Myunghyun Oh, Kevin Zumbrun
We establish instability of periodic traveling waves arising in conservation laws featuring phase transition. The analysis uses the Evans function framework introduced by R.A. Gardner in the periodic case. The main new tool is a periodic generalization of the stability index introduced by Gardner and Zumbrun in the traveling front or pulse case.
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We establish one-dimensional spectral stability of small amplitude viscous and relaxation shock profiles using Evans function techniques to perform a series of reductions and normal forms to reduce to the case of the scalar Burgers equation. In multidimensions, the canonical behavior is described, rather by a 2x2 viscous conservation law; this case will be treated in a companion paper by similar techniques.
In this paper, we investigate the asymptotic behavior of solutions to the Cauchy problem for the scalar viscous conservation law where the far field states are prescribed. Especially, we deal with the case when the viscosity is of non-Newtonian type, including a pseudo-plastic case. When the corresponding Riemann problem for the hyperbolic part admits a Riemann solution which consists of single rarefaction wave, under a condition on nonlinearity of the viscosity, it is proved that the solution of the Cauchy problem tends toward the rarefaction wave as time goes to infinity, without any smallne…
Shock formation in 1D conservation laws II: Vanishing viscosity
本文研究一维双曲守恒律中弱粘性对激波形成的影响，通过匹配渐近展开确定无粘激波形成前强范数的尖锐收敛速率，并识别奇点附近的普适粘性行为，适用于可压缩纳维-斯托克斯方程。
In this paper, we investigate the asymptotic behavior of solutions toward a multiwave pattern of the Cauchy problem for the scalar viscous conservation law where the far field states are prescribed. Especially, we deal with the case when the flux function is convex or concave but linearly degenerate on some interval, and also the viscosity is a nonlinearly degenerate one (p-Laplacian type viscosity). When the corresponding Riemann problem admits a Riemann solution which consists of rarefaction waves and contact discontinuity, it is proved that the solution of the Cauchy problem tends toward th…
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.…
Decay properties of solutions toward shock waves of the scalar conservation law with linear and $p$-Laplacian viscosity
研究标量粘性守恒律弱解向粘性激波的渐近衰减，分别在 Burgers 通量下的退化 $p$-Laplacian 粘性与线性粘性、以及一般通量线性粘性情形中给出 $L^\infty$ 时间衰减率，且无需初始数据小性假设。
Kružkov-type uniqueness theorem for the chemical flood conservation law system with local vanishing viscosity admissibility
本文针对化学驱守恒律系统初边值问题，在分段光滑函数类中利用局部消失粘性方法判定激波容许性，并结合拉格朗日坐标变换与Kružkov型熵不等式证明了解的唯一性。
Energy Conservation and Vanishing Viscosity Limit for the Primitive Equations
本文针对有界域中无黏原始方程弱解的能量守恒问题，引入新思路克服垂直速度信息缺失、区域含角点的困难，在Onsager型假设下证明带边界条件的能量守恒，并给出消失黏性极限中无反常能量耗散的充分条件。
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.
In this paper, we study the decay rate in time to solutions of the Cauchy problem for the one-dimensional viscous conservation law where the far field states are prescribed. Especially, we deal with the case that the flux function which is convex and also the viscosity is a nonlinearly degenerate one (p-Laplacian type viscosity). As the corresponding Riemann problem admits a Riemann solution as the constant state or the single rarefaction wave, it has already been proved by Matsumura-Nishihara that the solution to the Cauchy problem tends toward the constant state or the single rarefaction wav…
In this paper we study a non strictly system of conservation law when viscosity is present and viscosity is zero, which is studied in [10]. We show the existence and uniqueness of the solution in the space of generalized functions of Colombeau for the viscous problem and construct a solution to the inviscid system in the sense of association. Also we construct a solution using shadow wave approach [5] and Volpert product which was partly determined as vanishing viscosity limit in [10].
