We show the multidimensional stability of subsonic phase transitions in a non-isothermal van der Waals fluid. Based on the existence result of planar waves in our previous work [1], a jump condition is posed on non-isothermal phase boundaries which makes the argument possible. Stability of planar waves both in one dimensional and multidi-mensional spaces are proved.
The motion of a 2-dimensional non-isothermal van der Waals fluid is governed by the following Euler equations
where, is the density, is the velocity with, p is the pressure satisfying the following state equation
with being the specific volume, being the temperature, R being the perfect gas constant and a, b being positive constants, e is the specific internal energy given by
and i is the specific enthalpy given by
Otherwise, according to the second law of thermodynamics, the specific entropy s and the specific free energy f of the fluid is defined by
and
respectively. Regarding as independent variables and denoting,
and
where is the sound speed, we can rewrite (1) as
or
When, the state Equation (2) is not monotonic with respect to, which means that there exist and such that
The fluid is in liquid phase in the region, while it is in vapor phase in the region. The region is a highly unstable region (spinodal region) where no state can be found in experiments [
A subsonic phase transition is a discontinuous solution to the Euler Equation (1) with a single discontinuity, which changes phases across the discontinuity and satisfies certain subsonic condition on both sides of the discontinuity. To explain the concept with more detail, let us consider the following planar subsonic phase transition
where are constant states of the flow, is the constant speed of the discontinuity and belong to different phases. The solution (10) satisfies the Rankine-Hugoniot condition
and the subsonic condition
where denotes the difference of a function across the discontinuity, and are the Mach number and the sound speed on each side of the discontinuity respectively.
Due to the subsonic property (12), the well-known Lax entropy inequality [
Compared with isothermal phase transitions, there is much less knowledge on non-isothermal phase transitions and there are fewer papers available. Slemrod [
The purpose of this paper is to study the multidimensional stability of non-isothermal phase transitions. With straightforward computation, we show that the corresponding linearized initial boundary problem for the planar phase transition satisfies the uniform Lopatinski condition [14,15]. Without giving much detail, here we briefly state the main result of this paper Theorem 1.1 There exists and K1 > 0 depending on the bounds of and given in (10) and given in (18), such that for and 0 < K < K1, the -admissible phase transition (10) is uniformly stable.
The definitions of the parameters, K, and - admissible will be given in Section 2, and the uniform stability will be described in detail in Section 4.
The paper is arranged as follows. Section 2 is a brief recall of the viscosity capillarity criterion for phase transitions and related existence results of traveling waves. In Section 3, we propose the main problem and prove the stability of phase transitions in one dimensional spaces. The multidimensional stability of phase transitions is presented and proved in Section 4.
For the simplicity of notations, we will need the following quantities in the coming arguments.
Considering the planar subsonic phase transition (10), we denote by the mass transfer flux, and and. Then, the Rankine-Hugoniot condition (11) and the subsonic condition (12) can be rewritten as
and
respectively.
Analogue to the traveling wave method for viscous shocks, the viscosity capillarity criterion is applied to find the planar wave (10) which admits the existence of the following traveling wave
satisfying and the Navier-Stokes equations
where is the Laplace operator, is the viscosity coefficient, is the capillarity coefficient and is the heat conductivity coefficient with, ,. Substituting (15) into (16) and noticing the Rankine-Hugoniot conditions (11), we can derive the following heteroclinic problem for the unknown functions,
where the prime ' denotes the derivative of a function with respect to.
In order to deal with the above problem by the center manifold method, we proposed the following assumption in [
which was later simplified as
with M being a positive constant and. Employing the Rankine-Hugoniot conditions (13), the hecteroclinic problem (17) becomes
Therefore, the admissibility of subsonic phase transitions can be defined by Definition 2.1 The planar phase transition (10) is admissible if and only if the problem (19) has a solution. The solution is called the viscosity capillarity profile, or -profile for simplicity. The pair, is called -admissible.
To state the existence result of -profile, we will need the following quantities. As usual for fixed , the Maxwell equilibrium is defined by the equal area rule
Then there exists a unique point, which satisfies that the chord connecting the points
and is tangent to the graph of at the point . Denote
When, the -profile satisfies
which implies. Setting, there exists satisfying the first equation of (20) by the generalized equal area rule as in [
Moreover, for every and , a unique pair can be found such that and can be connected by the -profile with the parameters j and.
Based on the existence of -profile, the following theorem shows the existence of -profile for small and small K in [
Theorem 2.1 For every and, there exist, and neighborhoods, , of, , respectively, such that, for, there are unique pair and, for which and are -admissible with the parameters j and.
