Maria Colombo, Gianluca Crippa, Marie Graff, Laura V. Spinolo
We provide an informal overview of recent developments concerning the singular local limit of nonlocal conservation laws. In particular, we discuss some counterexamples to convergence and we highlight the role of numerical viscosity in the numerical investigation of the nonlocal-to-local limit. We also state some open questions and describe recent related progress.
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We deal with the problem of approximating a scalar conservation law by a conservation law with nonlocal flux. As convolution kernel in the nonlocal flux, we consider an exponential-type approximation of the Dirac distribution. This enables us to obtain a total variation bound on the nonlocal term. By using this, we prove that the (unique) weak solution of the nonlocal problem converges strongly in $C(L^{1}_{\text{loc}})$ to the entropy solution of the local conservation law. We conclude with several numerical illustrations which underline the main results and, in particular, the difference bet…
We are interested in a nonlocal conservation law which describes the morphodynamics of sand dunes sheared by a fluid flow, recently proposed by Andrew C. Fowler. We prove that constant solutions of Fowler's equation are non-linearly unstable. We also illustrate this fact using a finite difference scheme.
In this note, we introduce some models of pedestrian traffic and prove existence and uniqueness for these models.
In some models involving nonlinear conservation laws, physical mechanisms exist which prevent the formation of shocks. This gives rise to conservation laws with a constraint on the gradient of the solution. We approach this problem by studying a related conservation law with a spatial nonlocal term. We prove existence, uniqueness and stability of solution of the Cauchy problem for this nonlocal conservation law. In turn, this allows us to provide a notion of solution to the conservation law with a gradient constraint. The proof of existence is based on a time-stepping technique, and an $L^1$-c…
A note on nonlocal approximations of sign-unrestricted solutions of conservation laws
本文研究非局部守恒律的奇异极限，证明当核趋于狄拉克分布时非局部解收敛到局部熵解，推广了符号限制情形，涵盖广义Burgers方程和Keyfitz--Kranzer系统。
In this work we consider companion conservation laws to general systems of conservation laws. We investigate sufficient regularity for weak solutions to satisfy companion laws, assuming the fluxes to be $C^{1,γ}$, $0<γ<1$, regular. We discuss applications of the general framework to nonlinear elasticity and compressible Euler system.
In this contribution we study the singular limit problem of a nonlocal conservation law with a discontinuity in space. The specific choice of the nonlocal kernel involving the spatial discontinuity as well enables it to obtain a maximum principle for the nonlocal equation. The corresponding local equation can be transformed diffeomorphically to a classical scalar conservation law where the well-know Kružkov theory can be applied. However, the nonlocal equation does not scale that way which is why the study of convergence is interesting to pursue. For exponential kernels in the nonlocal operato…
We consider conservation laws with nonlocal velocity and show for nonlocal weights of exponential type that the unique solutions converge in a weak or strong sense (dependent on the regularity of the velocity) to the entropy solution of the local conservation law when the nonlocal weight approaches a Dirac distribution. To this end, we establish first a uniform total variation estimate on the nonlocal velocity which enables it to prove that the nonlocal solution is entropy admissible in the limit. For the entropy solution, we use a tailored entropy flux pair which allows the usage of only one …
This work is devoted to examine the uniqueness and existence of kinetic solutions for a class of scalar conservation laws involving a nonlocal super-critical diffusion operator and a multiplicative noise. Our proof for uniqueness is based upon the analysis on double variables method and the existence is enabled by a parabolic approximation.
Conservation laws with nonlocality in density and velocity and their applicability in traffic flow modelling
本文提出速度依赖空间局部积分项的非局部守恒律，统一已有模型并刻画交通流新动力学，证明弱解存在唯一性，且给出数值离散与算例。
On the singular limit problem for nonlocal conservation laws: A general approximation result for kernels with fixed support
本文针对固定支撑且非单调的非局部核，证明当核趋近狄拉克分布时非局部守恒律解收敛至局部熵解，通过尖锐估计与熵条件分析完成收敛性证明。
Singular limit for nonlocal conservation laws with non-locality in density and velocity
研究具有密度和速度非局部性的非局部守恒律的奇异极限，通过推导得出基础Burgers方程的经典解作为奇异极限，此推导在激波形成前有效，回答了相关文献提出的问题。
Nonlocal Approximation Principle for Entropy Solutions of Scalar Conservation Laws
本文提出非局部近似原理，用于标量守恒律的熵解。通过非局部守恒律的弱星极限，证明非负初值的熵解，并探讨凸通量的收敛估计。
On the singular limit problem in nonlocal balance laws: Applications to nonlocal lane-changing traffic flow models
本文研究了非局部平衡律系统的奇异极限问题，证明了当非局部算子收敛到Dirac分布时，系统向局部熵解收敛，并应用于非局部换道交通流模型。
We investigate a class of nonlocal conservation laws in several space dimensions, where the continuum average of weighted nonlocal interactions are considered over a finite horizon. We establish well-posedness for a broad class of flux functions and initial data via semigroup theory in Banach spaces and, in particular, via the celebrated Crandall-Liggett Theorem. We also show that the unique mild solution satisfies a Kružkov-type nonlocal entropy inequality. Similarly to the local case, we demonstrate an efficient way of proving various desirable qualitative properties of the unique solution.