In this paper, we investigate large amplitude solutions to a system of conservation laws which is transformed, by a change of variable, from the well-known Keller-Segel model describing cell (bacteria) movement toward the concentration gradient of the chemical that is consumed by the cells. For the Cauchy problem and initial-boundary value problem, the global unique solvability is proved based on the energy method. In particular, our main purpose is to investigate the convergence rates as the diffusion parameter $ε$ goes to zero. It is shown that the convergence rates in $L^\infty$-norm are of…
In this paper we investigate spectral stability of traveling wave solutions to 1-$D$ quantum hydrodynamics system with nonlinear viscosity in the $(ρ,u)$, that is, density and velocity, variables. We derive a sufficient condition for the stability of the essential spectrum and we estimate the maximum modulus of eigenvalues with non-negative real part. In addition, we present numerical computations of the Evans function in sufficiently large domain of the unstable half-plane and show numerically that its winding number is (approximately) zero, thus giving a numerical evidence of point spectrum …
Vanishing viscosity limit for $n\times n$ hyperbolic system of conservation laws in 1-d with nonlinear viscosity: Part-I Uniform BV estimates
针对具有非奇异粘性矩阵且严格双曲的一维守恒律系统，在初始数据小BV且矩阵可交换条件下，证明了解的一致BV界，并得到粘性消失极限的唯一全局弱解，应用于Navier-Stokes-Korteweg系统的粘色散极限。
We are concerned with multidimensional stochastic balance laws driven by Lévy processes. Using bounded variation (BV) estimates for vanishing viscosity approximations, we derive an explicit continuous dependence estimate on the nonlinearities of the entropy solutions under the assumption that Lévy noise only depends on the solution. This result is used to show the error estimate for the stochastic vanishing viscosity method. In addition, we establish fractional $BV$ estimate for vanishing viscosity approximations in case the noise coefficient depends on both the solution and spatial variable.
We study a nonlocal regularisation of a scalar conservation law given by a fractional derivative of order between one and two. The nonlocal operator is of Riesz-Feller type with skewness two minus its order. This equation describes the internal structure of hydraulic jumps in a shallow water model. The main purpose of the paper is the study of the vanishing viscosity limit of the Cauchy problem for this equation. First, we study the properties of the solution of the regularised problem and then we show that solutions converge to the entropy solution of the scalar conservation law in this limit…
In this paper, we study the precise decay rate in time to solutions of the Cauchy problem for the one-dimensional conservation law with a nonlinearly degenerate viscosity where the far field states are prescribed. Especially, we deal with the case when the flux function is convex or concave but linearly degenerate on some interval. As the corresponding Riemann problem admits a Riemann solution as a multiwave pattern which consists of the rarefaction waves and the contact discontinuity, it has already been proved by Yoshida that the solution to the Cauchy problem tends toward the linear combina…
Existence and Blow-up for Non-autonomous Scalar Conservation Laws With Viscosity
本文完整解答了带黏性非自治标量守恒律在特定系数条件下的爆破问题，给出了解行为的精确刻画。
For any open set $Ω\subset\mathbb R^n$ and $n\ge 2$, we establish everywhere differentiability of viscosity solutions to the Aronsson equation $$ <D_x(H(x, Du)), D_p H(x, Du)>=0 \quad \rm in\ \ Ω, $$ where $H$ is given by $$H(x,\,p)=<A(x)p,p>=\sum_{i,\,j=1}^na^{ij}(x)p_i p_j,\ x\inΩ, \ p\in\mathbb R^n, $$ and $A=(a^{ij}(x))\in C^{1,1}(\barΩ,\mathbb R^{n\times n})$ is uniformly elliptic. This extends an earlier theorem by Evans and Smart \cite{es11a} on infinity harmonic functions.
Vanishing viscosity solution to a 2 x 2 system of conservation laws with linear damping
本文针对源自欧拉液滴模型的带线性阻尼2×2非严格双曲守恒律方程组，研究其变黏度正则化系统的弱渐近解与Colombeau代数解，证明消失黏度极限为分布解，还分析了大时间渐近行为，突破了此前仅研究黎曼型初值的局限。
Large time behaviour of solutions to the $N-$dimensional scalar conservation law under periodic perturbations with nonlinear degenerate viscosity
研究含非线性退化粘性的 $N$ 维标量守恒律，证明周期初值下解的存在唯一性、正则性与衰减率，并分析周期扰动趋近平面稀疏波的长时间行为。
Numerical spectral analysis of standing waves in quantum hydrodynamics with viscosity
本文利用Evans函数技术研究了带线性和非线性粘性的量子流体力学系统在驻波附近的线性化谱，数值发现存在实不稳定特征值，为谱不稳定性提供了数值证据。
Existence of vanishing physical viscosity solutions of characteristic initial-boundary value problems for systems of conservation laws
针对单空间变量非线性守恒律方程组的边界特征型初边值问题，研究者在小总变差框架下，通过精细边界层分析和新的波前追踪算法，证明了满足物理黏性消失近似边界条件的容许解的存在性，结果适用于Navier-Stokes等方程。
Global Existence, Hamiltonian Conservation and Vanishing Viscosity for the Surface Quasi-Geostrophic Equation
针对表面准地转方程，证明了任意初始数据下全局弱解的存在性，该解保持哈密顿量（$\dot{H}^{-1/2}$范数）守恒，并通过消失粘性极限获得。
On the role of numerical viscosity in the study of the local limit of nonlocal conservation laws
本文研究了非局部守恒律局部极限的数值研究中数值粘性的作用，指出数值粘性可能影响数值方案的可靠性，并通过实验验证了不同数值方案在不同情况下的表现。
In this paper we study existence and stability of shock profiles for a 1-D compressible Euler system in the context of Quantum Hydrodynamic models. The dispersive term is originated by the quantum effects described through the Bohm potential; moreover we introduce a (linear) viscosity to analyze its interplay with the former while proving existence, monotonicity and stability of travelling waves connecting a Lax shock for the underlying Euler system. The existence of monotone profiles is proved for sufficiently small shocks; while the case of large shocks leads to the (global) existence for an…
We present a quantitative compensated compactness estimate for stochastic conservation laws, which generalises a previous result of Golse & Perthame (2013) for deterministic equations. With a stochastic modification of Kruzkov's interpolation lemma, this estimate provides bounds on the rate at which a sequence of vanishing viscosity solutions becomes compact. This contribution is for the Proceedings of HYP2022.
