-like friction. For the Riemann problem, we show that it explicitly exhibits two kinds of different structures and the delta shock wave appears in some certain situations. The generalized Rankine–Hugoniot conditions of the delta shock wave are established and the exact position, propagation speed and strength of the delta shock wave are given explicitly. Unlike the homogeneous case, it is shown that the Coulomb-like friction term makes contact discontinuities and the delta shock wave bend into parabolic shapes and the Riemann solutions are not self-similar anymore. For the perturbed Riemann problem with delta initial data, not only the delta shock wave but also the delta contact discontinuity are found in solutions and the friction term makes them bent. Under the generalized Rankine–Hugoniot conditions and the entropy condition, by taking variable substitution, we constructively obtain the global existence of generalized solutions which explicitly exhibit four kinds of different structures. The results in this paper yield a way of studying the wave interaction involving the delta shock wave for conservation laws with source terms and will give us some insights into the research on the Chaplygin pressure Aw–Rascle model, the pressureless Euler equations and the Chaplygin Euler equations with various kinds of source terms. [ABSTRACT FROM AUTHOR]

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 -Hölder 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. [ABSTRACT FROM AUTHOR]

We study approximate solutions to a hyperbolic system of conservation laws, constructed by a backward Euler scheme, where time is discretized while space is still described by a continuous variable x ∈ R . We prove the global existence and uniqueness of these approximate solutions, and the invariance of suitable subdomains. Furthermore, given a left and a right state u l , u r connected by an entropy-admissible shock, we construct a traveling wave profile for the backward Euler scheme connecting these two asymptotic states in two main cases. Namely: (1) a scalar conservation law, where the jump u l - u r can be arbitrarily large, and (2) a strictly hyperbolic system, assuming that the jump u l - u r occurs in a genuinely nonlinear family and is sufficiently small. [ABSTRACT FROM AUTHOR]

The paper contains a solution to the Riemann problem for the generalized Chaplygin gas with a general source term defined by the Cargo–LeRoux potential. Besides the classical waves (rarefaction waves, shocks, and contact discontinuities), we have used the shadow waves coupled with the rarefaction ones to completely solve the above problem. [ABSTRACT FROM AUTHOR]

Both propagation and collision behavior of δ ′ waves in a model of three partial differential equations are investigated. These are Cauchy problems for this model with initial values involving the first-order derivative of Dirac measure. Based on an α -solution concept defined in the framework of multiplication of distributions, we obtain a unique α -solution for the propagation process of an δ ′ wave. This α -solution shows rigorously the location, velocity and strengths of the δ ′ wave. Besides, with the help of this result, we also derive an α -solution for the collision behavior of two δ ′ waves. Such α -solution not only reveals, respectively, the location, speed and strengths of each one of the two δ ′ waves before their interaction, but also describes the location, velocity and strengths of a new emerged δ ′ wave after their collision. Moreover, this work can be applied to investigate the Cauchy problems for a system of partial differential equations with general initial values involving the derivative of the Dirac measure. [ABSTRACT FROM AUTHOR]

This paper is concerned with Riemann problem for a class of non-strictly hyperbolic systems of conservation laws. The two kinds of Riemann solutions including vacuum and delta shock wave are constructed. Under the generalized Rankine-Hugoniot relation and entropy condition, we establish the existence and uniqueness of delta shock wave solutions. Furthermore, by studying the interactions among of the delta shock wave and vacuum as well as contact discontinuity, the Riemann solutions with four kinds of different structures are obtained. Additionally, the stability of the Riemann solutions is obtained under certain perturbation of the initial data. [ABSTRACT FROM AUTHOR]

In this paper, we have considered the van der Waals equation of state to study the one-dimensional spherically symmetric self-gravitating interstellar gas clouds. The transport equation for the acceleration waves is obtained. The evolution and propagation of the characteristic shock are considered to study its interaction with the acceleration wave. The amplitudes of the reflected and the transmitted waves are also determined. The effect of non-ideal parameter on the amplitude of the acceleration waves is discussed in detail. [ABSTRACT FROM AUTHOR]

