Fig 5 - uploaded by Marko Nedeljkov
Content may be subject to copyright.
Source publication
We study interaction of generalized solutions to a strictly hyperbolic system of two conservation laws from magnetohydrodynamics. The Riemann problem of this system is known to be solvable by a combination of shock waves, contact discontinuities, rarefaction waves and delta shock waves. We derive how the solutions continue beyond points of interact...
Similar publications
This note reports the derivation of an alternative form of Andrews' conservation law for small-amplitude, quasi-geostrophic, transient eddies on a steady but spatially nonuniform flow.
Traveling wave solutions of viscous conservation laws, that are associated to Lax shocks of the inviscid equation, have generically a transversal viscous profile. In the case of a non-transversal viscous profile we show by using Melnikov theory that a parametrized perturbation of the profile equation leads generically to a saddle-node bifurcation o...
Based on the relation between kinetic Boltzmann-like transport equations and non-linear hyperbolic conservation laws, we derive kinetic-induced moment systems for the spatially one-dimensional shallow water equations (the Saint-Venant equations). Using Chapman-Enskog-like asymptotic expansion techniques in terms of the relaxation parameter of the k...
We describe a way to apply the true Riemann solver with seven characteristic waves to solve the system of eight conservation laws for multi-dimensional magnetohydrodynamics.
Citations
... Extensive research has been conducted on delta-waves and related topics. For instance, Mitrović and Nedeljkov [18], Nedeljkov and Oberguggenberger [19], as well as Zhang and Zhang [20] have made significant contributions in this area. ...
We consider the well‐posedness of a class of initial‐boundary value problems of the spherically symmetric isentropic compressible Euler equations for polytropic gas. We find that a singularity always occurs on the sonic circle near the origin, even in the case of steady flow, regardless of whether the speed of the incoming flow at far field is subsonic or supersonic. This inherent difficulty makes it impossible to extend the solution towards the origin pass across the sonic circle in the sense of classical solutions. To address the singularity, our idea is to fix a delta‐wave on the sonic circle in the framework of the Radon measure solution. As a result, we obtain the existence of the Radon measure solution to the initial‐boundary value problem and prove that a concentration will form on the sonic circle. More importantly, we deduce an exact formula for pressure distribution on the sonic circle. In particular, there is a concentration at the origin as the Mach number of incoming flow goes to infinity, which corresponds to the case of pressureless gas. Additionally, we investigate the case of Chaplygin gas.
... In the field of scientific research, numerous more advanced methods have been proposed to address the issue of delta shock wave phenomena. These methods include the vanishing viscosity method, 24 the weak asymptotic method, 25 the shadow wave method, 26 the method of split delta function, 27 and the method of distributional product, 28 among others. For more detail of delta shock, please refer to the papers by Danilov and Shelkovich, 29 Li and Yang, 30 Shen, 31 Zhang and Zhang, 32 Zhang and Wang, 33 and Shah et al. 34 The present work introduces a new dimension to the existing scientific literature as we inquire into the discussion of the concentration and cavitation phenomena within the Aw-Rascle model. ...
In this manuscript, we explore the concentration and cavitation phenomena in the Riemann problem for the Aw–Rascle model coupled with an anti van der Waals Chaplygin gas while considering a two-parameter flux approximation. We investigate the presence of a δ-shock and a vacuum state within the Riemann problem for this specific system. Additionally, we incorporate a perturbed flux approximation scheme and analyze the Riemann solution as the values of α1 and α2 approach 0. Our findings demonstrate that the δ-shock solution to the simplified equations can be achieved by examining the Riemann solution that involves two shock waves in the perturbed flux approximation system. This occurs when the flux approximation linked to the anti van der Waals Chaplygin gas model vanishes. Furthermore, the Riemann solution that includes two rarefaction waves converges to the vacuum state solution of the simplified equations.