Aim of this paper is to review some basic ideas and recent developments in the theory of strictly hyperbolic systems of conservation laws in one space dimension. The main focus will be on the uniqueness and stability of entropy weak solutions and on the convergence of vanishing viscosity approximations.
For a general class of hyperbolic-parabolic systems including the compressible Navier-Stokes and compressible MHD equations, we prove existence and stability of noncharacteristic viscous boundary layers for a variety of boundary conditions including classical Navier-Stokes boundary conditions. Our first main result, using the abstract framework established by the authors in the companion work \cite{GMWZ6}, is to show that existence and stability of arbitrary amplitude exact boundary-layer solutions follow from a uniform spectral stability condition on layer profiles that is expressible in term…
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.
On the spectral vanishing viscosity method for periodic fractional conservation laws
针对周期分数阶守恒律，提出并分析了一种谱消失粘性逼近方法，通过非单调的傅里叶伽辽金方法结合谱粘性项，证明了该方法收敛于熵解，保持谱精度，并显著降低分数阶项的计算成本。
Local Asymptotic Patterns for Viscous Approximations of Conservation Laws
研究粘性消失近似在激波附近的行为，通过局部坐标缩放获得极限解，揭示近似方程的永恒解特性。
We prove the well-posedness of entropy weak solutions for a class of space-discontinuous scalar conservation laws with non-local flux arising in traffic modeling. We approximate the problem adding a viscosity term and we provide $L^\infty$ and BV estimates for the approximate solutions. We use the doubling of variable technique to prove the stability with respect to the initial data from the entropy condition.
Spectral stability of shock profiles for hyperbolically regularized systems of conservation laws
本文证明在自然假设且激波振幅足够小时，双曲正则化守恒律系统的异宿行波激波剖面谱稳定，Evans 函数仅在原点有一个单零点。
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.
We propose new entropy admissibility conditions for multidimensional hyperbolic scalar conservation laws with discontinuous flux which generalize one-dimensional Karlsen-Risebro-Towers entropy conditions. These new conditions are designed, in particular, in order to characterize the limit of vanishing viscosity approximations. On the one hand, they comply quite naturally with a certain class of physical and numerical modeling assumptions; on the other hand, their mathematical assessment turns out to be intricate. \smallskip The generalization we propose is not only with respect to the space di…
In this paper, we investigate the global structure of solutions to the Cauchy problem for the scalar viscous conservation law where the far field states are prescribed. Especially, we deal with the case when the viscous/diffusive flux $σ(v) \sim |\, v \,|^p$ is of non-Newtonian type (i.e., $p>0$), including a pseudo-plastic case (i.e., $p<0$). When the corresponding Riemann problem for the hyperbolic part admits a Riemann solution which consists of single rarefaction wave, under a condition on nonlinearity of the viscosity, it has been recently proved by Matsumura-Yoshida \cite{matsumura-yoshi…
Germs for scalar conservation laws: the Hamilton-Jacobi equation point of view
本文通过Hamilton-Jacobi方程视角证明具有空间不连续流函数的标量守恒律熵解与相应HJ方程解的导数一致，并刻画了AKR理论中的极大胚、完全胚及粘性解框架中的松弛算子。
We establish the well-posedness of the Neumann problem for stochastic conservation laws with multiplicative noise. As a major step for establishing the uniqueness of the kinetic solution to the referred problem we establish the new strong trace property for stochastic conservation laws. Existence of kinetic solutions is proved through the vanishing viscosity method and the detailed analysis of the corresponding stochastic parabolic problem is also made here for the first time, as far as the authors know.
