Stability of scalar radiative shock profiles††thanks: This work was supported in part by the National Science Foundation award number DMS-0300487. CL, CM and RGP are warmly grateful to the Department of Mathematics, Indiana University, for their hospitality and financial support during two short visits in May 2008 and April 2009, when this research was carried out. The research of RGP was partially supported by DGAPA-UNAM through the program PAPIIT, grant IN-109008.
This work establishes nonlinear orbital asymptotic stability of scalar radiative shock profiles, namely, traveling wave solutions to the simplified model system of radiating gas , 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 standard fashion by linear estimates derived from these pointwise bounds, combined with nonlinear-damping type energy estimates.
yperbolic-elliptic coupled systems, Radiative shock, pointwise Green function bounds, Evans function.
35B35 (34B27 35M20 76N15)
The one-dimensional motion of a radiating gas (due to high-temperature effects) can be modeled by the compressible Euler equations coupled with an elliptic equation for the radiative flux term [8, 39]. The present work considers the following simplified model system of a radiating gas
consisting of a single regularized conservation law coupled with a scalar elliptic equation. In (1), , and are scalar functions of , is a constant, and are scalar functions of . Typically, and represent velocity and heat flux of the gas, respectively. When the velocity flux is the Burgers flux function, , and the coupling term is linear ( constant), this system constitutes a good approximation of the physical Euler system with radiation , and it has been extensively studied by Kawashima and Nishibata [16, 17, 18], Serre  and Ito , among others. For the details of such approximation the reader may refer to [17, 19, 8].
Formally, one may express in terms of as , where , so that system (1) represents some regularization of the hyperbolic (inviscid) associated conservation law for . Thus, a fundamental assumption in the study of such systems is that
for all under consideration, conveying the right sign in the diffusion coming from Chapman–Enskog expansion (see ).
We are interested in traveling wave solutions to system (1) of the form
where the triple is a shock front of Lax type of the underlying scalar conservation law for the velocity,
satisfying Rankine-Hugoniot condition , and Lax entropy condition . Morover, we assume genuine nonlinearity of the conservation law (4), namely, that the velocity flux is strictly convex,
for all under consideration, for which the entropy condition reduces to . We refer to weak solutions of the form (3) to the system (1), under the Lax shock assumption for the scalar conservation law, as radiative shock profiles. The existence and regularity of traveling waves of this type under hypotheses (2) is known [16, 22], even for non-convex velocity fluxes .
According to custom and without loss of generality, we can reduce to the case of a stationary profile , by introducing a convenient change of variable and relabeling the flux function accordingly. Therefore, and after substitution, we consider a stationary radiative shock profile solution to (1), satisfying
(here denotes differentiation with respect to ), connecting endpoints at , that is,
Therefore, we summarize our main structural assumptions as follows:
|(Lax entropy condition),||(A3)|
where . For concreteness let us denote
and assume (up to translation) that . Besides the previous structural assumptions we further suppose that
Under assumption (A4), the radiative shock profile is monotone, and, as shown later on, the spectral stability condition holds. Let us stress that, within the analysis of the linearized problem and of the nonlinear stability, we only need (A4) to hold at the end states and at the degenerating value .
Hypotheses (A5) are a set of additional technical assumptions inherited from the present stability analysis (see the establishment of energy estimates of Section 6 below, and of pointwise reduction bounds in Lemma 3.3) and are not necessarily sharp. It is worth mentioning, however, that assumptions (A5), with , are satisfied, for instance, for all profiles with small-amplitude , in view of (2) and .
In the present paper, we establish the asymptotic stability of the shock profile under small initial perturbation. Nonlinear wave behavior for system (1) and its generalizations has been the subject of thorough research over the last decade. The well-posedness theory is the object of study in [21, 14, 15, 12] and , both for the simplified model system and more general cases. The stability of constant states , rarefaction waves [19, 5], asymptotic profiles [24, 4, 3] for the model system with Burgers flux has been addressed in the literature.