... These type of systems occur in nonlinear elasticity and gas dynamics and have been studied by various authors. [8][9][10] By approximating the equations of magnetohydrodynamics, one can obtain the system (1.5) for sufficient details we refer. 8,9,11 There are some nonclassical situations where in contrast to Lax's 12 and Glimm's 13 results, the Cauchy problem for a system of conservation laws fails to contain a weak L ∞ -solution except for some particular initial data. ...
... [8][9][10] By approximating the equations of magnetohydrodynamics, one can obtain the system (1.5) for sufficient details we refer. 8,9,11 There are some nonclassical situations where in contrast to Lax's 12 and Glimm's 13 results, the Cauchy problem for a system of conservation laws fails to contain a weak L ∞ -solution except for some particular initial data. For that when we solve Cauchy problem in this nonclassical situation, it is necessary to introduce a new singular solution called as delta shock wave. ...
... For the system (1.5), stability and various feasible wave interactions containing delta shock wave have been analyzed by Nedeljkov and Oberguggenberger. 8 But for some values of the initial data, it is impossible to achieve classical solution of Riemann problem. So we may turn our attention to nonclassical waves like delta shock wave. ...
Here, we study the Riemann problem for a strictly hyperbolic system of conservation laws, which occurs in gas dynamics and nonlinear elasticity. We establish the existence and uniqueness of the solution of Riemann problem containing delta shock wave by employing self‐similar vanishing viscosity approach. We prove that delta shock is stable under self‐similar viscosity perturbation, which ensures that delta shock wave is a unique entropy solution.
... Extensive investigations have been carried out on the system (1.1) due to the fact that the singular δ-shock wave is involved in the Riemann solution under some suitable initial conditions (1.2). More specifically, Nedeljkov and Oberguggenberger [2] investigated the interaction problem of δ-shock wave for the system (1.1) carefully by using the approach of splitting δ-function. Sen and Raja Sekhar [3] obtained the existence and uniqueness of δ-shock solution for the Riemann problem (1.1)-(1.2) with the help of self-similar viscosity vanishing method. ...
... Therefore, if the suitable linear substitutions for the two state variables u and v are adopted here, then our problem in the current work can be further simplified to a certain extent. In fact, it is not difficult to find that our perturbed system (1.4) can be changed into ⎧ ⎨ ⎩ (u + εα) t + (u + εα) 2 2 + εβ(v + εγ ) ...
The stability of delta shock solution for the simplified magnetohydrodynamics equations is investigated carefully under the linear flux-function perturbation. Five different structures of Riemann solutions for our perturbed equations are solved in completely explicit situations. By analyzing all the arisen circumstances carefully, we prove rigorously that the limits of Riemann solutions for our perturbed equations are in well agreement with those for the original ones as the perturbation parameter vanishes. In particular, the formation of delta shock wave is well observed in the ultimate limiting state.
... Over the past three decades, such research studies have become very active. There are lots of authors who have obtained a great many excellent achievements; see [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] and references cited therein. Regrettably, all the delta waves in these investigations are overcompressible. ...
... Relations in equation(3.4) are called the generalized Rankine-Hugoniot relation of the delta wave, which reflects the exact relationship among the location, propagation speed, weight, and assignment of v and q on the discontinuity.Let us consider the delta wave in the form (3.2) with the assumption that ρ v ) = , and w 0 0 ( ) = . In view of knowledge concerning delta waves[1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16], it is found that v One can observe that if v δ is determined, then the remaining is determined accordingly. ...
Most previous studies concerning delta waves have been focused on the overcompressible ones. To study the non-overcompressible delta waves, this article is concerned with the isentropic relativistic Euler system coupled with an advection equation for Chaplygin gas. The Riemann problem is completely solved. The solutions exhibit four kinds of wave patterns: the first contains three contact discontinuities; the second includes a single overcompressible delta wave, and the third and fourth involve a contact discontinuity and a non-overcompressible delta wave.