The complete set of exact solutions to the Riemann problem for a novel second-order continuous macroscopic traffic flow model proposed by Hwang and Yu (J Comput Phys 350:927–950, 2017) is constructively solved in explicit forms by choosing the specific driving function of momentum. Especially, a hyperbolic composite wave is found in certain Riemann solution under the suitable initial condition, where a delta contact discontinuity is attached on the head of a rarefaction wave. Moreover, the asymptotic behavior of different Riemann solutions is explored and discussed, respectively, to analyze the influence of the two perturbed parameters in this model comprehensively. Additionally, the offered numerical experiments are well identical with our theoretic results. [ABSTRACT FROM AUTHOR]

This article discusses the Riemann problem for the Chaplygin gas Euler equations that include the presence of two source terms. By means of variable substitution, two kinds of non-self-similar Riemann solutions involving delta-shock are constructed explicitly. For the delta-shock, the generalized Rankine–Hugoniot relations and the over-compressive entropy condition are clarified. Moreover, the position, propagation speed and strength of the delta-shock are given explicitly. It is discovered that the position of the delta-shock is a combination of an exponential function and a linear function, and the weight of the delta-shock is an exponential function of the time. Interestingly, even when the delta-shock is a straight line, the weight of the delta-shock is no longer a linear function of the time t. In addition, it is proved that the Riemann solutions converge to the corresponding ones of Chaplygin gas Euler equations with friction as k drops to zero, and the Riemann solutions converge to the corresponding ones of Chaplygin gas Euler equations as k and β tend to zero simultaneously. Furthermore, it is also shown that the limits of Riemann solutions are just the Riemann solutions to the transport equations with same source terms as the Chaplygin gas pressure falls to zero. [ABSTRACT FROM AUTHOR]

We continue our previous study on the sub-shock formation in a binary mixture of gases (Ruggeri in Phys. Fluids 34: 066116, 2022) by considering the dissipation due to the bulk viscosity in both constituents. We use the model of a mixture of gases proposed by T. Arima, T. Ruggeri, M. Sugiyama, and S. Taniguchi (Arima in Rend. Lincei Mat. Appl. 28: 495, 2017). For prescribed values of the mass ratio of the constituents and the degrees of freedom of molecules, we classify the regions depending on the concentration and the Mach number for which the sub-shock may exist inside the shock profile of one or both constituents or not. Furthermore, we perform for some mixtures of gas numerical calculations of the profile, showing the role of the dissipation with respect to the Eulerian gases. As we expect, the shock profile is more regularized, and in the case that it exists sub-shocks, the amplitude of the sub-shock is reduced compared to those of non-dissipative gases. [ABSTRACT FROM AUTHOR]

Introducing a logarithmic pressure, we analyze the phenomenon of concentration and the formation of delta-shocks for the generalized Chaplygin gas dynamics. We first solve the Riemann problem for the logarithmic perturbed model and construct the solutions with four kinds of structures R 1 + R 2 , R 1 + S 2 , S 1 + R 2 and S 1 + S 2 . Then it is shown that when the logarithmic pressure vanishes, the limits of the Riemann solutions for the logarithmic perturbed model are just these of the generalized Chaplygin gas dynamics. In particular, when the initial data satisfy some certain conditions, the S 1 + S 2 solution of the logarithmic perturbed model tends to the delta-shock solution of the generalized Chaplygin gas dynamics. Finally, some numerical results exhibit the process of formation of delta-shocks. [ABSTRACT FROM AUTHOR]

Shock waves propagating through an ideal radiating gas containing solid dust particles of arbitrary strength are studied in this work. To analyze shock front kinematics, an infinite set of transport equations determining shock strength and induced discontinuities is obtained. A truncation approach of the infinite system of transport equations yields an efficient shock propagation system of finite ordinary differential equations. The analysis appropriately reflects the nonlinear steepening effects of the flow behind shock fronts due to the dynamic interaction between shock fronts and the flow behind them. The effects due to dust parameters and radiation on shock propagation are discussed with the help of a graphical representation. The decay laws for weak shocks in a non-radiating gas are precisely recovered by the second-order truncation approximation. The characteristic rule and the first- and second-order approximations are compared for shock waves of arbitrary strength. [ABSTRACT FROM AUTHOR]

