Research Article | Open Access
The Inviscid Limits to Piecewise Smooth Solutions for a General Parabolic System
We study the viscous limit problem for a general system of conservation laws. We prove that if the solution of the underlying inviscid problem is piecewise smooth with finitely many noninteracting shocks satisfying the entropy condition, then there exist solutions to the corresponding viscous system which converge to the inviscid solutions away from shock discontinuities at a rate of as the viscosity coefficient vanishes.
We consider the relation between the solutions, , of the system of viscous conservation laws and the distributional solution, , of the corresponding system of conservation laws without viscosity
We assume that (1.2) is strictly hyperbolic, then by normalization, we have the decomposition where with is a matrix whose rows are left eigenvectors of , and is a matrix whose columns are right eigenvectors of .
For the zero dissipation limit problem, there are many significant works. When the Euler flow contains a single shock, Hoff and Liu  studied the isentropic case, they established the limit process from the solutions of the compressible Navier-Stokes equations to the single shock-wave solution of the corresponding compressible Euler system (so-called p-system). They show that the solutions to the isentropic Navier-Stokes equations with shock data exist and converge to the inviscid shocks as the viscosity vanishes, uniformly away from the shocks. Ignoring the initial layers, Goodman and Xin  gave a very detailed description of the asymptotic behavior of solutions for the general viscous systems as the viscosity tends to zero, via a method of matching asymptotics. This method can be applied to the Navier-Stokes equations (1.1), such as [3–5]. Later, Yu  revealed the rich structure of nonlinear wave interactions due to the presence of shocks and initial layers by a detailed pointwise analysis. As far as rarefaction wave is concerned, Xin in  has obtained that the solutions for the isentropic Navier-Stokes equations with weak centered rarefaction wave data exist for all time and converge to the weak centered rarefaction wave solution of the corresponding Euler system, as the viscosity tends to zero, uniformly away from the initial discontinuity. Moreover, in the case that either the initial layers are ignored or the rarefaction waves are smooth, he also obtains a rate of convergence which is valid uniformly for all time. Recently, Jiang et al.  improve the first part with weak centered srarefaction waves data and Zeng  improve the other results, respectively, in  to the full compressible Navier-Stokes equations, provided that the viscosity and heat-conductivity coefficients are in the same order. Furthermore, by a spectral analysis and Evans function method, Kevin Zumbrun and his collaborators have obtained many important results even for large amplitude and multidimensional case [10–14], and so forth. The case that the solutions to the Euler system containing contact discontinuity is much more subtle, there are few results in this respect[15–17].
In this paper, motivated by Goodman and Xin’s work , we establish that the piecewise smooth solutions, , of (1.2), with finitely many noninteracting shocks satisfying the entropy condition, are strong limits as of solutions, , of (1.1) when the matrix is positive definite.
For simplicity of presentation, we only consider the case in which is a single-shock solution.
Definition 1.1. A function is called a single-shock solution of (1.2) up to time if:(i) is a distributional solution of the hyperbolic system (1.2) in the region ;(ii)there is a smooth curve, the shock, , so that is sufficiently smooth at any point ;(iii)the limits exist and are finite for and ;(iv)the Lax geometrical entropy condition  is satisfied at , that is,
The main results of this paper are as follows.
Theorem 1.2. Suppose that the system (1.2) is strictly hyperbolic and that the th characteristic family is genuinely nonlinear. There exist positive constants, and , such that if is a single-shock solution up to time with then for each , there is a smooth solution, , of (1.1) with Moreover, for any given , where is a positive constant depending only on .
Notation 1. In this paper, we use to denote the usual Sobolev space with the norm , and denotes the usual -norm. We also use to denote any positive bounded function which is independent of .
2. Construction of the Approximate Solution
In this section, following the method of Goodman and Xin, in , we construct the approximate solution through different scaling and asymptotic expansions in the region near and away from the shock respectively, such that approximate the piecewise smooth inviscid solution away from the shock and has a sharp change near the shock.
2.1. Outer and Inner Expansions and the Matching Conditions
In the region away from the shock, , we approximate the solution of (1.1) by truncation of the formal series Substituting this into (1.1) and comparing the coefficients of powers of , we get, for , that and so forth. The outer functions , are generally discontinuous at the shock, , but smooth up to the shock. The leading term, , is the single-shock solution of (1.2) which is given in the theorem.
Near the shock, should be represented by an inner expansion: where and is the perturbation of the shock position to be determined later.
In a matching zone, we expect the outer and the inner expansion agree with each other. Using the Taylor series to express the outer solutions in terms of , we obtain the following “matching conditions” as : and so forth.
2.2. The Structure of Viscous Shock Profiles
Our construction of the approximate solution depends on the properties of the viscous shock profiles, which are the solutions of the ordinary differential equation satisfying the boundary conditions and moving with speed : Integrate the differential equation to reduce that It is well known that for a given state and the wave family, if is sufficiently small, then there exists a shock profile , which connects and from left to right. Using the genuine nonlinearity, by similar arguments in , we can obtain And as , As ,
2.3. Solutions of the Outer and Inner Problems
Now we construct and order by order.
