Viscous Conservation Laws Research Articles

On the Extinction of Multiple Shocks in Scalar Viscous Conservation Laws

On the Extinction of Multiple Shocks in Scalar Viscous Conservation Laws

Asymptotic stability of planar rarefaction wave to a 2D hyperbolic-elliptic coupling system of the radiating gas on half-space

This paper studies the asymptotic stability of solutions to an initial-boundary value problem for a hyperbolic-elliptic coupling system on the two-dimensional half-space, where the data on the boundary and at the far field are prescribed as u_{-} and u_{+} , respectively. We show that the solution to the problem converges to the corresponding planar rarefaction wave for 0\le u_{-}<u_{+} as time tends to infinity. To the best of our knowledge, the stability results of planar rarefaction waves on half-space focus primarily on the single viscous conservation law because the rarefaction wave (one-dimensional diffusion wave) of the corresponding one-dimensional problem to scalar viscous conservation law is known. In other words, for a general high-dimensional system of equations, we cannot obtain the stability of planar rarefaction waves on half-space because we cannot construct the rarefaction wave of the corresponding one-dimensional problem. In this paper, we use the structure of the hyperbolic-elliptic coupling system to obtain the monotonic rarefaction wave of the corresponding one-dimensional hyperbolic-elliptic coupling system, and hence give the stability of the planar rarefaction wave on half-space. This can be viewed as the first result for the system of equations on the stability of planar rarefaction waves on half-space.

Pushed, pulled and pushmi-pullyu fronts of the Burgers-FKPP equation

We consider the long time behavior of the solutions to the Burgers-FKPP equation with advection of a strength \beta\in\mathbb{R} . This equation exhibits a transition from pulled to pushed front behavior at \beta_{c}=2 . We prove convergence of the solutions to a traveling wave in a reference frame centered at a position m_{\beta}(t) and study the asymptotics of the front location m_{\beta}(t) . When \beta < 2 , it has the same form as for the standard Fisher-KPP equation established by Bramson: m_{\beta}(t) = 2t - (3/2)\log t + x_{\infty} + o(1) as t\to\infty . This form is typical of pulled fronts. When \beta > 2 , the front is located at the position m_{\beta}(t)=c_{*}(\beta)t+x_{\infty}+o(1) with c_{*}(\beta)=\beta/2+2/\beta , which is the typical form of pushed fronts. However, at the critical value \beta_{c} = 2 , the expansion changes to m_{\beta}(t) = 2t - (1/2)\log t + x_{\infty} + o(1) , reflecting the “pushmi-pullyu” nature of the front. The arguments for \beta<2 rely on a new weighted Hopf–Cole transform that allows one to control the advection term, when combined with additional steepness comparison arguments. The case \beta>2 relies on standard pushed front techniques. The proof in the case \beta=\beta_{c} is much more intricate and involves arguments not usually encountered in the study of the Bramson correction. It relies on a somewhat hidden viscous conservation law structure of the Burgers-FKPP equation at \beta_{c}=2 and utilizes a dissipation inequality, which comes from a relative entropy type computation, together with a weighted Nash inequality involving dynamically changing weights.

Sharp bounds on enstrophy growth for viscous scalar conservation laws

We prove sharp bounds on the enstrophy growth in viscous scalar conservation laws. The upper bound is, up to a prefactor, the enstrophy created by the steepest viscous shock admissible by the L∞ and total variation bounds and viscosity. This answers a conjecture by Ayala and Protas (2011 Physica D 240 1553–63), based on numerical evidence, for the viscous Burgers equation.

Planar Viscous Shocks with Periodic Perturbations for Scalar Multidimensional Viscous Conservation Laws

This paper studies a Cauchy problem for a scalar multidimensional (multi-d) viscous convex conservation law, in which the initial data is a planar viscous shock with a multi-d periodic perturbation. We show that if the wave-strength and perturbation are both small, then the viscous shock is stable in the space with an exponential decay rate. One contribution of the paper is to establish a new framework to study the stability of viscous shocks in multiple dimensions, where the elementary energy method with the antiderivative technique can be used. The idea is to decompose the multi-d perturbation into a one-dimensional function and a multi-d remainder, where the former can well define its antiderivative and the latter satisfies the Poincaré inequality over a periodic domain.

Partial dissipation and sub-shock

To study the dissipation property of a physical system one first considers infinitesimal waves for the analysis of weakly nonlinear phenomena. For some physical systems, the dissipation is partial and there is the appearance of sub-shocks in a strong traveling trajectory. The phenomenon of partial dissipation can occur for systems of hyperbolic balance laws and also for viscous conservation laws in continuum physics. We illustrate the phenomenon for a simple relaxation model and for the Navier-Stokes equations for compressible media. The admissibility criteria and the formation of sub-shocks are studied through the zero viscosity limit.