In this article, we explore some of the main mathematical problems connected to multidimensional fractional conservation laws driven by Lévy processes. Making use of an adapted entropy formulation, a result of existence and uniqueness of a solution is established. Moreover, using bounded variation (BV) estimates for vanishing viscosity approximations, we derive an explicit continuous dependence estimate on the nonlinearities of the entropy solutions under the assumption that Lévy noise depends only on the solution. This result is used to show the error estimate for the stochastic vanishing vis…
Diffusion stabilises time-periodic solutions in conservation laws coupled to a relaxation oscillator
研究与快速常微分方程耦合的粘性一维守恒律，利用平均策略和谱理论方法，证明对称初始条件下小正粘性会选择唯一线性稳定的周期解，数值模拟验证结果。
In this work we introduce the obstacle-mass constraint problem for a multidimensional scalar hyperbolic conservation law. We prove existence of an entropy solution to this problem by a penalization/viscosity method. The mass constraint introduces a nonlocal Lagrange multiplier in the penalized equation, giving rise to a nonlocal parabolic problem. We determine conditions on the initial data and on the obstacle function which ensure global in time existence of solution. These are not smoothness conditions, but relate to the propagation of the support of the initial data.
We study by a combination of numerical and analytical Evans function techniques the stability of solitary wave solutions of the St. Venant equations for viscous shallow-water flow down an incline, and related models. Our main result is to exhibit examples of metastable solitary waves for the St. Venant equations, with stable point spectrum indicating coherence of the wave profile but unstable essential spectrum indicating oscillatory convective instabilities shed in its wake. We propose a mechanism based on ``dynamic spectrum'' of the wave profile, by which a wave train of solitary pulses can …
Let $H$ be a bounded and Lipschitz continuous function. We consider discontinuous viscosity solutions of the Hamilton-Jacobi equation $U_{t}+H(U_x)=0$ and signed Radon measure valued entropy solutions of the conservation law $u_{t}+[H(u)]_x=0$. After having proved a precise statement of the formal relation $U_x=u$, we establish estimates for the (strictly positive!) times at which singularities of the solutions disappear. Here singularities are jump discontinuities in case of the Hamilton-Jacobi equation and signed singular measures in case of the conservation law.
We consider multidimensional conservation laws perturbed by multiplicative Lévy noise. We establish existence and uniqueness results for entropy solutions. The entropy inequalities are formally obtained by the Itô-Lévy chain rule. The multidimensionality requires a generalized interpretation of the entropy inequalities to accommodate Young measure-valued solutions. We first establish the existence of entropy solutions in the generalized sense via the vanishing viscosity method, and then establish the $L^1$-contraction principle. Finally, the $L^1$ contraction principle is used to argue that th…
In this paper, we consider the Cauchy problem for the nonlinear fractional conservation laws driven by a multiplicative noise. In particular, we are concerned with the well-posedness theory and the study of the long-time behavior of solutions for such equations. We show the existence of desired kinetic solution by using the vanishing viscosity method. In fact, we establish strong convergence of the approximate viscous solutions to a kinetic solution. Moreover, under a nonlinearity-diffusivity condition, we prove the existence of an invariant measure using the well-known Krylov-Bogoliubov theor…
Accurate numerical schemes for approximating initial-boundary value problems for systems of conservation laws
针对守恒律系统初边值问题解依赖于粘性机制的问题，设计了融入显式粘性信息的数值格式以逼近物理相关解，并通过数值实验验证了鲁棒性。
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。
Shock formation in 1D conservation laws I: Inviscid structure
本文研究一维双曲守恒律中激波形成的稳定性与结构，证明其在简单波附近稳定，刻画经典发展边界，并用分数次齐次函数展开描述非退化激波形成的精确性质。
In this paper we show existence of a trace for functions of bounded variation on Riemannian manifolds with boundary. The trace, which is bounded in $L^\infty$, is reached via $L^1$-convergence and allows an integration by parts formula. We apply these results in order to show well-posedness and total variation estimates for the initial boundary value problem for a scalar conservation law on compact Riemannian manifolds with boundary in the context of functions of bounded variation via the vanishing viscosity method. The flux function is assumed to be time-dependent and divergence-free.
