V. G. Danilov, D. Mitrovic
We describe $δ$ shock wave arising from continuous initial data in the case of triangular conservation law system arising from "generalized pressureless gas dynamics model". We use the weak asymptotic method.
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We study the one-dimensional Riemann problem for a hyperbolic system of three conservation laws of Temple class. This systems it is a simplification of a recently propose system of five conservations laws by Bouchut and Boyaval that model viscoelastic fluids. An important issues is that the considered $3 \times 3$ system is such that every characteristic field is linearly degenerate. We show the Riemann problem for this system. Under suitable generalized Rankine-Hugoniot relation and entropy condition, both existence and uniqueness of particular delta-shock type solutions are established.
In this paper, we propose a time-dependent viscous system and by using the vanishing viscosity method we show the existence of delta shock solution for a particular $2 \times 2$ system of conservation laws with linear damping.
We introduce integral identities to define delta-shock wave type solutions for the multidimensional zero-pressure gas dynamics Using these integral identities, the Rankine-Hugoniot conditions for delta-shocks are obtained. We derive the balance laws describing mass, momentum, and energy transport from the area outside the delta-shock wave front onto this front. These processes are going on in such a way that the total mass, momentum, and energy are conserved and at the same time mass and energy of the moving delta-shock wave front are increasing quantities. In addition, the total kinetic energ…
The paper deals with scalar conservation laws having a flux discontinuity at $x=0$ without a weak solution that satisfies the classical Rankine--Hugoniot jump condition at $x=0$. We are using unbounded solutions in the form of shadow waves supported by the origin for solving that problem. The shadow waves are nets of piecewise constant functions approximating a shock wave with added a delta function and sometimes another unbounded part.
Viscous approximation of triangular system in 1-d with nonlinear viscosity
本文研究了带非线性粘性的$2\times2$双曲守恒律三角系统的粘性消失极限，在粘性矩阵与对流部分交换的假设下，证明了小$BV$初始数据下全局光滑解的存在性及全变差一致有界性，推广了先前$B(u)=I$情形的结果。
An overview on the local limit of non-local conservation laws, and a new proof of a compactness estimate
本文概述了非局部守恒律向局部守恒律收敛极限的最新进展，重点讨论了在交通模型中极具应用价值的各向异性核情况，并给出了相关紧性估计的新证明。
In this paper we introduce a concept of "regulated function" $v(t,x)$ of two variables, which reduces to the classical definition when $v$ is independent of $t$. We then consider a scalar conservation law of the form $u_t+F(v(t,x),u)_x=0$, where $F$ is smooth and $v$ is a regulated function, possibly discontinuous w.r.t.both $t$ and $x$. By adding a small viscosity, one obtains a well posed parabolic equation. As the viscous term goes to zero, the uniqueness of the vanishing viscosity limit is proved, relying on comparison estimates for solutions to the corresponding Hamilton--Jacobi equation.…
We give an answer to a question posed in [P. Amorim, R. Colombo, and A. Teixeira, ESAIM Math. Model. Numerics. Anal. 2015], which can be loosely speaking formulated as follows. Consider a family of continuity equations where the velocity depends on the solution via the convolution by a regular kernel. In the singular limit where the convolution kernel is replaced by a Dirac delta, one formally recovers a conservation law: can we rigorously justify this formal limit? We exhibit counterexamples showing that, despite numerical evidence suggesting a positive answer, one in general does not have co…
For general hyperbolic systems of conservation laws we show that dissipative weak solutions belonging to an appropriate Besov space $B^{α,\infty}_q$ and satisfying a one-sided bound condition are unique within the class of dissipative solutions. The exponent $α>1/2$ is universal independently of the nature of the nonlinearity and the Besov regularity need only be imposed in space when the system is expressed in appropriate variables. The proof utilises a commutator estimate which allows for an extension of the relative entropy method to the required regularity setting. The systems of elasticit…