Weak and strong error analysis for mean-field rank-based particle approximations of one-dimensional viscous scalar conservation laws

In this paper, we analyse the rate of convergence of a system of N interacting particles with mean-field rank-based interaction in the drift coefficient and constant diffusion coefficient. We first adapt arguments by Kolli and Shkolnikov (Ann. Probab. 46 (2018) 1042–1069) to check trajectorial propagation of chaos with optimal rate N−1/2 to the associated stochastic differential equations nonlinear in the sense of McKean. We next relax the assumptions needed by Bossy (Math. Comp. 73 (2004) 777–812) to check the convergence in L1(R) with rate O(1N+h) of the empirical cumulative distribution function of the Euler discretization with step h of the particle system to the solution of a one-dimensional viscous scalar conservation law. Last, we prove that the bias of this stochastic particle method behaves as O(1N+h). We provide numerical results which confirm our theoretical estimates.

Asymptotic traveling wave solutions to anisotropic higher-order traffic flow models with viscosities

The boundary layer correction method is applied to study asymptotic traveling wave solutions to anisotropic higher-order traffic flow model with viscous coefficient, which clearly indicates that the truncation error between the original and approximate equations is an equivalent infinitesimal of the coefficient. Therefore, the solution is exact in the case when the coefficient vanishes, which corresponds to a certain nonviscous model following the standard hyperbolic and viscous conservation laws. Numerical simulation is implemented to reproduce a typical traveling wave or wide moving jam consisting of a transitional layer (the downstream front) and a shock (the upstream front) profiles, of which the parameters converge not only for refined meshes but for refined viscous coefficient as well. These are highly in agreement with the analytical results.

Asymptotic Stability of Solutions to a Hyperbolic-Elliptic Coupled System of Radiating Gas on the Half Line

This paper is concerned with the asymptotic stability of the solution to an initial-boundary value problem on the half line for a hyperbolic-elliptic coupled system of radiating gas, where the data on the boundary and at the far field state are defined as $u_-$ and $u_+$ satisfying $u_-<u_+$. For the scalar viscous conservation law case, it is known by the work of Liu, Matsumura, and Nishihara [SIAM J. Math. Anal., 29 (1998), pp. 293--308] that the solution tends toward a rarefaction wave or stationary solution or superposition of these two kinds of waves depending on the distribution of $u_\pm$. Motivated by their work, we prove the stability of the above three types of wave patterns for the hyperbolic-elliptic coupled system of radiating gas with small perturbation. A singular phase plane analysis method is introduced to show the existence and the precise asymptotic behavior of the stationary solution, especially for the degenerate case: $u_-<u_+=0$ such that the system has inevitable singularities. The stability of the rarefaction wave, the stationary solution, and their superposition is proved by applying the standard $L^2$-energy method.

Time decay rate of solutions toward the viscous shock waves to the Cauchy problem for the scalar conservation law with nonlinear viscosity and discontinuous initial data

In this paper, it is shown that the weak solution to the Cauchy problem for the scalar viscous conservation law, with nonlinear viscosity and different far field states, exists globally in time. Especially, this existence holds with discontinuous initial data. Furthermore, when the corresponding Riemann problem for the hyperbolic part has a single shock wave solution, it is proved that the weak solution converges toward the viscous shock wave as time goes to infinity, and the decay rate is also obtained. The proof is given by a technical energy method.

Propagating fronts for a viscous Hamer-type system

&lt;p style='text-indent:20px;'&gt;Motivated by radiation hydrodynamics, we analyse a &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ 2\times2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; system consisting of a one-dimensional viscous conservation law with strictly convex flux –the viscous Burgers' equation being a paradigmatic example– coupled with an elliptic equation, named &lt;b&gt;viscous Hamer-type system&lt;/b&gt;. In the regime of small viscosity and for large shocks, namely when the profile of the corresponding underlying inviscid model undergoes a discontinuity –usually called &lt;i&gt;sub-shock&lt;/i&gt;– it is proved the existence of a smooth propagating front, regularising the jump of the corresponding inviscid equation. The proof is based on &lt;i&gt;Geometric Singular Perturbation Theory&lt;/i&gt; (GSPT) as introduced in the pioneering work of Fenichel [&lt;xref ref-type="bibr" rid="b5"&gt;5&lt;/xref&gt;] and subsequently developed by Szmolyan [&lt;xref ref-type="bibr" rid="b21"&gt;21&lt;/xref&gt;]. In addition, the case of small shocks and large viscosity is also addressed via a standard bifurcation argument.&lt;/p&gt;

Navier‐Stokes Equations in Gas Dynamics: Green's Function, Singularity, and Well‐Posedness

The purpose of the present article is to study weak solutions of viscous conservation laws in physics. We are interested in the well‐posedness theory and the propagation of singularity in the weak solutions for the initial value problem. Our approach is to convert the differential equations into integral equations on the level of weak solutions. This depends on exact analysis of the associated linear equations and their Green's functions. We carry out our approach for the Navier‐Stokes equations in gas dynamics. Local in time as well as time‐asymptotic behaviors of weak solutions, and the continuous dependence of the solutions on their initial data are established. © 2022 Wiley Periodicals LLC.