The leading order outer function, , is the single-shock solution of the theorem. For any fixed , the leading order inner solution is exactly the viscous shock profile with , and . So Here we take the shift to be zero since it can be absorbed into .
Next we determine , and together. Substituting (2.21) into (2.9) gives By the matching condition (2.12), we expect that So we set where is a smooth function satisfying Then inserting (2.24) into (2.22) and using (2.19)–(2.20) and the identity we obtain where for large . Define . Then we have where are integration constants to be determined later. We express in terms of the basis, , of the right eigenvectors of . We write Here the are for and the are for , and so forth. Then the matching conditions (2.12) are transformed into Define . Multiplying (2.28) by , and using (2.29), we obtain and then we have the following result.
Lemma 2.1. There is a smooth solution, , to (2.31) with the following property: for , where , and is a positive constant.
Proof. We use the standard iteration argument. Define , and
Then from the lax entropy condition (1.5), we can obtain
where is independent of . And then for suitably small , we have
and converges uniformly to a smooth bounded function, , which is a solution to (2.31). The asymptotic behavior of the solution, , follows from the formulas (2.33).
With Lemma 2.1 and the matching condition (2.30) at hand, we can determine, completely the same as in , , and , which guarantees the existence of and . We give the sketch of this process. First, we use Lemma 2.1 and (2.30) for incoming indices to get a system of () equations for unknowns, that is, Then we can solve for from (2.37)-(2.38). Substituting the resulting expression into (2.39), by writing , we arrive at an ordinary differential equation for : provided that is suitably small. Here , and are smooth known functions, and and remain bounded even as . Solving for from (2.37) up to a constant, we obtain uniquely in terms of . Then substitute the expression of and into the equation of the matching condition for outgoing indices to yield the linear relations where , and is a smooth known function, is a smooth matrix and remains bounded even as . Then the theory of linear hyperbolic equations [19, 20] shows that the problem (2.3), (2.41) has a solution smooth up to the shock provided that the initial value, , is chosen to satisfy the appropriate compatibility conditions at . Thus is completely determined, which in turn gives and by (2.37)-(2.38), and therefore .
Now we summarize the above discussion to achieve the following.
Proposition 2.2. If is suitably small, then and can be established such that(i) and its derivatives are uniformly continuous up to , and(ii) and are smooth functions, and there are an , such that
The above constructions can be carried out to any order. In particular, we can determine and simultaneously and similar results as in Proposition 2.2 hold for them.
2.4. Approximate Solutions
Now we can construct an approximate solution to (1.1) by patching the truncated outer and inner solutions in the previous discussion as in . Define Let satisfy , and Let be a constant. Then we define the approximate solution to (1.1) as where is a higher-order correction term to be determined.
Using the structures of the various orders of inner and outer solutions, we compute that where where denote the truncated Taylor's expansion of , respectively, at , including all the terms of , and denote the truncated Taylor's expansion of , respectively, at , including all the terms of .
In view of our construction, we have(i)supp , and (ii)supp , and (iii)supp , and
where we have used the estimates on .
We now choose to satisfy so that satisfies Since is smooth and positive definite, by the standard energy estimates for the linear parabolic system and Sobolev’s inequalities, we have the following results.
Lemma 2.3. Let be the solution of (2.52). Then the following estimates hold for all : Then for , we have
And by our construction, we obtain the following.
Lemma 2.4. One has
3. Stability Analysis
We now show that there exists an exact solution to (1.1) in a neighborhood of the approximate solution , and that the asymptotic behavior of the viscous solution is given by for small viscosity .
Set in (3.2) and integrate the resulting equation with respect to to give by making the following scalings, we transform (3.3) into Then we only need to show that for suitably small , (3.5) has a unique “small” smooth solution up to . By the standard existence and uniqueness theory, and the continuous induction argument for parabolic equations , this will follow from the following a priori estimate.
Proposition 3.1. Suppose that the Cauchy problem (3.5) has a solution for some . Then there exist positive constants and , which are independent of and , such that if then where is defined in Section 2.4.
The proof of the proposition occupies the rest of this section. We separate it into several parts. First we diagonalize the system (3.5). Define Then we have Using the identity (3.9), we can rewrite (3.5) as
In what follows, we use to denote any positive constant which is independent of , and ; to denote any positive constant which is independent of and . And we set .
Now we do the following estimates on transversal waves.
Lemma 3.2. There exist suitably small positive constants , , independent of and , such that provided that is bounded.
Proof. Using (3.10), we compute that for ,
We now estimate separately as follows:
By Lemma 2.4,
For the second term , since
which follows from Lemma 2.4. Then we have
Consequently, we obtain
Using the estimate , we find
Notice that the facts
we arrive at
where we have used the estimate . From Lemma 2.3,
Using Lemma 2.4 again, we have
Then it follows from Lemma 2.3 and (3.16) that
In view of (3.16)–(3.21),
Same bounds hold for and .
Applying Cauchy inequality and (3.21), can be estimated as Lemma 2.4 yields, where we used the fact . Similarly, Thus, combining the above two inequality together, we obtain