Extending to systems of hyperbolic--parabolic conservation laws results of Howard and Zumbrun for strictly parabolic systems, we show for viscous shock profiles of arbitrary amplitude and type that necessary spectral (Evans function) conditions for linearized stability established by Mascia and Zumbrun are also sufficient for linearized and nonlinear phase-asymptotic stability, yielding detailed pointwise estimates and sharp rates of convergence in $L^p$, $1\le p\le \infty$.
The Oleinik inequality for conservation laws and Aronson-Benilan type inequalities for porous medium or p-Laplacian equations are one-sided inequalities that provide the fundamental features of the solution such as the uniqueness and sharp regularity. In this paper such one-sided inequalities are unified and generalized for a wide class of first and second order equations in the form of $$ u_t=σ(t,u,u_x,u_{xx}),\quad u(x,0)=u^0(x)\ge0,\quad t>0,\,x\in\bfR, $$ where the non-strict parabolicity ${\partial\over\partial q} σ(t,z,p,q)\ge0$ is assumed. The generalization or unification of one-sided …
Under natural spectral stability assumptions motivated by previous investigations of the associated spectral stability problem, we determine sharp $L^p$ estimates on the linearized solution operator about a multidimensional planar periodic wave of a system of conservation laws with viscosity, yielding linearized $L^1\cap L^p\to L^p$ stability for all $p \ge 2$ and dimensions $d \ge 1$ and nonlinear $L^1\cap H^s\to L^p\cap H^s$ stability and $L^2$-asymptotic behavior for $p\ge 2$ and $d\ge 3$. The behavior can in general be rather complicated, involving both convective (i.e., wave-like) and dif…
A remark on the uniqueness of solutions to hyperbolic conservation laws
本文针对严格双曲守恒律方程组，证明了任何满足 Liu 容许性条件且取值于半群定义域内的弱解必与唯一的 Lipschitz 半群轨迹重合，从而强化了解的唯一性结论。
Strong time regularity and decay of $L^\infty$ solutions to $2\times 2$ systems of conservation laws
本文研究$2\times2$守恒律系统的$\mathbf L^\infty$解，利用动力学表述证明有限熵解的强时间正则性，并为消失黏性解建立色散型衰减估计。
We investigate a class of scalar conservation laws on manifolds driven by multiplicative Gaussian (Ito) noise. The Cauchy problem defined on a Riemannian manifold is shown to be well-posed. We prove existence of generalized kinetic solutions using the vanishing viscosity method. A rigidity result is derived, which implies that generalized solutions are kinetic solutions and that kinetic solutions are uniquely determined by their initial data ($L^1$ contraction principle). Deprived of noise, the equations we consider coincide with those analyzed by Ben-Artzi and LeFloch (2007), who worked with …
We give a full characterization of the range of the operator which associates, to any initial condition, the viscosity solution at time $T$ of a Hamilton-Jacobi equation with convex Hamiltonian. Our main motivation is to be able to treat the case of convex Hamiltonians with no further regularity assumptions. We give special attention to the case $H(p) = |p|$, for which we provide a rather geometrical description of the range of the viscosity operator by means of an interior ball condition on the sublevel sets. From our characterization of the reachable set, we are able to deduce further result…
We consider the asymptotic behavior of perturbations of Lax and overcompressive type viscous shock profiles arising in systems of regularized conservation laws with strictly parabolic viscosity, and also in systems of conservation laws with partially parabolic regularizations such as arise in the case of the compressible Navier--Stokes equations and in the equations of magnetohydrodynamics. Under the necessary conditions of spectral and hyperbolic stability, together with transversality of the connecting profile, we establish detailed pointwise estimates on perturbations from a sum of the visc…
This paper deals with a hyperbolic system of two nonlinear conservation laws, where the phase space contains two contact manifolds. The governing equations are modelling bidisperse suspensions, which consist of two types of small particles that are dispersed in a viscous fluid and differ in size and viscosity. For certain parameter choices quasi-umbilic points and a contact manifold in the interior of the phase space are detected. The dependance of the solutions structure on this contact manifold is examined. The elementary waves that start in the origin of the phase space are classified. Prot…
Selection problems of Z^2-periodic entropy solutions and viscosity solutions
本文研究了双曲守恒律的Z²周期熵解和哈密顿-雅可比方程的Z²周期粘性解在有限差分近似下的选择问题，提出不同于消去粘性法的选取准则，揭示两种近似方法在特征上的差异。
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