... For equations in nonconservative form, one may also consult [26] for ideas on definitions of products of measures and discontinuous functions. The methods of shadow waves, split -functions, Colombeau's generalized functions, as well as Sarrico's -product framework, could be found in [11,[31][32][33]35]. From these works, we notice that an underlying philosophy is this: to solve a Riemann problem not solvable in the usual sense of functions with minimal waves, which means that one has an over-determined ...
We solve the Riemann problems for isentropic compressible Euler equations of polytropic gases in the class of Radon measures, and the solutions admit the concentration of mass. It is found that, under the requirement of satisfying the over-compressing entropy condition: (i) there is a unique delta shock solution, corresponding to the case that has two strong classical Lax shocks; (ii) for the initial data that the classical Riemann solution contains a shock wave and a rarefaction wave, or two shocks with one being weak, there are infinitely many solutions, each consists of a delta shock and a rarefaction wave; (iii) there is no delta shocks for the case that the classical entropy weak solutions consist only of rarefaction waves. These solutions are self-similar. Furthermore, for the generalized Riemann problem with mass concentrated initially at the discontinuous point of initial data, there always exists a unique delta shock for at least a short time. It could be prolonged to a global solution. Not all the solutions are self-similar due to the initial velocity of the concentrated point-mass (particle). Whether the delta shock solutions constructed satisfy the over-compressing entropy condition is clarified. This is the first result on the construction of singular measure solutions to the compressible Euler system of polytropic gases, that is strictly hyperbolic, and whose characteristics are both genuinely nonlinear. We also discuss possible physical interpretations and applications of these new solutions.
... For equations in nonconservative form, one may also consult [37] for ideas on definitions of products of measures and discontinuous functions. The methods of shadow waves, split δfunctions, Colombeau's generalized functions, as well as Sarrico's α-product framework, could be found in [16,[45][46][47]50]. From these works, we notice that an underlying philosophy is this: to solve a Riemann problem not solvable in the usual sense of functions with minimal waves, which means that one has an over-determined (usually algebraic) problem. ...
We solve the Riemann problems for isentropic compressible Euler equations of polytropic gases in the class of Radon measures, and the solutions admit the concentration of mass. It is found that under the requirement of satisfying the over-compressing entropy condition: (i) there is a unique delta shock solution, corresponding to the case that has two strong classical Lax shocks; (ii) for the initial data that the classical Riemann solution contains a shock wave and a rarefaction wave, or two shocks with one being weak, there are infinitely many solutions, each consists of a delta shock and a rarefaction wave; (iii) there are no delta shocks for the case that the classical entropy weak solutions consist only of rarefaction waves. These solutions are self-similar. Furthermore, for the generalized Riemann problem with mass concentrated initially at the discontinuous point of initial data, there always exists a unique delta shock for at least a short time. It could be prolonged to a global solution. Not all the solutions are self-similar due to the initial velocity of the concentrated point-mass (particle). Whether the delta shock solutions constructed satisfy the over-compressing entropy condition is clarified. This is the first result on the construction of singular measure solutions to the compressible Euler system of polytropic gases, that is strictly hyperbolic, and whose characteristics are both genuinely nonlinear. We also discuss possible physical interpretations and applications of these new solutions.
... Page 20 of 31 M. Nedeljkov and S. Ružičić NoDEA delta contact discontinuity (see [15]). If u 0 = u δ = u 1 , the solution is a single delta contact discontinuity. ...
The main goal of the paper is to define and use a condition sufficient to choose a unique solution to conservation law systems with a singular measure in initial data. Different approximations can lead to solutions with different distributional limits. The new notion called backward energy condition is then to single out a proper approximation of the distributional initial data. The definition is based on the maximal energy dissipation defined in Dafermos (J Differ Equ 14:202–212, 1973). Suppose that a conservation law system admits a supplementary law in space–time divergent form where the time component is a (strictly or not) convex function. It could be an energy density or a mathematical entropy in gas dynamic models, for example. One of the admissibility conditions is that a proper weak solution should maximally dissipate the energy or the mathematical entropy. We show that it is consistent with other admissibility conditions in the case of Riemann problems for systems of isentropic gas dynamics with non-positive pressure in the first part of the paper. Singular solutions to these systems are described by shadow waves, nets of piecewise constant approximations with respect to the time variable. In the second part, we define and apply the backward energy condition for those systems when the initial data contains a delta measure approximated by piecewise constant functions.
