The Inviscid Limits to Piecewise Smooth Solutions for a General Parabolic System
Shixiang Ma1
Academic Editor: M. Znojil, U. Kulshreshtha
Received13 Oct 2011
Accepted26 Oct 2011
Published05 Jan 2012
Abstract
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.
1. Introduction
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 [1] 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 [2] 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 [3β5]. Later, Yu [6] 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 [7] 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. [8] improve the first part with weak centered srarefaction waves data and Zeng [9] improve the other results, respectively, in [7] 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 [10β14], 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[15β17].
In this paper, motivated by Goodman and Xinβs work [2], 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 [18] 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 [2], 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.
We assume that has the form
Substitute (2.5)β(2.7) into (1.1) to obtain
and so forth, where , and so forth. The inner approximation is supposed to be valid in a small zone of size around .
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 βmatching conditionsβ 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 [2], 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)β(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
Set
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 [2], , 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 [2]. 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 Sobolevβs 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 .
Suppose that is the exact solution to (1.1) with the initial data . We decompose the solution as
Then using the relation (2.53) for , we compute that
where satisfies for small .
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 βsmallβ smooth solution up to . By the standard existence and uniqueness theory, and the continuous induction argument for parabolic equations [21], 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)β(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
Finally,
Summing all the inequalities for , we arrive at
provided that is bounded. We complete the proof of Lemma 3.2.
Lemma 3.3. Suppose that the conditions in Proposition 3.1 be satisfied. Then
for all , where is independent of and .
Proof. Multiplying (3.10) on the left by and integrating over , we obtain after integration by parts that
Next we estimate each term on the right hand side above. First, it follows from (3.16) that
Here is the minimum of the eigenvalues, valued at and , of and . And
By virtue of (3.16)β(3.21), one finds
Now the remaining terms on the right hand side of (3.34) can be estimated in a similar way to that in the proof of Lemma 3.2. We list them below:
Lemma 2.3 leads to
Continuing, we compute that
Same bounds hold for . As before, using Lemma 2.4 and (3.16)-(3.20), we obtain
In view of (2.55),
Collecting all the estimates previously we have achieved, we get
provided that is bounded. Insert (3.11) into (3.43) to arrive at
which gives
Differentiating (3.5) with respect to , multiplying the resulting equation by on the left and integrating over , we obtain after integration by parts that
Using Sobolev's inequality, we have
and so
Lemma 2.3 gives
The last term can be estimated as
By taking to be sufficiently small, we get
Denote the constant in this inequality by . Insert (3.11) into (3.51) to give
Multiplying suitably small constants to (3.11) and (3.52), respectively, then adding the resulting inequalities to (3.45) and taking and sufficiently small, we can obtain the following inequality:
Integrating the above inequality to give
Take suitably small to yield
It follows from the Gronwall's inequality that
In particular,
By the fact and , we obtain
Lemma 3.4. Let the conditions in Proposition 3.1 be satisfied. Then
with the constant independent of and .
Proof. Applying to (3.5), multiplying on the left of the resulting equation by , and integrating over , we compute that
In the case , this gives
Using Cauchy inequality, we obtain
Lemma 2.3 and Sobolevβs inequality give
provided that is bounded. For the last two terms, we have
Choose sufficiently small to yield
Integrate the above inequality, and use Lemma 3.3 to give
Similarly, for the case , we can obtain
This finishes the proof of Lemma 3.4.
Combining the results of Lemmas 3.3-3.4, we complete the proof of Proposition 3.1.
Using Proposition 3.1 and the standard continuous induction argument, we conclude the following.
Proposition 4.1. There exist positive constants , and , which are independent of such that if , and , then the Cauchy problem (3.5) has a unique solution . Furthermore, the following inequality holds
Now we choose . Then due to (4.1), we have
On the other hand, it follows from Lemma 2.4 that
Consequently,
which gives (1.8). Finally,
This yields (1.9) by using Lemma 2.4 again. We complete the proof of Theorem 1.2.
Acknowledgment
The author is supported by Tianyuan Foundation (no. 11026093) and the National Natural Science Foundation of China (nos. 11101162, 11071086).