Stability of Large-Amplitude Viscous Shock Under Periodic Perturbation for 1-d Isentropic Navier–Stokes Equations

The stability of solutions under periodic perturbations for both inviscid and viscous conservation laws is an interesting and important problem. In this paper, a large-amplitude viscous shock under space-periodic perturbation for the isentropic Navier–Stokes equations is considered. It is shown that if the initial perturbation around the shock is suitably small and satisfies a zero-mass type condition (2.17), then the solution of the N–S equations tends to the viscous shock with a shift, which is partially determined by the periodic oscillations. In other words, the viscous shock is nonlinearly stable even though the perturbation oscillates at the far fields. The key point is to construct a suitable ansatz $$ (\tilde{v},\tilde{u}) $$ , which carries the same oscillations of the solution (v, u) at the far fields, so that the difference $$ (v-\tilde{v},u-\tilde{u}) $$ belongs to the $$ H^2(\mathbb {R}) $$ space for all $$ t\ge 0. $$

Finite-volume approximation of the invariant measure of a viscous stochastic scalar conservation law

Abstract We study the numerical approximation of the invariant measure of a viscous scalar conservation law, one-dimensional and periodic in the space variable and stochastically forced with a white-in-time but spatially correlated noise. The flux function is assumed to be locally Lipschitz continuous and to have at most polynomial growth. The numerical scheme we employ discretizes the stochastic partial differential equation (SPDE) according to a finite-volume method in space and a split-step backward Euler method in time. As a first result, we prove the well posedness as well as the existence and uniqueness of an invariant measure for both the semidiscrete and the split-step scheme. Our main result is then the convergence of the invariant measures of the discrete approximations, as the space and time steps go to zero, towards the invariant measure of the SPDE, with respect to the second-order Wasserstein distance. We investigate rates of convergence theoretically, in the case where the flux function is globally Lipschitz continuous with a small Lipschitz constant, and numerically for the Burgers equation.

L2-type contraction of viscous shocks for scalar conservation laws

We study small shocks of 1D scalar viscous conservation laws with uniformly convex flux and nonlinear dissipation. We show that such shocks are [Formula: see text] stable independently of the strength of the dissipation, even with large perturbations. The proof uses the relative entropy method with a spatially-inhomogeneous pseudo-norm.

Stability of planar rarefaction waves for scalar viscous conservation law under periodic perturbations

The large time behavior of the solutions to a multi-dimensional viscous conservation law is considered in this paper. It is shown that the solution time-asymptotically tends to the planar rarefaction wave if the initial perturbations are multi-dimensional periodic. The time-decay rate is also obtained. Moreover, a Gagliardo-Nirenberg type inequality is established in the domain $ \mathbb R \times \mathbb T^{n-1} (n\geq2) $, where $\mathbb T^{n-1}$ is the $ n-1 $-dimensional torus.

一类具有多个激波层结构粘性守恒律方程的粘性消失极限问题

一类具有多个激波层结构粘性守恒律方程的粘性消失极限问题

Asymptotic stability of shock profiles and rarefaction waves under periodic perturbations for 1-D convex scalar viscous conservation laws

This paper studies the asymptotic stability of shock profiles and rarefaction waves under space-periodic perturbations for one-dimensional convex scalar viscous conservation laws. For the shock profile, we show that the solution approaches the background shock profile with a constant shift in the $ L^\infty(\mathbb{R}) $ norm at exponential rates. The new phenomena contrasting to the case of localized perturbations is that the constant shift cannot be determined by the initial excessive mass in general, which indicates that the periodic oscillations at infinities make contributions to this shift. And the vanishing viscosity limit for the shift is also shown. The key elements of the poof consist of the construction of an ansatz which tends to two periodic solutions as $ x \rightarrow \pm\infty, $ respectively, and the anti-derivative variable argument, and an elaborate use of the maximum principle. For the rarefaction wave, we also show the stability in the $ L^\infty(\mathbb{R}) $ norm.

L2-type contraction for shocks of scalar viscous conservation laws with strictly convex flux

We study the L2-type contraction property of large perturbations around shock waves of scalar viscous conservation laws with strictly convex fluxes in one space dimension. The contraction holds up to a shift, and it is measured by a weighted relative entropy, for which we choose an appropriate entropy associated with the strictly convex flux. In particular, we handle shocks with small amplitude. This result improves the recent article [19] of the author and Vasseur on L2-contraction property of shocks to scalar viscous conservation laws with a special flux, that is almost the Burgers flux.

Local discontinuous Galerkin method for a nonlocal viscous conservation laws

SummaryThe purpose of this article is to investigate high‐order numerical approximations of scalar conservation laws with nonlocal viscous term. The viscous term is given in the form of convolution in space variable. With the help of the characteristic of viscous term, we design a semidiscrete local discontinuous Galerkin (LDG) method to solve the nonlocal model. We prove stability and convergence of semidiscrete LDG method in L2 norm. The theoretical analysis reveals that the present numerical scheme is stable with optimal convergence order for the linear case, and it is stable with sub‐optimal convergence order for nonlinear case. To demonstrate the validity and accuracy of our scheme, we test the Burgers equation with two typical nonlocal fractional viscous terms. The numerical results show the convergence order accuracy in space for both linear and nonlinear cases. Some numerical simulations are provided to show the robustness and effectiveness of the present numerical scheme.