Theory Of Conservation Laws Research Articles

Explicit solutions for fractional Chaffee–Infante reaction–diffusion coupled hierarchy system with conservation laws

This research investigates the explicit series solutions of the fractional Chaffee–Infante reaction–diffusion coupled hierarchy system utilizing Lie symmetry analysis in two‐dimensional space. This fractional system is a prominent model in fluid dynamics to characterize the surface‐stripe wave evolution of liquids in light of the theory of conservation laws. The governing fractional system is analytically solved by the well‐defined technique, called the power series method, so that the total derivative used is based on the Riemann–Liouville derivative. Simulations are also provided in this direction. The acquired results are systematically validated through a series of test cases. The results show that the proposed method is robust, accurate, and flexible in gaining more solutions, and it is effective in treating various fractional physical phenomena within the framework of conservation laws as well.

On Model Two-Dimensional Pressureless Gas Flows: Variational Description and Numerical Algorithm Based on Adhesion Dynamics

Weak solutions of the system of pressureless gas dynamics equations in two dimensions are studied. Theoretical issues are considered, namely, the general mathematical theory of conservation laws for the system is addressed. Emphasis is placed on an important distinctive feature of this system: the emergence of strong density singularities along manifolds of different dimensions. This property is characterized as the formation of a hierarchy of singularities. In earlier application-oriented works (e.g., by A.N. Krayko, et al., including more complicated cases of 3D two-phase flows), this property was studied at the physical level of rigor. In this paper, the formation of a hierarchy of singularities is examined mathematically, since, for example, the existence of a solution with a strong singularity at a point (in the 2D case) is rather difficult to prove rigorously. Accordingly, a special numerical algorithm is used to develop mathematical hypotheses concerning solution behavior. Approaches to the construction of a variational principle for weak solutions are considered theoretically. A numerical algorithm based on approximate adhesion dynamics in the multidimensional case is implemented. The algorithm is tested on several examples (2D Riemann problem) in terms of internal convergence and is compared with mathematical results, including those obtained by other authors.

A mathematical treatment of the bump structure of particle-laden flows with particle features

In this study, we consider particle-laden flows on an inclined plane under the effect of gravity. Previous experimental works have noted that a particle-rich ridge is generated near the contact line. We investigate the bump structure observed in particle-rich ridges in terms of Lax’s shock waves in the mathematical theory of conservation laws, explicitly considering the effect of particles with nontrivial radii on morphology of particle-laden flows. We also extract the dependence of radius and concentration of particles on the bump structure.

On group of Lie symmetry analysis, explicit series solutions and conservation laws for the time-fractional (2 + 1)-dimensional Zakharov-Kuznetsov (q,p,r) equation

This research article intends to utilize results on Lie symmetry analysis, explicit series solutions and conservation laws for the time-fractional Zakharov-Kuznetsov (q,p,r) equation in three-dimensional space. Such a fractional equation yields the mathematical model which describes an occurrence of stationary spatial stripe modalities in a three-dimensional system in the framework of the theory of conservation laws. The governing equation is solved analytically by the power series method, where the total derivative in the sense of Riemann-Liouville type. Simulation results are systematically validated through a series of test cases. Strong evidence shows that the model and the method are conservative and robust.

Exact Riemann solver for a nonlinear hyperbolic system of equations modeling a compressible two-phase flow in pipes

This paper is devoted to present an exact Riemann solver for a gas-liquid mixture in pipes. By applying the theory of conservation laws, we first determine the exact solution of the Riemann problem. We demonstrate that this analytical solution consists of three waves: 1-Wave (Shock/Rarefaction), 2-Wave (Contact Discontinuity) and 3-Wave (Shock/Rarefaction). Secondly, we derive an exact Riemann solver based on the solution of the Riemann problem. The performance of this solver is verified by performing a series of test problems, such as the Riemann problem and the sudden upstream valve closure.