Regarding the asymptotic stability of radiative shock profiles, the problem has been previously studied by Kawashima and Nishibata  in the particular case of Burgers velocity flux and for linear , which is one of the few available stability results for scalar radiative shocks in the literature111The other scalar result is the partial analysis of Serre  for the exact Rosenau model; in the case of systems, we mention the stability result of  for the full Euler radiating system under zero-mass perturbations, based on an adaptation of the classical energy method of Goodman-Matsumura-Nishihara [7, 30].. In , the authors establish asymptotic stability with basically the same rate of decay in and under fairly similar assumptions as we have here. Their method, though, relies on integrated coordinates and contraction property, a technique which may not work for the system case (i.e., , ). In contrast, we provide techniques which may be extrapolated to systems, enable us to handle variable , and provide a large-amplitude theory based on spectral stability assumptions in cases that linearized stability is not automatic (e.g., system case, or variable). These technical considerations are some of the main motivations for the present analysis.
The nonlinear asymptotic stability of traveling wave solutions to models in continuum mechanics, more specifically, of shock profiles under suitable regularizations of hyperbolic systems of conservation laws, has been the subject of intense research in recent years (see, e.g., [10, 43, 26, 27, 28, 40, 41, 42, 34, 32, 20]). The unifying methodological approach of these works consists of refined semigroup techniques and the establishment of sharp pointwise bounds on the Green function associated to the linearized operator around the wave, under the assumption of spectral stability. A key step in the analysis is the construction of the resolvent kernel, together with appropriate spectral bounds. The pointwise bounds on the Green function follow by the inverse Laplace transform (spectral resolution) formula [43, 27, 40]. The main novelty in the present case is the extension of the method to a situation in which the eigenvalue equations are written as a degenerate first order ODE system (see discussion in Section 1.3 below). Such extension, we hope, may serve as a blueprint to treat other model systems for which the resolvent equation becomes singular. This feature is also one of the main technical contributions of the present analysis.
1.1 Main results
where source is the initial data .
An evident necessary condition for orbital stability is the absence of solutions to (10) for values of in , being the eigenvalue associated to translation invariance. This strong spectral stability can be expressed in terms of the Evans function, an analytic function playing a role for differential operators analogous to that played by the characteristic polynomial for finite-dimensional operators (see [1, 35, 6, 43, 26, 27, 41, 40, 42] and the references therein). The main property of the Evans function is that, on the resolvent set of a certain operator , its zeroes coincide in both location and multiplicity with the eigenvalues of .
In the present case and due to the degenerate nature of system (10) (observe that vanishes at ) the number of decaying modes at , spanning possible eigenfunctions, depends on the region of space around the singularity (see Section 3 below, Remark 3.2). Therefore, we define the following stability criterion, where the analytic functions (see their definition in (54) below) denote the two Evans functions associated with the linearized operator about the profile in regions , correspondingly, analytic functions whose zeroes away from the essential spectrum agree in location and multiplicity with the eigenvalues of the linearized operator or solutions of (10):
Our main result is then as follows.
for initial perturbation that are sufficiently small in , for all , for some satisfying and
where denotes the derivative with respect to .
The time-decay rate of is not optimal. In fact, it can be improved as we observe that and is expected to decay like ; we omit, however, the details of the proof.
The second result of this paper is the verification of the spectral stability condition (D) under particular circumstances.
The spectral stability condition (D) holds under either
(i) is a constant; or,
(ii) is sufficiently small.
See Appendix B.
Combining Theorem 1.1 and Proposition 1.1, we partially recover the results of  for the Burgers flux and constant , , and at the same time extend them to general convex flux and quasilinear . We note that the stability result of  was for all smooth shock profiles, for which the boundary (see , Thm. 1.25(ii)(a)) is the condition ; that is, their results hold whenever . By comparison, our results hold on the smaller set of waves for which ; see Remark 1. By estimating high-frequency contributions explicitly, rather than by the simple energy estimates used here, we could at the expense of further effort reduce these conditions to the single condition
used to prove Lemma 3.3. Elsewhere in the analysis, we need only ; however, at the moment we do not see how to remove (11) to recover the full result of  in the special case considered there. The interest of our technique, rather, is in its generality —particularly the possibility to extend to the system case— and in the additional information afforded by the pointwise description of behavior, which seems interesting in its own right.
1.3 Abstract framework
Before beginning the analysis, we orient ourselves with a few simple observations framing the problem in a more standard way. Consider now the inhomogeneous version
of (8), with initial data . Defining the compact operator and the bounded operator
we may rewrite this as a nonlocal equation
in alone. The generator of (13) is a zero-order perturbation of the generator of a hyperbolic equation, so it generates a semigroup and an associated Green distribution . Moreover, and may be expressed through the inverse Laplace transform formulae
for all (for some ), where is the resolvent kernel of .