References
D. Hoff and T. Liu, βThe inviscid limit for the Navier-Stokes equations of compressible, isentropic flow with shock data,β Indiana University Mathematics Journal, vol. 38, no. 4, pp. 861β915, 1989.
J. Goodman and Z. Xin, βViscous limits for piecewise smooth solutions to systems of conservation laws,β Archive for Rational Mechanics and Analysis, vol. 121, no. 3, pp. 235β265, 1992.
S. Ma, βThe inviscid limit for an inflow problem of compressible viscous gas in presence of both shocks and boundary layers,β Journal of Mathematical Analysis and Applications, vol. 378, no. 1, pp. 268β288, 2011.
H. Wang, βViscous limits for piecewise smooth solutions of the p-system,β Journal of Mathematical Analysis and Applications, vol. 299, no. 2, pp. 411β432, 2004.
Y. Wang, βZero dissipation limit of the compressible heat-conducting navier-stokes equations in the presence of the shock,β Acta Mathematica Scientia. Series B. English Edition, vol. 28, no. 4, pp. 727β748, 2008.
S. Yu, βZero-dissipation limit of solutions with shocks for systems of hyperbolic conservation laws,β Archive for Rational Mechanics and Analysis, vol. 146, no. 4, pp. 275β370, 1999.
Z. Xin, βZero dissipation limit to rarefaction waves for the one-dimensional Navier-Stokes equations of compressible isentropic gases,β Communications on Pure and Applied Mathematics, vol. 46, no. 5, pp. 621β665, 1993.
S. Jiang, G. Ni, and W. Sun, βVanishing viscosity limit to rarefaction waves for the Navier-Stokes equations of one-dimensional compressible heat-conducting fluids,β SIAM Journal on Mathematical Analysis, vol. 38, no. 2, pp. 368β384, 2006.
H. Zeng, Asymptotic behavior of solutions to fluid dynamical equations, Doctoral thesis, The Chinese University of Hong Kong.
A. Szepessy and K. Zumbrun, βStability of rarefaction waves in viscous media,β Archive for Rational Mechanics and Analysis, vol. 133, no. 3, pp. 249β298, 1996.
C. Mascia and K. Zumbrun, βStability of large-amplitude shock profiles of general relaxation systems,β SIAM Journal on Mathematical Analysis, vol. 37, no. 3, pp. 889β913, 2006.
C. Mascia and K. Zumbrun, βStability of large-amplitude viscous shock profiles of hyperbolic-parabolic systems,β Archive for Rational Mechanics and Analysis, vol. 172, no. 1, pp. 93β131, 2004.
K. Zumbrun and P. Howard, βPointwise semigroup methods and stability of viscous shock waves,β Indiana University Mathematics Journal, vol. 47, no. 3, pp. 741β871, 1998.
F. Huang, Y. Wang, and T. Yang, βFluid dynamic limit to the Riemann solutions of Euler equations: I. Superposition of rarefaction waves and contact discontinuity,β Kinetic and Related Models, vol. 3, no. 4, pp. 685β728, 2010.
S. Ma, βZero dissipation limit to strong contact discontinuity for the 1-D compressible Navier-Stokes equations,β Journal of Differential Equations, vol. 248, no. 1, pp. 95β110, 2010.
S. Ma, βViscous limit to contact discontinuity for the 1-D compressible Navier-Stokes equations,β Journal of Mathematical Analysis and Applications, vol. 387, no. 2, pp. 1033β1043, 2012.
P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 1973.
H. O. Kreiss, βInitial boundary value problems for hyperbolic systems,β Communications on Pure and Applied Mathematics, vol. 23, pp. 277β298, 1970.
J. Rauch, β is a continuable initial condition for Kreiss' mixed problems,β Communications on Pure and Applied Mathematics, vol. 25, pp. 265β285, 1972.
S. Kawashima, Systems of hyperbolic-parabolic composite type, with appli-cations to the equations of magneto-hydrodynamics, Doctoral thesis, Kyoto University, 1983.