Pointwise Green's function bounds and stability of relaxation shocks
Corrado Mascia, Kevin Zumbrun
备注：
120 pages. Changes since original submission. Corrected typos, esp. energy estimates of Section 7, corrected bad forward references, expanded Remark 1.17, end of introduction
We establish sharp pointwise Green's function bounds and consequent linearized and nonlinear stability for smooth traveling front solutions, or relaxation shocks, of general hyperbolic relaxation systems of dissipative type, under the necessary assumptions ([G,Z.1,Z.4]) of spectral stability, i.e., stable point spectrum of the linearized operator about the wave, and hyperbolic stability of the corresponding ideal shock of the associated equilibrium system. This yields, in particular, nonlinear stability of weak relaxation shocks of the discrete kinetic Jin--Xin and Broadwell models. The techniques of this paper should have further application in the closely related case of traveling waves of systems with partial viscosity, for example in compressible gas dynamics or MHD.
The pointwise space-time behaviors of the Green's function and the global solution to the Vlasov-Poisson-Fokker-Planck (VPFP) system in spatial three dimension are studied in this paper. It is shown that the Green's function consists of the diffusion waves decaying exponentially in time but algebraically in space, and the singular kinetic waves which become smooth for all $(t,x,v)$ when $t>0.$ Furthermore, we establish the pointwise space-time behaviors of the global solution to the nonlinear VPFP system when the initial data is not necessarily smooth in terms of the Green's function.
Using pointwise semigroup techniques of Zumbrun--Howard and Mascia--Zumbrun, we obtain sharp global pointwise Green function bounds for noncharacteristic boundary layers of arbitrary amplitude. These estimates allow us to analyze linearized and nonlinearized stability of noncharacteristic boundary layers of one-dimensional systems of conservation laws, showing that both are equivalent to a numerically checkable Evans function condition. Our results extend to the large-amplitude case results obtained for small amplitudes by Matsumura, Nishihara and others using energy estimates.
The pointwise space-time behaviors of the Green's function and the global solution to the modified Vlasov- Poisson-Boltzmann (mVPB) system in one-dimensional space are studied in this paper. It is shown that, the Green's function admits the diffusion wave, the Huygens's type sound wave, the singular kinetic wave and the remainder term decaying exponentially in space-time. These behaviors are similar to the Boltzmann equation (Liu and Yu in Comm. Pure Appl. Math. 57: 1543-1608, 2004). Furthermore, we establish the pointwise space-time nonlinear diffusive behaviors of the global solution to the …
Green's Function and Pointwise Space-time Behaviors of the three-Dimensional modified Vlasov-Poisson-Boltzmann System
School of Mathematics and Information Sciences, Guangxi University（广西大学数学与信息科学学院）；Center for Applied Mathematical of Guangxi (Guangxi University)（广西应用数学中心（广西大学））
This article is about the $\mathbb{Z}^d$-periodic Green function $G_n(x,y)$ of the multiscale elliptic operator $Lu=-{\rm div}\left( A(n\cdot) \cdot \nabla u \right)$, where $A(x)$ is a $\mathbb{Z}^d$-periodic, coercive, and Hölder continuous matrix, and $n$ is a large integer. We prove here pointwise estimates on $G_n(x,y)$, $\nabla_x G_n(x,y)$, $\nabla_y G_n(x,y)$ and $\nabla_x \nabla_y G_n(x,y)$ in dimensions $d \geq 2$. Moreover, we derive an explicit decomposition of this Green function, which is of independent interest. These results also apply for systems.