Collecting information, we may write the solution of (12) using Duhamel’s principle/variation of constants as
where is determined through (14).
That is, the solution of the linearized problem reduces to finding the Green kernel for the -equation alone, which in turn amounts to solving the resolvent equation for with delta-function data, or, equivalently, solving the differential equation (9) with source . We shall do this in standard fashion by writing (9) as a first-order system and solving appropriate jump conditions at obtained by the requirement that be a distributional solution of the resolvent equations.
This procedure is greatly complicated by the circumstance that the resulting first-order system, given by
is singular at the special point where vanishes. However, in the end we find as usual that is uniquely determined by these criteria, not only for the values guaranteed by -semigroup theory/energy estimates, but, as in the usual nonsingular case , on the set of consistent splitting for the first-order system, which includes all of . This has the implication that the essential spectrum of is confined to .
The fact (obtained by energy-based resolvent estimates) that is coercive for shows by elliptic theory that the resolvent is well-defined and unique in class of distributions for large, and thus the resolvent kernel may be determined by the usual construction using appropriate jump conditions. That is, from standard considerations, we see that the construction must work, despite the apparent wrong dimensions of decaying manifolds (which happen for any ).
To deal with the singularity of the first-order system is the most delicate and novel part of the present analysis. It is our hope that the methods we use here may be of use also in other situations where the resolvent equation becomes singular, for example in the closely related situation of relaxation systems discussed in [26, 29].
Plan of the paper
This work is structured as follows. Section 2 collects some of the properties of radiative profiles, and contains a technical result which allows us to rigorously define the resolvent kernel near the singularity. The central Section 3 is devoted to the construction of the resolvent kernel, based on the analysis of solutions to the eigenvalue equations both near and away from the singularity. Section 4 establishes the crucial low frequency bounds for the resolvent kernel. The following Section 5 contains the desired pointwise bounds for the “low-frequency” Green function, based on the spectral resolution formulae. Section 6 establishes an auxiliary nonlinear damping energy estimate. Section 7 deals with the high-frequency region by establishing energy estimates on the solution operator directly. The final Section 8 blends all previous estimations into the proof of the main nonlinear stability result (Theorem (1.1)). We also include three Appendices, which contain, a pointwise extension of the Tracking lemma, the proof of spectral stability under linear coupling or small-amplitude assumptions, and the monotonicity of general scalar profiles, respectively.
2.1 Structure of profiles
Under definition (7), we may assume (thanks to translation invariance; see Remark C below) that vanishes exactly at one single point which we take as . Likewise, we know that the velocity profile is monotone decreasing (see [22, 23, 38] or Lemma C below), that is , which implies, in view of genuine nonlinearity (5), that
From the profile equations we obtain, after integration, that
for all , due to Lax condition. Therefore, substitution of the profile equations (6) yields the relation
which, evaluating at and from monotonicity of the profile, implies that
Next, we show that the waves decay exponentially to their end states, a crucial fact in the forthcoming analysis.
As , defining and , we consider the asymptotic system of (6), that is the constant coefficient linear system
which, by substituting into the second equation, becomes
which then gives the exponential decay estimate (16) for by the hyperbolicity of the matrix , that is, eigenvalues of are distinct and nonzero. Estimates for follow immediately from those for and the relation
obtained by integrating the first equation of (6).
2.2 Regularity of solutions near
In this section we establish some analytic properties of the solutions to system (10) near the singularity, which will be used during the construction of the resolvent kernel in the central Section 3 below. Introducing the variable , system (10) takes the form of a first-order system, which reads
For technical reasons which will be clear from the forthcoming analysis, in order to define the transmission conditions in the definition of the resolvent kernel (which is defined as solutions to the conservative form of system (17) in distributional sense with appropriate jump conditions; see Section 3.1 below), we need and to be regular across the singularity (having finite limits at both sides), to have (at most) an integrable singularity at that point, namely, that near zero (away from zero it is bounded, so this is trivially true), and that it verifies as . These properties are proved in the next technical lemma.
for and for some ;
is absolutely continuous and is (for ),
In particular, (for ) and as .
The proof will be done in two steps: (i) first, taking into account “elliptic regularity” in the equation for ,
we prove the bound for close to zero and the subsequent regularity for and ; and (ii), using such a bound, we then prove the pointwise control given in .