The accurate Riemann solutions in fully explicit forms are obtained for the one-dimensional inviscid, compressible, isothermal liquid–gas two-phase flow model of drift-flux type under the action of the unique external force of gravity for a pipeline placed on an inclined plane. It is also shown that the Riemann solution involves either a curved delta shock wave or the vacuum state for the pressureless case. Moreover, the asymptotic limits of Riemann solutions are investigated in detail by letting the pressure drop to zero, in which the formation of curved delta shock wave is obtained from the curved Riemann solution made up of backward shock wave, middle contact discontinuity and forward shock as well as the formation of vacuum state is also achieved from the curved Riemann solution consisting of backward rarefaction wave, middle contact discontinuity and forward rarefaction wave. In addition, some representative numerical results are offered to confirm the formation of delta shock wave and vacuum state. [ABSTRACT FROM AUTHOR]

This paper considers the one-dimensional model of heat conduction in solids at low temperature, the so called phonon-Bose model. The nonlinear model consists of a conservation equation for the energy density e and the heat flux Q with ∣Q∣ < e. We present a simple and accurate class of finite volume schemes for numerical simulation of heat flow in arteries. This scheme consists of predictor and corrector steps, the predictor step contains a parameter of control of the numerical diffusion of the scheme, which modulate by using limiter theory and Riemann invariant, the corrector step recovers the balance conservation equation, the scheme can compute the numerical flux corresponding the real state of solution without relying on Riemann problem solvers and it can thus be turned to order 1 in the regions where the flow has a strong variation and to order 2 in the regions where the flow is regular. The numerical test cases demonstrate high resolution of the proposed finite volume scheme (modified Rusanov) and confirm its capability to provide accurate simulations for heat flow under flow regimes with strong shocks. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the upper bound of the lifespan estimate for the compressible Euler system with time-dependent damping and small initial perturbations. We employ some techniques from the blow-up study of nonlinear wave equations. The novelty consists in the introduction of tools from the Orlicz spaces theory to handle the nonlinear term emerging from the pressure p ≡ p (ρ) , which admits different asymptotic behavior for large and small values of ρ - 1 , being ρ the density. Hence, we can establish, in dimensions n ∈ { 2 , 3 } , unified upper bounds of the lifespan estimate depending only on the dimension n and on the damping strength and independent of the adiabatic index γ > 1 . We conjecture our results to be optimal. The method employed here not only improves the known upper bounds of the lifespan for n ∈ { 2 , 3 } , but has potential application in the study of related problems. [ABSTRACT FROM AUTHOR]

An asymptotic preserving and energy stable scheme for the barotropic Euler system under the low Mach number scaling is designed and analysed. A velocity shift proportional to the pressure gradient is introduced in the convective fluxes, which leads to the dissipation of mechanical energy and the entropy stability at all Mach numbers. The resolution of the semi-implicit in time and upwind finite volume in space fully-discrete scheme involves two steps: the solution of an elliptic problem for the density and an explicit evaluation for the velocity. The proposed scheme possesses several physically relevant attributes, such as the positivity of density, the entropy stability and the consistency with the weak formulation of the continuous Euler system. The AP property of the scheme, i.e. the boundedness of the mesh parameters with respect to the Mach number and its consistency with the incompressible limit system, is shown rigorously. The results of extensive case studies are presented to substantiate the robustness and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

The Riemann problem for a pressureless hydrodynamic model is solved explicitly, whose solution is either a contact-vacuum-contact wave or a delta shock wave to connect the left and right constant states. Moreover, the global perturbed Riemann solutions for this model are also constructed by investigating carefully all the presentive delta shock interaction problems when the initial condition is made up of three piecewise constant states. Finally, the stability of Riemann solutions is also proved by letting the perturbed parameter tend to zero under this specially designated small perturbation of this step-like initial condition. [ABSTRACT FROM AUTHOR]

The solutions of the Riemann problem for the isentropic Euler equations with the specific three independent constitutive pressure laws are obtained constructively for four different structures. The formation of delta shock wave and the combination between two contact discontinuities in the Riemann solutions for the isentropic Chaplygin Euler equations are explored in detail by letting the isothermal and logarithmic-corrected equation of state tend to zero and simultaneously remaining the Chaplygin gas equation of state unchanged in the total pressure term. In addition, the numerical experiments are carried out to confirm the delta shock formation. [ABSTRACT FROM AUTHOR]

In this article, we consider the Riemann problem with arbitrary initial data for one-dimensional blood flow equations in the arterial circulation. Here, we establish the existence of the self-similar solution to the Riemann problem by limiting viscosity approach. We convert the Riemann problem to a boundary value problem by adding a suitable viscosity term and establish the existence of the solution. Finally, we construct the existence of the solution to this Riemann problem in the presence of vacuum state. [ABSTRACT FROM AUTHOR]