We introduce a kinetic formulation for scalar conservation laws with nonlocal and nonlinear diffusion terms. We deal with merely L 1 initial data, general self-adjoint pure jump L{é}vy operators, and locally Lipschitz nonlinearities of porous medium kind possibly strongly degenerate. The cornerstone of the formulation and the uniqueness proof is an adequate explicit representation of the nonlocal dissipation measure. This approach is inspired from the second order theory unlike the cutting technique previously introduced for bounded entropy solutions. The latter technique no longer seems to fi…
Infinitely many conservation laws for generalized nonlinear progressive wave equation
本文研究了广义非线性前进波方程，完整描述了其所有阶的非平凡局部守恒律，并证明了此类守恒律有无穷多个。
The obstacle problem for scalar conservation laws with nonlocal dynamics
通过推广局部守恒律的结果，提出一种松弛速度的方法，证明松弛问题解的存在性，并展示当松弛映射趋近于Heaviside型函数时解收敛到不连续非局部守恒律的弱解，同时表征极限通量。
This work is devoted to examine the uniqueness and existence of kinetic solutions for a class of scalar conservation laws involving a nonlocal super-critical diffusion operator. Our proof for uniqueness is based upon the analysis on a microscopic contraction functional and the existence is enabled by a parabolic approximation. As an illustration, we obtain the existence and uniqueness of kinetic solutions for the generalized fractional Burgers-Fisher type equations. Moreover, we demonstrate the kinetic solutions' Lipschitz continuity in time, and continuous dependence on nonlinearities and Lév…
We determine the Lie point symmetries of a Gardner type system and establish its nonlinear self-adjointness. We then construct conservation laws via Ibragimov's Theorem.
On multidimensional nonlocal conservation laws with BV kernels
本文研究多维非局部守恒律在 Sobolev 或 BV 弱正则卷积核下的局部适定性，并通过反例说明有限时间爆破后不同光滑化核会导致不同极限测度。
This paper deals with some qualitative properties of entropy solutions to hyperbolic conservation laws. In [11] the jump set of entropy solution to conservation laws has been introduced. We find an entropy solution to scalar conservation laws for which the jump set is not closed, in particular, it is dense in a space-time domain. In the later part of this article, we obtain a similar result for the hyperbolic system. We give two different approaches for scalar conservation laws and hyperbolic system to obtain the results. For the scalar case, obtained solutions are more explicitly calculated.
We consider an initial value problem for a nonlocal differential equation with a bistable nonlinearity in several space dimensions. The equation is an ordinary differential equation with respect to the time variable t, while the nonlocal term is expressed in terms of spatial integration. We discuss the large time behavior of solutions and prove, among other things, the convergence to steady-states. The proof that the solution orbits are relatively compact is based upon the rearrangement theory.
For nonlinear Schroedinger equations with a power nonlinearity, a new approach to derive the conservation law of the momentum and the pseudo conformal conservation law is obtained. Since this approach does not contain approximating procedure, the argument is simplified to derive these conservation laws.
Under a precise genuine nonlinearity assumption we establish the decay of entropy solutions of a multidimensional scalar conservation law with merely continuous flux.
We obtain conservation laws at negative regularity for the Benjamin-Ono equation on the line and on the circle. These conserved quantities control the $H^s$ norm of the solution for $-\frac{1}{2} < s < 0$. The conservation laws are obtained from a study of the perturbation determinant associated to the Lax pair of the equation.
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 are concerned with a control problem related to the vanishing \emph{fractional} viscosity approximation to scalar conservation laws. We investigate the $Γ$-convergence of the control cost functional, as the viscosity coefficient tends to zero.
We prove the decay of the L 2-distance from the solution u(t) of a hyperbolic scalar conservation law, to some convex, flow-invariant target sets.
We present a general method for obtaining conservation laws for integrable PDE at negative regularity and exhibit its application to KdV, NLS, and mKdV. Our method works uniformly for these problems posed both on the line and on the circle.