Refining previous work in \cite{Z.3, MaZ.3, Ra, HZ, HR}, we derive sharp pointwise bounds on behavior of perturbed viscous shock profiles for large-amplitude Lax or overcompressive type shocks and physical viscosity. These extend well-known results of Liu \cite{Liu97} obtained by somewhat different techniques for small-amplitude Lax type shocks and artificial viscosity, completing a program set out in \cite{ZH}. As pointed out in \cite{Liu91, Liu97}, the key to obtaining sharp bounds is to take account of cancellation associated with the property that the solution decays faster along character…
We establish global pointwise bounds for the Green's matrix for divergence form, second order elliptic systems in a domain under the assumption that weak solutions of the system vanishing on a portion of the boundary satisfy a certain local boundedness estimate. Moreover, we prove that such a local boundedness estimate for weak solutions of the system is equivalent to the usual global pointwise bound for the Green's matrix. In the scalar case, such an estimate is a consequence of De Giorgi-Moser-Nash theory and holds for equations with bounded measurable coefficients in arbitrary domains. In t…
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…
We prove the first ever pointwise estimates of the (unrestricted) Green tensor and the associated pressure tensor of the nonstationary Stokes system in the half-space, for every space dimension greater than one. The force field is not necessarily assumed to be solenoidal. The key is to find a suitable Green tensor formula which maximizes the tangential decay, showing in particular the integrability of Green tensor derivatives. With its pointwise estimates, we show the symmetry of the Green tensor, which in turn improves pointwise estimates. We also study how the solutions converge to the initi…
We prove the existence and pointwise bounds of the Green functions for stationary Stokes systems with measurable coefficients in two dimensional domains. We also establish pointwise bounds of the derivatives of the Green functions under a regularity assumption on the $L_1$-mean oscillations of the coefficients.
We construct the Green function for second-order elliptic equations in non-divergence form when the mean oscillations of the coefficients satisfy the Dini condition. We show that the Green's function is BMO in the domain and establish logarithmic pointwise bounds. We also obtain pointwise bounds for first and second derivatives of the Green's function.
Generalizing similar results for viscous shock and relaxation waves, we establish sharp pointwise Green function bounds and linearized and nonlinear stability for traveling wave solutions of an abstract viscous combustion model including both Majda's model and the full reacting compressible Navier--Stokes equations with artificial viscosity with general multi-species reaction and reaction-dependent equation of state, % under the necessary conditions of strong spectral stability, i.e., stable point spectrum of the linearized operator about the wave, transversality of the profile as a connection…
In the previous paper \cite{J1}, we established pointwise bounds for the Green function of the linearized equation associated with spatially periodic traveling waves $\bar u$ of a system of reaction diffusion equations, and also obtained pointwise nonlinear stability and behavior of $\bar u$ under small perturbations. In this paper, using periodic resolvent kernels and the Bloch-decomposition, we establish pointwise bounds for the Green function of the linearized equation associated with periodic standing waves $\bar u$ of a system of conservation laws. We also show pointwise nonlinear stabili…
In this paper we focus on the global-in-time existence and the pointwise estimates of solutions to the initial value problem for the semilinear dissipative wave equation in multi-dimensions. By using the method of Green function combined with the energy estimates, we obtain the pointwise estimates of the solution.
The paper deals with a mixed boundary value problem for the Stokes system in a polyhedral cone. Here different boundary conditions (in particular, Dirichlet, Neumann, free surface conditions) are prescribed on the sides of the polyhedron. The authors obtain regularity results for weak solutions in weighted $L_2$ Sobolev spaces and prove point estimates of Green's matrix.
In this paper, we investigate the pointwise space-time behavior of two-phase fluid model derived by Choi \cite{Choi} [SIAM J. Math. Anal., 48(2016), pp. 3090-3122], which is the compressible damped Euler equations coupled with compressible Naiver-Stokes equations. Based on Green's function method together with frequency analysis and nonlinear coupling of different wave patterns, it shows that both of two densities and momentums obey the generalized Huygens' principle as the compressible Navier-Stokes equations \cite{LW}, however, it is different from the compressible damped Euler equations \ci…
This work establishes nonlinear orbital asymptotic stability of scalar radiative shock profiles, namely, traveling wave solutions to the simplified model system of radiating gas \cite{Hm}, consisting of a scalar conservation law coupled with an elliptic equation for the radiation flux. The method is based on the derivation of pointwise Green function bounds and description of the linearized solution operator. A new feature in the present analysis is the construction of the resolvent kernel for the case of an eigenvalue system of equations of degenerate type. Nonlinear stability then follows in…
In this paper, we investigate and prove the nonlinear stability of viscous shock wave solutions of a scalar viscous conservation law, using the methods developed for general systems of conservation laws by Howard, Mascia, Zumbrun and others, based on instantaneous tracking of the location of the perturbed viscous shock wave. In some sense, this paper extends the treatment in a previous expository work of Zumbrun ["Instantaneous shock location ..."] on Burgers equation to the general case, giving an exposition of these methods in the simplest setting of scalar equations. In particular we give b…
We prove a pointwise control for the Green's function of polyharmonic operators with holes: this control is uniform while holes shrink. For the usual Laplacian, such a control is given by the maximum principle; the techniques developed here applies to general polyharmonic operators for which there is no comparison principle.