We consider the $L^2$-critical nonlinear Schrödinger equation (NLS) with the delta potential $$i\partial_tu +\partial^2_x u + μδu +|u|^{4}u=0, \, \, t\in \R, \, x\in \R , $$ where $ μ\in \R$, and $δ$ is the Dirac delta distribution at $x=0$. Local well-posedness theory together with sharp Gagliardo-Nirenberg inequality and the conservation laws of mass and energy implies that the solution with mass less than $\|Q\|_{2}$ is global existence in $H^1(\R)$, where $Q$ is the ground state of the $L^2$-critical NLS without the delta potential (i.e. $μ=0$). We are interested in the dynamics of the sol…
We prove a quadratic interaction estimate for wavefront approximate solutions to the triangular system of conservation laws \[ \begin{cases} u_t + \tilde f(u,v)_x = 0, \cr v_t - v_x = 0. \end{cases} \] This quadratic estimate has been used in the literature to prove the convergence rate of the Glimm scheme. Our aim is to extend the analysis, done for scalar conservation laws, in the presence of transversal interactions among wavefronts of different families. The proof is based on the introduction of a quadratic functional $\mathfrak Q(t)$, decreasing at every interaction, and such that its tot…
In 1977 Korchinski presented a new type of shock discontinuity in conservation laws. These singular solutions were coined $δ$-shocks since there is a time dependent Dirac delta involved. A naive description is that such $δ$-shock is of the overcompressive type: a two-family shock wave the four characteristic lines of which impinge into the shock itself. In this work, we open the fan of solutions by studying two-family waves without intermediate constant states but, possessing central rarefactions and also comprising $δ$-shocks.
Scalar Approach to ARZ-Type Systems of Conservation Laws
针对含GARZ模型等的2×2守恒律系统，提出结合Panov的线性输运方程理论与不连续通量理论的分裂方法，为该类系统的求解提供新思路。
In this paper, we propose a time-dependent viscous system and by using the vanishing viscosity method we show the existence of %delta shock solution solutions for the Riemann problem to a particular $2 \times 2$ system of conservation laws with linear damping.
Ole\uınik-type estimates for nonlocal conservation laws and applications to the nonlocal-to-local limit
研究带指数核的非局部守恒律，证明非局部项满足 Ole\uınik 型熵条件，并推导出核缩放至 Dirac delta 时非局部弱解收敛到局部守恒律的唯一熵容许解。
We explore new ways of regulating defect behavior in systems of conservation laws. Contrary to usual regularization schemes (such as a vanishing viscosity limit), which attempt to control defects by making them smoother, our schemes result in defects which are \textit{more singular}, and we thus refer to such schemes as "irregularizations". In particular, we seek to produce \textit{delta shock} defects which satisfy a condition of \textit{stationarity}. We are motivated to pursue such exotic defects by a physical example arising from dislocation dynamics in materials physics, which we describe…
We consider the $2 \times 2$ parabolic systems \begin{equation*} u^ε_t + A(u^ε) u^ε_x = εu^ε_{xx} \end{equation*} on a domain $(t, x) \in ]0, + \infty[ \times ]0, l[$ with Dirichlet boundary conditions imposed at $x=0$ and at $x=l$. The matrix $A$ is assumed to be in triangular form and strictly hyperbolic, and the boundary is not characteristic, i.e. the eigenvalues of $A$ are different from 0. We show that, if the initial and boundary data have sufficiently small total variation, then the solution $u^ε$ exists for all $t \geq 0$ and depends Lipschitz continuously in $L^1$ on the initial and …
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 \cite{CD_1973}. Suppose that a conservation law system admits a supplementary law in space--time divergent form where the time component is a (strict…
An Analysis of the Riemann Problem for a $2 \times 2$ System of Keyfitz-Kranzer Type Conservation Laws Using Shadow Waves and Dafermos Regularization