... During the process of analysis, we find a new kind of nonclassical solution-delta contact discontinuity, which is a kind of solution containing δ−function supported on contact discontinuity. This kind of nonclassical solution has also appeared in wave interactions of magnetic dynamic equations and the pressureless Euler equations [23,29]. ...
... Here, δ J is a new kind of nonclassical solution-δ-contact discontinuity, which is a kind of solution containing δ-function supported on a contact discontinuity. This kind of nonclassical solution has also appeared in the wave interaction of the magnetic dynamic equations and the pressureless Euler equations [23,29]. ...
... This illustration is relatively rare in the literature. The third one lies in that compared with other methods dealing with impingements involving δ-shocks for conservation laws, such as splitting δ-functions in [23,29], our method here is based on the local perturbation of initial condition and passing to the limit according to the stability theory of weak solutions. We have already successfully used this method dealing with similar problem for degenerate conservation laws in [37]. ...
In this paper, we are concerned with a general Riemann problem for the 1-D Eulerian droplet model with delta initial condition. The main novelty of this paper lies in that, by providing a global analytical solution, it fills a gap in reference \cite{Keita-Bourgault} in the literature, in which it was shown that a solution exists for the associated generalized Rankine-Hugoniot conditions (GRHC) with delta initial condition, but no analytical solution was given. By constructing solutions to the perturbed initial value problem, and then passing to the limit to recover the delta initial condition and the limit analytical solution, we successfully develop the global analytical solution to the delta initial value problem.
During this process, four different cases of wave interactions are studied and a new type of wave (called delta contact discontinuity) is introduced. The above analysis gives a nice illustration of wave interactions for degenerate hyperbolic conservation laws with source term, which illustrations are
relatively rare in the literature. Moreover, with above methods and results, it will give us some new insights into our future study on 1-D or 2-D conservation laws with source term.
... Let us briefly review the concept of left-and right-hand side delta functions, which we shall use extensively. A detailed study on this subject can be found in [25,26]. Let R 2 + be divided into finitely many disjoint open sets Ω i , i = 1, . . . ...
... The continuously emerging number of interesting works [3,4,[6][7][8][9]12,18,21,[23][24][25][26]28,31,[34][35][36]39,41] (see, e.g., [10,16,27,30,32,33] for very recent results) on the theoretical, the applications and the numerical analysis of nonlinear phenomena related to hyperbolic systems of conservation laws involving delta shock published in recent years seems to be only scratching the surface and a testament to the vitality of this research topic. Indeed, the fact that these works complement each other, as they differ in scope and the wide range of applications in intricate natural events with solid mathematics supporting the physics, is indicative of the breadth of occurrences of delta shock as a genuine and fundamental nonlinear phenomena. ...
... There are many papers devoted to the interaction of delta shock waves and contact discontinuities for systems of Keyfitz-Kranzer type and other related models. For instance, Shen and Sun [34] also use the method of splitting of delta functions [25,26] to study the interaction of delta shock waves and vacuum states for the transport equations that corresponds to the pressureless case of (1) when φ(ρ, u) = u, i.e. ...
In this work, the mechanism for the formation of the delta shock wave is analyzed to deal with interaction of delta shock waves and contact discontinuities for a system of Keyfitz–Kranzer type by means of analysis and solutions of Riemann problems. A set of numerical experiments are provided, illustrating the theoretical findings numerically. A brief survey of the Keyfitz–Kranzer systems as a base model of fundamental nonlinear phenomena in applications is provided aiming to shed light on the intricate wave structure for other related models of conservation laws appearing in applied sciences.