$\mathcal{A}$-Quasiconvexity, Gårding Inequalities, and Applications in PDE Constrained Problems in Dynamics and Statics

A Gårding-type inequality is proved for a quadratic form associated to $\mathcal{A}$-quasiconvex functions. This quadratic form appears as the relative entropy in the theory of conservation laws and it is related to the Weierstrass excess function in the calculus of variations. The former provides weak-strong uniqueness results, whereas the latter has been used to provide sufficiency theorems for local minimizers. Using this new Gårding inequality we provide an extension of these results to PDE constrained problems in dynamics and statics under $\mathcal{A}$-quasiconvexity assumptions. The application in statics improves existing results by proving uniqueness of $L^p$-local minimizers in the classical $\mathcal{A}={curl}$ case.

A system of conservation laws with discontinuous flux modelling flotation with sedimentation

Abstract The continuous unit operation of flotation is extensively used in mineral processing, wastewater treatment and other applications for selectively separating hydrophobic particles (or droplets) from hydrophilic ones, where both are suspended in a viscous fluid. Within a flotation column, the hydrophobic particles are attached to gas bubbles that are injected and float as aggregates forming a foam or froth at the top that is skimmed. The hydrophilic particles sediment and are discharged at the bottom. The hydrodynamics of a flotation column is described in simplified form by studying three phases, namely the fluid, the aggregates and solid particles, in one space dimension. The relative movements between the phases are given by constitutive drift-flux functions. The resulting model is a system of two scalar conservation laws with a multiply discontinuous flux for the aggregates and solids volume fractions as functions of height and time. The model is of triangular nature since one equation can be solved independently of the other. Based on the theory of conservation laws with discontinuous flux, steady-state solutions that satisfy all jump and entropy conditions are constructed. For the existence of the industrially relevant steady states, conditions on feed flows and concentrations are established and mapped as ‘operating charts’. A numerical method that exploits the triangular structure is formulated on a pair of staggered grids and is employed for the simulation of the fill-up and transitions between steady states of the flotation column.

Existence and uniqueness of singular solutions for a conservation law arising in magnetohydrodynamics

The Brio system is a two-by-two system of conservation laws arising as a simplified model in ideal magnetohydrodynamics. The system has the form It was found in previous works that the standard theory of hyperbolic conservation laws does not apply to this system since the characteristic fields are not genuinely nonlinear on the set . As a consequence, certain Riemann problems have no weak solutions in the traditional Lax admissible sense.It was argued in Hayes and LeFloch (1996 Nonlinearity 9 1547–63) that in order to solve the system, singular solutions containing Dirac masses along the shock waves might have to be used. Solutions of this type were exhibited in Kalisch and Mitrović (2012 Proc. Edinburgh Math. Soc. 55 711–29) and Sarrico (2015 Russ. J. Math. Phys. 22 518–27), but uniqueness was not obtained.In the current work, we introduce a nonlinear change of variables which makes it possible to solve the Riemann problem in the framework of the standard theory of conservation laws. In addition, we develop a criterion which leads to an admissibility condition for singular solutions of the original system, and it can be shown that admissible solutions are unique in the framework developed here.

One-dimensional, forward-forward mean-field games with congestion

Here, we consider one-dimensional forward-forward mean-field games (MFGs) with congestion, which were introduced to approximate stationary MFGs. We use methods from the theory of conservation laws to examine the qualitative properties of these games. First, by computing Riemann invariants and corresponding invariant regions, we develop a method to prove lower bounds for the density. Next, by combining the lower bound with an entropy function, we prove the existence of global solutions for parabolic forward-forward MFGs. Finally, we construct traveling-wave solutions, which settles in a negative way the convergence problem for forward-forward MFGs. A similar technique gives the existence of time-periodic solutions for non-monotonic MFGs.

Stability of contact discontinuities to 1-D piston problem for the compressible Euler equations

We consider 1-D piston problem for the compressible Euler equations when the piston is static relatively to the gas in the tube. By a modified wave front tracking method, we prove that a contact discontinuity is structurally stable under the assumptions that the total variation of the initial data and the perturbation of the piston velocity are both sufficiently small. Meanwhile, we study the asymptotic behavior of the solutions by the generalized characteristic method and approximate conservation law theory as t→+∞.