Nonlocal conservation laws with p-norm, the singular limit problem and applications to traffic flow
本文研究了具有p-范数的非局部守恒定律，探讨了奇异极限问题，并展示了其在交通流中的应用。
A common feature of systems of conservation laws of continuum physics is that they are endowed with natural companion laws which are in such case most often related to the second law of thermodynamics. This observation easily generalizes to any symmetrizable system of conservation laws. They are endowed with nontrivial companion conservation laws, which are immediately satisfied by classical solutions. Not surprisingly, weak solutions may fail to satisfy companion laws, which are then often relaxed from equality to inequality and overtake a role of a physical admissibility condition for weak s…
We provide an informal overview on the theory of transport equations with non smooth velocity fields, and on some applications of this theory to the well-posedness of hyperbolic systems of conservation laws.
We analize the Nonlinear Schrödinger Equation.
Solutions to a class of conservation laws with discontinuous flux are constructed relying on the Crandall-Liggett theory of nonlinear contractive semigroups~\cite{CL}. In particular, the paper studies the existence of backward Euler approximations, and their convergence to a unique entropy-admissible solution to the Cauchy problem. The proofs are achieved through the study of the backward Euler approximations to the viscous conservation laws.
A theorem due to Nail H. Ibragimov (2007) provides a connection between symmetries and conservation laws for arbitrary differential equations. The theorem is valid for any system of differential equations provided that the number of equations is equal to the number of dependent variables. In this paper we use the theorem to determine conservation laws for a nonlinear system of differential equations that represents a mathematical model for HIV transmission.
We establish conservation laws for the second order Kudryashov-Sinelshchikov equation, which models pressure waves in liquid with bubbles. For this purpose we use the method of Nail Ibragimov based on the notion of nonlinear self-adjointness.
Motivated by the work of P.L. Lions and J-C. Rochet [12], concerning multi-time Hamilton-Jacobi equations, we introduce the theory of multi-time systems of conservation laws. We show the existence and uniqueness of solution to the Cauchy problem for a system of multi-time conservation laws with two independent time variables in one space dimension. Our proof relies on a suitable generalization of the Lax-Oleinik formula.
Stochastic perturbation and zero noise limit for scalar conservation laws
受平均场博弈启发，研究提出标量守恒律的新型随机扰动，证明其适定性、零噪声收敛至唯一熵解，首次为非线性双曲守恒律建立噪声作为解选择准则的结果。
We determine the Lie point symmetries of a class of BBM-KdV systems and establish its nonlinear self-adjointness. We then construct conservation laws via Ibragimov's Theorem.
We propose a new sufficient non-degeneracy condition for the strong precompactness of bounded sequences satisfying the nonlinear first-order differential constraints. This result is applied to establish the decay property for periodic entropy solutions to multidimensional scalar conservation laws.
Using the complete group classification of semilinear differential equations on the three-dimensional Heisenberg group carried out in a preceding work, we establish the conservation laws for the critical Kohn-Laplace equations via the Noether's Theorem.
Existence and uniqueness of nonlocal nonlinear conservation laws via fixed-point methods
本文研究了非局部非线性守恒律的well-posedness，通过定点方法结合熵解稳定性估计，证明了弱熵解在有限时间内的存在性和唯一性，并通过数值模拟展示了记忆效应对解动力学的影响。
We prove the convergence of quasilinear parabolic viscous approximations to the entropy solution (in the sense of Bardos-Leroux-Nedelec) of a scalar conservation law, considered on a bounded domain in $\R^d$.
Following Rivière's study of conservation laws for second order quasilinear systems with critical nonlinearty and Lamm/Rivière's generalization to fourth order, we consider similar systems of order $2m$. Typical examples are $m$-polyharmonic maps. Under natural conditions, we find a conservation law for weak solutions on $2m$-dimensional domains. This implies continuity of weak solutions.
We study nonlocal conservation laws with a discontinuous flux function of regularity $\mathsf{L}^{\infty}(\mathbb{R})$ in the spatial variable and show existence and uniqueness of weak solutions in $\mathsf{C}\big([0,T];\mathsf{L}^{1}_{\text{loc}}(\mathbb{R})\big)$, as well as related maximum principles. We achieve this well-posedness by a proper reformulation in terms of a fixed-point problem. This fixed-point problem itself necessitates the study of existence, uniqueness and stability of a class of discontinuous ordinary differential equations. On the ODE level, we compare the solution type …
We discuss properties of kinetic solutions of scalar conservation laws in the variational approach developed by Eu. Panov and Y. Brenier. Our main result shows that such solutions can be considered as curves in a suitable Hilbert space with tangents that are unique minimizers of an interaction functional.