Alternatively, one can explicitly solve the above elliptic equation for and get directly the pointwise result for by plugging the relation into the Duhamel formula for . Finally, such a control gives the property for and all other regularity properties.
[Proof of Lemma 2.2] Let us consider the case , the case being similar. Consider a fixed , to be chosen afterwards and let be any solution of (17) emanating from that point. Therefore, from (18) we know that
for a given (regular) kernel . Therefore there exists a constant such that for any
Note that the constant is uniform on and it stays bounded as approaches zero and it depends only on the initial values , . Moreover, the Duhamel principle gives for any :
From (5) we obtain
Hence, for ,
Hence the first term of (2.2) is integrable in provided , being (our argument applies for ; for all functions in the integrals above are indeed bounded at zero and the proof of the lemma is even simpler). Thus, for a constant as above,
Finally, for a sufficiently small, but fixed , from the above relation we conclude
Once we have obtained the property of at zero, we know in particular is bounded. Hence we can repeat all estimates on the integral terms of (2.2) to obtain part 1. of the lemma. Finally,
is again a consequence of .
From condition (15) it is clear that, for , but sufficiently close to zero, is not blowing up for , but it vanishes in that limit, regardless of the shock strength (the negative term approaches zero as the strength of the shock tends to zero).
3 Construction of the resolvent kernel
Let us now construct the resolvent kernel for , or equivalently, the solution of (17) with delta-function source in the component. The novelty in the present case is the extension of this standard method to a situation in which the spectral problem can only be written as a degenerate first order ODE. Unlike the real viscosity and relaxation cases [26, 27, 28, 29] (where the operator , although degenerate, yields a non-degenerate first order ODE in an appropriate reduced space), here we deal with the resolvent system for the unknown
that degenerates at .
To construct the resolvent kernel , we solve
in the distributional sense, so that
for all with appropriate jump conditions (to be determined) at . The first element in the first row of the matrix-valued function is the resolvent kernel of that we seek.
3.2 Asymptotic behavior
First, we study the asymptotic behavior of solutions to the spectral system
away from the singularity at , and for values of , . We pay special attention to the small frequency regime, . Denote the limits of the coefficients as
From the structure of the wave we already have that . The asymptotic system can be written as
To determine the dimensions of the stable/unstable eigenspaces, let , . The characteristic polynomial reads
has one negative and one positive zero, regardless of the sign of , for each ; they are local extrema of . Since as , has the opposite sign with respect to and
so that is positive/negative at some negative/positive value of , we get two positive and one negative zeroes for , and two negative and one positive zeroes for , whenever , .
We readily conclude that for each , there exist two unstable eigenvalues and with , and one stable eigenvalue with . The stable and unstable manifolds (solutions which decay, respectively, grow at ) have, thus, dimensions
in . Likewise, there exist two unstable eigenvalues with , and one stable eigenvalue with , so that the stable (solutions which grow at ) and unstable (solutions which decay at ) manifolds have dimensions
Notice that, unlike customary situations in the Evans function literature [1, 43, 6, 26, 27, 35], here the dimensions of the stable (resp. unstable) manifolds and (resp. and ) do not agree. Under these considerations, we look at the dispersion relation
For each , the -roots of last equation define algebraic curves
touching the origin at . Denote as the open connected subset of bounded on the left by the two curves , . Since by assumption (A4), the set is properly contained in . By connectedness the dimensions of and do not change in . We define as the set of (not so) consistent splitting , in which the matrices remain hyperbolic, with not necessarily agreeing dimensions of stable (resp. unstable) manifolds.
In the low frequency regime , we notice, by taking , that the eigenvalues behave like those of . If we define
as the decay/growth rates for the fast modes (notice that , ), then the latter are given by
The associated eigenvectors are given by
Since the highest order coefficient of as a polynomial in is different from zero, then is a regular point and whence, by standard algebraic curves theory, there exist convergent series in powers of for the eigenvalues. For low frequency the eigenvalues of have analytic expansions of the form
corresponding to a slow varying mode and two fast modes, respectively, for low frequencies. By inspection, the associated eigenvectors can be chosen as
Notice, in particular, that for this choice of bases, there hold, for ,
Moreover, for , we can find analytic representations for and , which consist of two slow modes
and four fast modes,
with associated eigenvectors (29).