In this article, we study the delta shock wave for zero-pressure gasdynamics system with energy conservation laws in the frame of α -solutions defined in the setting of distributional products. By reformulating the system, we construct within a convenient space of distributions, all solutions which include discontinuous solutions and Dirac delta measures. We also establish the generalized Rankine–Hugoniot jump conditions for delta shock waves. The α -solutions which we constructed coincide with the solution obtained through different methods. [ABSTRACT FROM AUTHOR]

In this paper, long-time behaviours of classical solutions to relativistic Euler–Poisson equations in radial symmetry (REPE) are investigated. First, we establish the finite propagation speed property of the REPE system and the propagation speed σ is found to be σ : = 8 π M R (1 + e / c 2) + c 2 , where all constants are given parameters or initial data to the system. Subsequently, we show that if the initial functional ∫ 0 R v (0 , r) d r is sufficiently large, then the classical solutions of the REPE will blow up on finite time, where v(0, r) is the initial velocity component of REPE and R is the radius of support of the initial mass-energy density. [ABSTRACT FROM AUTHOR]

We investigate the global structures of the non-selfsimilar solutions for n-dimensional (n-D) non-homogeneous Burgers equation, in which the initial data has two different constant states, which are separated by a (n − 1)-dimensional sphere. We first obtain the expressions of n-D shock waves and rarefaction waves emitting from the initial discontinuity. Then, by estimating the new kind of interactions of the related elementary waves, we obtain the global structures of the non-selfsimilar solutions, in which ingenious techniques are proposed to construct the n-D shock waves. The asymptotic behaviors with geometric structures are also proved. [ABSTRACT FROM AUTHOR]

It is interesting and challenging to study conservation laws with discontinuous source terms and explore how the delta shock wave is influenced by the discontinuous source term. However, so far, few results have been obtained about it. In this paper, the Riemann problem for the pressureless Euler equations with a discontinuous source term is considered. The delta shock wave solution is obtained by combining the generalized Rankine-Hugoniot conditions, together with the method of characteristics for various kinds of different situations, and the impact of the discontinuous source term on the delta shock front are precisely illustrated. Moreover, during the process of constructing the Riemann solution, some interesting phenomena are also observed, such as the disappearance of the delta shock wave and the occurrence of the vacuum state, etc. [ABSTRACT FROM AUTHOR]

In this paper travelling waves with distributional profiles for the Camassa–Holm equation are studied. Using a product of distributions a new solution concept is introduced which extends the classical one. As a consequence, necessary and sufficient conditions for the propagation of a distributional profile are established and we present examples with discontinuous solutions, measures and even distributions that are not measures. One of these examples may be interpreted as a simple model for a "tsunami" in the setting of shallow water theory. We also prove that, under natural assumptions, profiles belonging to the Sobolev space H loc 1 (R) usually considered in the classical weak formulation can be seen as particular cases of our distributional solution concept. [ABSTRACT FROM AUTHOR]

We study the local-in-time well-posedness for an interface that separates an anisotropic plasma from a vacuum. The plasma flow is governed by the ideal Chew–Goldberger–Low (CGL) equations, which are the simplest collisionless fluid model with anisotropic pressure. The vacuum magnetic and electric fields are supposed to satisfy the pre-Maxwell equations. The plasma and vacuum magnetic fields are tangential to the interface. This represents a nonlinear hyperbolic-elliptic coupled problem with a characteristic free boundary. By a suitable symmetrization of the linearized CGL equations, we reduce the linearized free boundary problem to a problem analogous to that in isotropic magnetohydrodynamics (MHD). This enables us to prove the local existence and uniqueness of solutions to the nonlinear free boundary problem under the same non-collinearity condition for the plasma and vacuum magnetic fields on the initial interface required by Secchi and Trakhinin (Nonlinearity 27:105–169, 2014) in isotropic MHD. [ABSTRACT FROM AUTHOR]