Extending previous work with Lattanzio and Mascia on the scalar (in fluid-dynamical variables) Hamer model for a radiative gas, we show nonlinear orbital asymptotic stability of small-amplitude shock profiles of general systems of coupled hyperbolic--eliptic equations of the type modeling a radiative gas, that is, systems of conservation laws coupled with an elliptic equation for the radiation flux, including in particular the standard Euler--Poisson model for a radiating gas. The method is based on the derivation of pointwise Green function bounds and description of the linearized solution op…
We construct the Green function for second order elliptic equations in non-divergence form when the mean oscillations of the coefficients satisfy the Dini condition and the domain has $C^{1,1}$ boundary. We also obtain pointwise bounds for the Green functions and its derivatives.
We present a new method for the existence and pointwise estimates of a Green's function of non-divergence form elliptic operator with Dini mean oscillation coefficients. We also present a sharp comparison with the corresponding Green's function for constant coefficients equations.
The classical Orr-Sommerfeld equations are the resolvent equations of the linearized Navier Stokes equations around a stationary shear layer profile in the half plane. In this paper, we derive pointwise bounds on the Green function of the Orr Sommerfeld problem away from its critical layers.
We construct Green's function for second order elliptic operators of the form $Lu=-\nabla \cdot (\mathbf{A} \nabla u + \boldsymbol{b} u)+ \boldsymbol c \cdot \nabla u+ du$ in a domain and obtain pointwise bounds, as well as Lorentz space bounds. We assume that the matrix of principal coefficients $\mathbf A$ is uniformly elliptic and bounded and the lower order coefficients $\boldsymbol{b}$, $\boldsymbol{c}$, and $d$ belong to certain Lebesgue classes and satisfy the condition $d - \nabla \cdot \boldsymbol{b} \ge 0$. In particular, we allow the lower order coefficients to be singular. We also …
This paper is the continuation of a program, initiated in Grenier-Nguyen [8,9], to derive pointwise estimates on the Green function of Orr Sommerfeld equations. In this paper we focus on long wavelength perturbations, more precisely horizontal wavenumbers $α$ of order $ν^{1/4}$, which correspond to the lower boundary of the instability area for monotonic profiles.
We establish pointwise bounds for the Green function and consequent linearized stability for multidimensional planar relaxation shocks of general relaxation systems whose equilibrium model is scalar, under the necessary assumption of spectral stability. Moreover, we obtain nonlinear $L^{2}$ asymptotic behavior/sharp decay rate of perturbed weak shocks of general simultaneously symmetrizable relaxation systems, under small $L^{1}\cap H^{[d/2]+3}$ perturbations with first moment in the normal direction to the front.
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 $L^p$, $1\le p\le \infty$.