Special issue on contemporary topics in conservation laws

During last 20 years the theory of Conservation Laws underwent adramatic developmen. Networks and Heterogeneous Media isdedicating two consecutive Special Issues to this topic.Researchers belonging to some of the major schools in thissubject contribute to these two issues, offering a view on the currentstate of the art, as well pointing to new research themeswithin areas already exposed to more traditional methodologies.For more information please click the “Full Text” above.

Dynamics in the fundamental solution of a non-convex conservation law

There is a huge jump in the theory of conservation laws if the convexity assumption is dropped. We study a scalar conservation law without the convexity assumption by monitoring the dynamics in the fundamental solution. We introduce three shock types in addition to the usual genuine shock: left-, right- and double-sided contacts. There are three kinds of phenomenon for these shocks, called branching, merging and transforming. All of these shocks and phenomena can be observed if the flux function has two inflection points. A comprehensive picture of a global dynamics of a non-convex flux is discussed in terms of characteristic maps and dynamical convex–concave envelopes.

Analytical solutions for flow in porous media with multicomponent cation exchange reactions

AbstractMulticomponent cation exchange reactions have important applications in groundwater remediation, disposal of nuclear wastes as well as enhanced oil recovery. The hyperbolic theory of conservation laws can be used to explain the nature of displacements observed during flow with cation exchange reactions between flowing aqueous phase and stationary solid phase. Analytical solutions have been developed to predict the effluent profiles for a particular case of heterovalent cations (Na+, Ca2+ and Mg2+) and an anion (Cl−) for any combination of constant injection and constant initial composition using this theory. We assume local equilibrium, neglect dispersion and model the displacement as a Riemann problem using mass action laws, the charge conservation equation and the cation exchange capacity equation. The theoretical predictions have been compared with experimental data available at two scales—the laboratory scale and the field scale. The theory agrees well with the experimental data at both scales. Analytical theory predictions show good agreement with numerical model, developed using finite differences.

RIEMANN SOLUTIONS FOR VERTICAL FLOW OF THREE PHASES IN POROUS MEDIA: SIMPLE CASES

We study Riemann solutions for a system of two nonlinear conservation laws that models buoyancy-driven flow of three immiscible fluids in a porous medium, which do not exchange mass. We also assume that the fluids are incompressible and the flow occurs in the vertical spatial dimension. We consider the simplified case in which two of the three fluids have equal densities, obtaining the Riemann solutions by the wave curve method. As expected, the solutions contain waves traveling both upwards and downwards. The sequences of waves contain rarefactions, shocks (sometimes traveling with characteristic speed), and constant states. The shocks found in this work are proper or generalized Lax shock waves. The solutions we found are L1-continuous with respect to the initial data. Waves involving only two fluids often take part in three-phase flow Riemann solutions; this is the basis of a useful tool (the wedge construction) to obtain shocks separating states in distinct two-phase regimes having a common fluid. This tool is similar to fractional flow theory, or Oleinik's convex construction. In this investigation, the wave curve method from the theory of conservation laws is combined with numerical calculations.

Well-Posedness for a One-Dimensional Fluid-Particle Interaction Model

The fluid-particle interaction model introduced by the three last authors in [J. Differential Equations, 245 (2008), pp. 3503-3544] is the object of our study. This system consists of the Burgers equation with a singular source term (term that models the interaction via a drag force with a moving point particle) and of an ODE for the particle path. The notion of entropy solution for the singular Burgers equation is inspired by the theory of conservation laws with discontinuous flux developed by the first author, Kenneth Hvistendahl Karlsen and Nils Henrik Risebro in [Arch. Ration. Mech. Anal., 201 (2011), pp. 26-86]. In this paper, we prove well-posedness and justify an approximation strategy for the particle-in-Burgers system in the case of initial data of bounded variation. Existence result for L∞ data is also given.