In this paper, we develop new second-order low-dissipation central-upwind (LDCU) schemes for hyperbolic systems of conservation laws. Like all of the Godunov-type schemes, the proposed LDCU schemes are developed in three steps: reconstruction, evolution, and projection. A major novelty of our approach is in the projection step, which is based on a subcell resolution and designed to sharper approximate contact waves while ensuring a non-oscillatory property of the projected solution. In order to achieve this goal, we take into account properties of the contact waves. We design the LDCU schemes for both the one- and two-dimensional Euler equations of gas dynamics. The new schemes are tested on a variety of numerical examples. The obtained results clearly demonstrate that the proposed LDCU schemes contain substantially smaller amount of numerical dissipation and achieve higher resolution compared with their existing counterparts. [ABSTRACT FROM AUTHOR]

for stationary hypersonic-limit Euler flows passing a solid body in three-dimensional space, the shock-front coincides with the upwind surface of the body, hence there is an infinite-thin layer of concentrated mass, in which all particles hitting the body move along its upwind surface. By proposing a concept of Radon measure solutions of boundary value problems of the multi-dimensional compressible Euler equations, which incorporates the large-scale of three-dimensional distributions of upcoming hypersonic flows and the small-scale of particles moving on two-dimensional surfaces, the authors derive the compressible Euler equations for flows in concentration layers, which is a stationary pressureless compressible Euler system with source terms and independent variables on curved surface. As a by-product, they obtain a formula for pressure distribution on surfaces of general obstacles in hypersonic flows, which is a generalization of the classical Newton-Busemann law for drag/lift in hypersonic aerodynamics. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Riemann solutions of the non-isentropic Euler equations for the modified Chaplygin gas and the pure Chaplygin gas, which are the set of all rarefaction waves, shock waves, contact discontinuity and δ-shock waves. Under some appropriate conditions, by studing the limiting behavior, we find that the Riemann solutions of modified Chaplygin gas is the same as pure Chaplygin gas including the (δ-shock waves. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the global solution and the structures of interaction between two dimensional non-selfsimilar shock wave and rarefaction wave of general two-dimensional scalar conservation law in which flux functions f(u) and g(u) do not need to be convex, and the initial value contains three constant states which are respectively separated by two general initial discontinuities. When initial value contains three constant states, the cases of selfsimilar shock wave and rarefaction wave had been studied before, but no results of the cases of neither non-selfsimilar shock wave or non-selfsimilar rarefaction wave. Under the assumption that Condition H which is generalization of one dimensional convex condition, and some weak conditions of initial discontinuity, according to all the kinds of combination of elementary waves respectively staring from two initial discontinuities, we get four cases of wave interactions as S + S, S + R, R + S and R + R. By studying these interactions between non-selfsimilar elementary waves, we obtain and prove all structures of non-selfsimilar global solutions for all cases. [ABSTRACT FROM AUTHOR]

This article considers the Cauchy problem for compressible Euler system in R{\bf{R}} with damping, in which the coefficient depends on the space variable. Assuming the initial density has a small perturbation around a constant state and both the small perturbation and the small initial velocity field are compact supported, finite-time blow-up result will be established. This result reveals the fact that if the space-dependent damping coefficient decays fast enough in the far field (belongs to L1(R){L}^{1}\left({\bf{R}})), then the damping is non-effective to the long-time behavior of the solution.

We consider equations of Müller–Israel–Stewart type describing a relativistic viscous fluid with bulk viscosity in four-dimensional Minkowski space. We show that there exists a class of smooth initial data that are localized perturbations of constant states for which the corresponding unique solutions to the Cauchy problem break down in finite time. Specifically, we prove that in finite time such solutions develop a singularity or become unphysical in a sense that we make precise. We also show that in general Riemann invariants do not exist in 1+1 dimensions for physically relevant equations of state and viscosity coefficients. Finally, we present a more general version of a result by Y. Guo and A.S. Tahvildar-Zadeh: We prove large-data singularity formation results for perfect fluids under very general assumptions on the equation of state, allowing any value for the fluid sound speed strictly less than the speed of light. [ABSTRACT FROM AUTHOR]

In this work, we study the existence of solutions to a 2 × 2 non-conservative and non-strictly hyperbolic system in one space dimension related to the dynamics of dislocation densities in crystallography, propagating in two opposite directions. For such systems, existence results are mainly established in the sense of viscosity solutions for Hamilton-Jacobi equations. We study this problem for large initial data using the notions of the theory of conservation laws by constructing entropy solutions through the means of an adapted Godunov scheme, where the associated Riemann problem enjoys new features, more elementary waves than usual, and loss of uniqueness in many cases. The existence is obtained in spaces of functions with bounded fractional total variation B V s , for all 0 < s ≤ 1 , such that B V 1 = B V . [ABSTRACT FROM AUTHOR]