本文研究了关键拟izer-kranzer 类守恒律系统中 $2 \times 2$ 系统的里曼问题，利用影波和达夫罗斯正则化分析了delta激波的存在，并通过数值模拟验证了其在运输动态中的应用。
In the paper, we consider the large time behavior of solutions to the convection-diffusion equation u_t - Delta u + nabla cdot f(u) = 0 in R^n times [0,infinity), where f(u) ~ u^q as u --> 0. Under the assumption that q >= 1+1/(n+beta) and the initial condition u_0 satisfies: u_0 in L^1(R^n), integral_{R^n} u_0(x) dx = 0, and |e^{t Delta}u_0|_{L^1(R^n)} <= Ct^{-beta/2} for fixed beta in (0,1), all t>0, and a constant C, we show that the L^1-norm of the solution to the convection-diffusion equation decays with the rate t^{-beta/2} as t --> infinity. Moreover, we prove that, for small initial co…
In this paper we study the one-dimensional Riemann problem for a new hyperbolic system of three conservation laws of Temple class. This systems it is a simplification of a recently propose system of five conservations laws by Bouchut and Boyaval that model viscoelastic fluids. An important issues is that the considered $3 \times 3$ system is such that every characteristic field is linearly degenerate. Then, in despite of the fact that it is of Temple class, the analysis of the Cauchy problem is more involved since general results for such a systems are not yet available. We show a explicit sol…
Upwind filtering of scalar conservation laws
本文提出一种基于平均迎风散度算子的非局部方法，将通量分解为单调分量并应用各向异性核，处理非单调通量的多维标量守恒律，建立适定性和熵估计，并恢复局部守恒律的熵解。
Singular limit for a class of nonlocal conservation laws via compensated compactness
针对一类建模交通流的非局部守恒律，在核集中为狄拉克δ分布时，不依赖全变差界等传统条件，通过补偿紧性理论证明弱解强收敛到对应局部熵容许解，解决了非凸核相关的长期开放问题。
We establish the equivalence of free piston and delta shock, for the one-space-dimensional pressureless compressible Euler equations. The delta shock appearing in the singular Riemann problem is exactly the piston that may move freely forward or backward in a straight tube, driven by the pressureless Euler flows on two sides of it in the tube. This result not only helps to understand the physics of the somewhat mysterious delta shocks, but also provides a way to reduce the fluid-solid interaction problem, which consists of several initial-boundary value problems coupled with moving boundaries,…
By the flux-approximation method, we study limits of Riemann solutions to the Brio system with two independent parameters. The Riemann problem of the perturbed system is solved analytically, and four kinds of solutions are obtained constructively. It is shown that, as the two-parameter flux perturbation vanishes, any two-shock-wave and two-rarefaction-wave solutions of the perturbed Brio system converge to the delta-shock and vacuum solutions of the transport equations, respectively. In addition, we specially pay attention to the Riemann problem of a simplified system of conservation laws deri…
{\it $δ$-Shock wave type solutions} in the multidimensional system of conservation laws $$ ρ_t + \nabla\cdot(ρF(U))=0, \qquad (ρU)_t + \nabla\cdot(ρN(U))=0, \quad x\in \bR^n, $$ are studied, where $F=(F_j)$ is a given vector field, $N=(N_{jk})$ is a given tensor field, $F_j, N_{kj}:\bR^n \to \bR$, $j,k=1,...,n$; $ρ(x,t)\in \bR$, $U(x,t)\in \bR^n$. The well-known particular cases of this system are zero-pressure gas dynamics in a standard form $$ ρ_t + \nabla\cdot(ρU)=0, \quad (ρU)_t + \nabla\cdot(ρU\otimes U)=0, $$ and in the relativistic form $$ ρ_t + \nabla\cdot(ρC(U))=0, \quad (ρU)_t + \nab…
We study nonlinear hyperbolic conservation laws with non-convex flux in one space dimension and, for a broad class of numerical methods based on summation by parts operators, we compute numerically the kinetic functions associated with each scheme. As established by LeFloch and collaborators, kinetic functions (for continuous or discrete models) uniquely characterize the macro-scale dynamics of small-scale dependent, undercompressive, nonclassical shock waves. We show here that various entropy-dissipative numerical schemes can yield nonclassical solutions containing classical shocks, including…
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