In this paper, we investigate the pointwise behavior of the solution for the compressible Navier-Stokes equations with mixed boundary condition in half space. Our results show that the leading order of Green's function for the linear system in half space are heat kernels propagating with sound speed in two opposite directions and reflected heat kernel (due to the boundary effect) propagating with positive sound speed. With the strong wave interactions, the nonlinear analysis exhibits the rich wave structure: the diffusion waves interact with each other and consequently, the solution decays wit…
Space-time behaviors for 3D compressible bipolar Navier-Stokes-Poisson system (BNSP) with unequal viscosities are given. The space-time estimate of electric field $\nablaϕ$ is the most important thing when deducing generalized Huygens' principle for BNSP since this estimate only can be obtained by $\nablaϕ=\frac{\nabla}Δ(ρ-n)$ from the Poisson equation. Thus, it requires to prove that the space-time estimate of $ρ-n$ only contains diffusion wave. The appearance of these unequal coefficients results that one cannot follow ideas for the special case, where the original system was rewritten as a …
We study Green functions for the pressure of stationary Stokes systems in a (possibly unbounded) domain $Ω\subset \mathbb{R}^d$, where $d\ge 2$. We construct the Green function when coefficients are merely measurable in one direction and have Dini mean oscillation in the other directions, and $Ω$ is such that the divergence equation is solvable there. We also establish global pointwise bounds for the Green function and its derivatives when coefficients have Dini mean oscillation and $Ω$ has a $C^{1,\rm{Dini}}$ boundary. Green functions for the flow velocity of Stokes systems are also considere…
We establish existence and pointwise estimates of fundamental solutions and Green's matrices for divergence form, second order strongly elliptic systems in a domain $Ω\subseteq \mathbb{R}^n$, $n \geq 3$, under the assumption that solutions of the system satisfy De Giorgi-Nash type local Hölder continuity estimates. In particular, our results apply to perturbations of diagonal systems, and thus especially to complex perturbations of a single real equation.
We study Green functions for stationary Stokes systems satisfying the conormal derivative boundary condition. We establish existence, uniqueness, and various estimates for the Green function under the assumption that weak solutions of the Stokes system are continuous in the interior of the domain. Also, we establish the global pointwise bound for the Green function under the additional assumption that weak solutions of the conormal derivative problem for the Stokes system are locally bounded up to the boundary. We provide some examples satisfying such continuity and boundedness properties.
By working with the periodic resolvent kernel and Bloch-decomposition, we establish pointwise bounds for the Green function of the linearized equation associated with spatially periodic traveling waves of a system of reaction diffusion equations.With our linearized estimates together with a nonlinear iteration scheme developed by Johnson-Zumbrun, we obtain $L^p$- behavior($p \geq 1$) of a nonlinear solution to a perturbation equation of a reaction-diffusion equation with respect to initial data in $L^1 \cap H^1$ recovering and slightly sharpening results obtained by Schneider using weighted en…
Optimal pointwise estimates are derived for the biharmonic Green function under Dirichlet boundary conditions in arbitrary $C^{4,γ}$-smooth domains. Maximum principles do not exist for fourth order elliptic equations and the Green function may change sign. It prevents using a Harnack inequality as for second order problems and hence complicates the derivation of optimal estimates. The present estimate is obtained by an asymptotic analysis. The estimate shows that this Green function is positive near the singularity and that a possible negative part is small in the sense that it is bounded by t…
This is a continuation and completion of the program (initiated in \cite{GrN1,GrN2}) to derive pointwise estimates on the Green function and sharp bounds on the semigroup of linearized Navier-Stokes around a generic stationary boundary layer profile. This is done via a spectral analysis approach and a careful study of the Orr-Sommerfeld equations, or equivalently the Navier-Stokes resolvent operator $(λ- L)^{-1}$. The earlier work (\cite{GrN1,GrN2}) treats the Orr-Sommerfeld equations away from critical layers: this is the case when the phase velocity is away from the range of the background p…
We study pointwise behavior of positive solutions to nonlinear integral equations, and related inequalities, of the type \begin{equation*} u(x) - \int_ΩG(x, y) \, g(u(y)) d σ(y) = h, \end{equation*} where $(Ω, σ)$ is a locally compact measure space, $G(x, y)\colon Ω\times Ω\to [0, +\infty]$ is a kernel, $h \ge 0$ is a measurable function, and $g\colon [0, \infty)\to [0, \infty)$ is a monotone function. This problem is motivated by the semilinear fractional Laplace equation \begin{equation*} (-Δ)^{\fracα{2}} u - g(u) σ= μ\quad \text{in} \, \, Ω, \quad u=0 \, \, \, \text{in} \, \, Ω^c, \end{equa…