Asymptotic stability of traveling wave solutions for nonlocal viscous conservation laws with explicit decay rates
本文研究 Riesz-Feller 型非局部扩散标量守恒律的单调递减行波解，通过构造 Lyapunov 泛函证明其在 Sobolev 空间中的局部渐近稳定性，并给出扰动范数的显式时间代数衰减率。
This paper is concerned with singular shocks for a system of conservation laws modeling incompressible two-phase fluid flow. We prove the existence of viscous profiles using the Geometric Singular Perturbation Theory. Weak convergence and growth rates of the unbounded family of solutions are also obtained.
For hyperbolic systems of conservation laws, uniqueness of solutions is still largely open. We aim to expand the theory of uniqueness for systems of conservation laws. One difficulty is that many systems have only one entropy. This contrasts with scalar conservation laws, where many entropies exist. It took until 1994 to show that one entropy is enough to ensure uniqueness of solutions for the scalar conservation laws (see Panov [Mat. Zametki, 55(5):116--129, 159, 1994]). This single entropy result was proven again by De Lellis, Otto and Westdickenberg about 10 years later [Quart. Appl. Math.,…
We extend the recent work of Oliveras arXiv:2008.00940 and Oliveras & Calatola-Young arXiv:2105.07580 to develop a new nonlocal formulation of the water-wave problem for a three-dimensional fluid with a two-dimensional free surface for an inviscid and irrotational fluid over a flat bathymetry. Using this new formulation, we show how one can systematically derive Benjamin & Olver's twelve conservation laws without explicitly relying on the underlying Lie symmetries. This allows us to make draw new conclusions about conservation laws and posit the potential existence of additional, nonlocal, con…
In this paper we study large time behaviors toward shock waves and rarefaction waves under periodic perturbations for 1-D convex scalar conservation laws. The asymptotic stabilities and decay rates of shock waves and rarefaction waves under periodic perturbations are proved.
Nonlocal conservation laws for the two-dimensional Euler equation in vorticity form
本研究结合标准守恒律构造与非局部余对称性，推导出涡量形式二维欧拉方程的非局部守恒律，并通过自变量复旋转简化计算与结果呈现。
I consider the existence and structure of conservation laws for the general class of evolutionary scalar second-order differential equations with parabolic symbol. First I calculate the linearized characteristic cohomology for such equations. This provides an auxiliary differential equation satisfied by the conservation laws of a given parabolic equation. This is used to show that conservation laws for any evolutionary parabolic equation depend on at most second derivatives of solutions. As a corollary, it is shown that the only evolutionary parabolic equations with at least one non-trivial co…
In this paper, we study the output feedback stabilization for a scalar conservation law with a nonlocal velocity, that models a highly re-entrant manufacturing system as encountered in semi-conductor production. By spectral analysis, we obtain a complete result on the exponential stabilization for the linearized control system. Moreover, by using a Lyapunov function approach, we also prove the exponential stabilization results for the nonlinear control system in certain cases.
Control of Conservation Laws in the Nonlocal-to-Local Limit
该研究针对初始数据为控制、状态由守恒律熵解给出的控制问题，通过非局部到局部极限策略，刻画了非局部控制问题极小化子的子序列极限为对应局部问题的极小解，并在离散层面用Eulerian-Lagrangian方案验证了类似结果。
A set of infinitely many nonlocal conservation laws are revealed for (1+1)-dimensional evolution equations. For some special known integrable systems, say, the KdV and Dym equations, it is found that different nonlocal conservation laws can lead to same new integrable systems via reciprocal transformation. On the other hand, it can be considered as one solution of the new model obtained via reciprocal transformation(s) can be changed to different solutions of the original model. The fact indicates also that two or more different (local and nonlocal) conservation laws can be used to find implic…
We are concerned with a control problem related to the vanishing viscosity approximation to scalar conservation laws. We investigate the $Γ$-convergence of the control cost functional, as the viscosity coefficient tends to zero. A first order $Γ$-limit is established, which characterizes the measure-valued solutions to the conservation laws as the zeros of the $Γ$-limit. A second order $Γ$-limit is then investigated, providing a characterization of entropic solutions to conservation laws as the zeros of the $Γ$-limit.
We prove a path-by-path regularization by noise result for scalar conservation laws. In particular, this proves regularizing properties for scalar conservation laws driven by fractional Brownian motion and generalizes the respective results obtained in [Gess, Souganidis; Comm. Pure Appl. Math. (2017)]. In addition, we introduce a new path-by-path scaling property which is shown to be sufficient to imply regularizing effects.
We investigate the quasi-potential problem for the entropy cost functionals of non-entropic solutions to scalar conservation laws with smooth fluxes. We prove that the quasi-potentials coincide with the integral of a suitable Einstein entropy.
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