Extensions for Systems of Conservation Laws

Extensions are central in the theory of conservation laws by providing intrinsic selection criteria for weak solutions. Given a system u t + f(u) x = 0 the extensions solve certain second order PDEs which are typically overdetermined. Determining the size of the set of extensions can be a challenging task. Instead of analyzing this second order system directly, we consider the equations satisfied by the lengths β i of the eigenvectors r i of the Jacobian matrix Df, measured with the inner product defined by an extension. For a given eigen-frame {r i } the extensions are determined uniquely, up to trivial affine parts, by these lengths. The β i solve a first order algebraic-differential system (the β-system) to which standard integrability theorems can be applied. The number of extensions is determined by determining the number of free constants and functions in the general solution to the β-system. We provide a complete breakdown for 3 × 3-systems, and for rich frames whose β-system has trivial algebraic part. Our framework is motivated by the work [16] where the authors consider conservative systems with prescribed eigen-frames. It is natural to ask how many extensions the resulting systems have, and the answer depends in an essential manner on the prescribed frame.

AN INTEGRO-DIFFERENTIAL CONSERVATION LAW ARISING IN A MODEL OF GRANULAR FLOW

We study a scalar integro-differential conservation law which was recently derived by the authors as the slow erosion limit of a granular flow. Considering a set of more general erosion functions, we study the initial boundary value problem for which one cannot adapt the standard theory of conservation laws. We construct approximate solutions with a fractional step method, by recomputing the integral term at each time step. A prioriL∞bound and total variation estimates yield the convergence and global existence of solutions with bounded variation. Furthermore, we present a well-posedness analysis which establishes that these solutions are stable in the L1norm with respect to the initial data.

Characterization of Radially Symmetric Finite Time Blowup in Multidimensional Aggregation Equations

This paper studies the transport of a mass $\mu$ in $\mathbb{R}^d, d \geq 2,$ by a flow field $v= -\nabla K*\mu$. We focus on kernels $K=|x|^\alpha/ \alpha$ for $2-d\leq \alpha<2$ for which the smooth densities are known to develop singularities in finite time. For this range we prove the existence for all time of radially symmetric measure solutions that are monotone decreasing as a function of the radius, thus allowing for continuation of the solution past the blowup time. The monotone constraint on the data is consistent with the typical blowup profiles observed in recent numerical studies of these singularities. We prove monotonicity is preserved for all time, even after blowup, in contrast to the case $\alpha >2$ where radially symmetric solutions are known to lose monotonicity. In the case of the Newtonian potential ($\alpha=2-d$), under the assumption of radial symmetry the equation can be transformed into the inviscid Burgers equation on a half line. This enables us to prove preservation of monotonicity using the classical theory of conservation laws. In the case $2 -d < \alpha < 2$ and at the critical exponent p we exhibit initial data in $L^p$ for which the solution immediately develops a Dirac mass singularity. This extends recent work on the local ill-posedness of solutions at the critical exponent.

Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes

We define entropy weak solutions and establish well-posedness for the Cauchy problem for the formal equation $$\partial_t u(t,x) + \partial_x \frac{u^2}2(t,x) = - \lambda \, u(t,x)\,\delta_0(x),$$ which can be seen as two Burgers equations coupled in a non-conservative way through the interface located at $x=0$. This problem appears as an important auxiliary step in the theoretical and numerical study of the one-dimensional particle-in-fluid model developed by Lagoutière, Seguin and Takahashi [30]. The interpretation of the non-conservative product '$ u(t,x) \, \delta_0(x)$' follows the analysis of [30]; we can describe the associated interface coupling in terms of one-sided traces on the interface. Well-posedness is established using the tools of the theory of conservation laws with discontinuous flux ([4]). For proving existence and for practical computation of solutions, we construct a finite volume scheme, which turns out to be a well-balanced scheme and which allows a simple and efficient treatment of the interface coupling. Numerical illustrations are given.

On the generalization of conservation law theory to certain degenerate parabolic systems of equations describing processes of compressible two-phase multicomponent filtration

A degenerate parabolic system of equations of two-phase multicomponent filtration is considered. It is shown that this system can be treated as a system of conservation laws and the notions developed in the corresponding theory, such as hyperbolicity, shock waves, Hugoniot relations, stability conditions, Riemann problem, entropy, etc., can be applied to this system. The specific character of the use of such notions in the case of multicomponent filtration is demonstrated. An example of two-component mixture is used to describe the specific properties of solutions of the Riemann problem.