We investigate the $L^p $ asymptotic behavior $(1\le p \le \infty)$ of a perturbation of a Lax or overcompressive type shock wave solution to a system of conservation law in one dimension. The system of the equations can be strictly parabolic, or have real viscosity matrix (partially parabolic, e.g., compressible Navier--Stokes equations or equations of Magnetohydrodynamics). We use known pointwise Green function bounds for the linearized equation around the shock to show that the perturbation of such a solution can be decomposed into a part corresponding to shift in shock position or shape, a…
We construct Green's functions for divergence form, second order parabolic systems in non-smooth time-varying domains whose boundaries are locally represented as graph of functions that are Lipschitz continuous in the spatial variables and 1/2-Hölder continuous in the time variable, under the assumption that weak solutions of the system satisfy an interior Hölder continuity estimate. We also derive global pointwise estimates for Green's function in such time-varying domains under the assumption that weak solutions of the system vanishing on a portion of the boundary satisfy a certain local bou…
Motivated by the study of conditional stability of traveling waves, we give an elementary $H^2$ center stable manifold construction for quasilinear parabolic PDE, sidestepping apparently delicate regularity issues by the combination of a carefully chosen implicit fixed-point scheme and "damping-type" $H^s$ energy estimates of a type familiar from the study of hyperbolic--parabolic and relaxation systems. An important feature of these methods is that they generalize to situations such as the hyperbolic--parabolic or relaxation case for which parabolic-type smoothing estimates are unavailable. A…
We study the Green function for the stationary Stokes system with bounded measurable coefficients in a bounded Lipschitz domain $Ω\subset \mathbb{R}^n$, $n\ge 3$. We construct the Green function in $Ω$ under the condition $(\bf{A1})$ that weak solutions of the system enjoy interior Hölder continuity. We also prove that $(\bf{A1})$ holds, for example, when the coefficients are $\mathrm{VMO}$. Moreover, we obtain the global pointwise estimate for the Green function under the additional assumption $(\bf{A2})$ that weak solutions of Dirichlet problems are locally bounded up to the boundary of the …
Motivated by physical and numerical observations of time oscillatory ``galloping'', ``spinning'', and ``cellular'' instabilities of detonation waves, we study Poincaré--Hopf bifurcation of traveling-wave solutions of viscous conservation laws. The main difficulty is the absence of a spectral gap between oscillatory modes and essential spectrum, preventing standard reduction to a finite-dimensional center manifold. We overcome this by direct Lyapunov--Schmidt reduction, using detailed pointwise bounds on the linearized solution operator to carry out a nonstandard implicit function construction …
We construct Green's functions for elliptic operators of the form $\mathcal{L}u=-\text{div}(A\nabla u+bu)+c\nabla u+du$ in domains $Ω\subseteq\mathbb R^n$, under the assumption $d\geq\text{div}b$, or $d\geq\text{div}c$. We show that, in the setting of Lorentz spaces, the assumption $b-c\in L^{n,1}(Ω)$ is both necessary and optimal to obtain pointwise bounds for Green's functions. We also show weak type bounds for Green's functions and their gradients. Our estimates are scale invariant and hold for general domains $Ω\subseteq\mathbb R^n$. Moreover, there is no smallness assumption on the norms …
Pointwise arbitrarily high-order interior estimates for mixed local and nonlocal elliptic equations
School of Mathematics and Statistics, Xinyang Normal University（信阳师范大学数学与统计学院）；School of Mathematics, South China University of Technology（华南理工大学数学学院）；School of Mathematical Sciences, Fudan University（复旦大学数学科学学院）
We establish the existence and the pointwise bound of the fundamental solution for the stationary Stokes system with measurable coefficients in the whole space $\mathbb{R}^d$, $d \ge 3$, under the assumption that weak solutions of the system are locally Hölder continuous. We also discuss the existence and the pointwise bound of the Green function for the Stokes system with measurable coefficients on $Ω$, where $Ω$ is an unbounded domain such that the divergence equation is solvable. Such a domain includes, for example, half space and an exterior domain.
In order to understand the nonlinear stability of many types of time-periodic travelling waves on unbounded domains, one must overcome two main difficulties: the presence of embedded neutral eigenvalues and the time-dependence of the associated linear operator. This problem is studied in the context of time-periodic Lax shocks in systems of viscous conservation laws. Using spatial dynamics and a decomposition into separate Floquet eigenmodes, it is shown that the linear evolution for the time-dependent operator can be represented using a contour integral similar to that of the standard time-in…