Moreover, an additional jump condition can be derived for (10), which plays an essential role in the study of the stability of phase transitions. In the isothermal case [
By multiplying the first equation of (19) with and integrating from to with respect to, we get the following jump condition on the phase boundary
where
with, and being the -profile with the parameters j and.
Remark 2.1 is a bounded function which also possesses a uniform limit as,. Indeed, from the second equation of (19), we have
where
.
Simple calculation yields
When, , has a uniform limit
where is the -profile.
Moreover, from the following jump conditions
the functions of can be determined by the implicity function theorem for near and every satisfying the conditions given in Theorem 2.1. The following identities can be easily verified
where − denotes the value of a function for and
,
,
.
In this section, we propose the nonlinear problem for a multidimensional subsonic phase transition and derive the corresponding linearized problem. Then we prove the 1-dimensional stability for the linear problem.
Endow the Euler Equation (1) with the following initial data
where is the initial discontinuity and belong to different phases. If the initial data (24) satisfies certain compatibility conditions, then we can expect to construct the following multidimensional subsonic phase transition
which satisfies the following nonlinear initial boundary value problem
where the third equation is a reformulation of the jump condition (21) with,
,
and, satisfying
.
Following [
Then the problem (26) becomes
where we have dropped the tildes for simplicity of notations.
Consider the perturbation, , of the planar phase transition (10), which satisfies the problem (26) and. Denote
. Then, the following linearized problem for the unknowns can be derived from (26),
where
with
and
Remark 3.1 Simple calculation yields
Noticing that the boundary conditions of (29) involve the quantities, , and, we will need the following lemma to deal with these quantities.
Lemma 3.1 For all, the functions and are continuously differentiable. Moreover, their derivatives are continuous with respect to at and are bounded depending on the bounds of and given in (10) and the constant M given in (18). There exists such that for all
Proof. The estimate (30) is immediate from Lemma 2 in [
By Taylor’s formula, we have
where and is an infinitesimal as r goes to zero. Substituting the above identities into (31) and employing the calculations (23), we get
which implies
Similar arguments yield the continuity of and as the following
The one dimensional stability concerns the stability of the problem (29) without terms of y-derivatives, namely,
The following theorem shows the stability of planar phase transitions in one dimensional spaces.
Theorem 3.2 There exists depending on the bounds of and given in (10) and the constant given in (18), such that for any fixed (is given in Lemma 3.1.) and, the subsonic phase transition (10) is stable with respect to perturbations in the x-direction, which means the problem (32) is well-posed.
Proof. The main idea of the proof is to show that the boundary values of outgoing characteristics and the free boundary can be determined by the boundary conditions, for which we need to investigate the eigenvalues and the eigenvectors of the matrix. The eigenvalues of are
of multiplicity 1 and
of multiplicity 2. The corresponding right eigenvectors are
and
respectively.
Denote by
the decompositions of on the bases
respectively. Since the mass transfer flux is nonzero, we assume. Then the subsonic condition (14) becomes
Accordingly, we rewrite the boundary condition of (32) as
to separate the outgoing characteristics together with the free boundary from the incoming characteristics. The necessary and sufficient condition for the well-posedness of the problem (32) is that the determinant
does not vanish. Direct computation yields
where denotes a bounded term depending on the bounds of, given in (10) and M given in (19). The determinant on the right side of (35) takes the value
for with given in Lemma 3.1.
Therefore, we can find depending on the bounds of, given in (10) and the constant given in (18) such that for and, the problem (32) is well-posed. Similar arguments can be carried out for the case.
First let us introduce the uniform stability in [
and
the Laplace-Fourier transform of V in -variables with. Then, from (29) we know that satisfies
where
and
.
Denote by all the distinct eigenvalues of
with multiplicity being mj. Obviously, we have
Introduce
the space of boundary values of all bounded solutions of the special form
to (36) with.
Thus, we can state the uniform stability result in detail as follows Theorem 4.1 There exist and depending on the bounds of, given in (10) and the constant given in (18), such that for any fixed and the viscosity-capillarity admissible phase transition (10) is uniformly stable, i.e. there is such that the estimate
holds for all and.
For simplicity, we shall only consider the case
and the other case can be studied similarly.
Taking the Laplace-Fourier transform on the equation of (29) with yields
where
As in [
then (38) is equivalent to
where
The eigenvalues of with negative real part for are
of multiplicity 2 and
of multiplicity 1, where the denotes the positive real part square root of a complex value. The corresponding eigenvectors are
and
respectively. The eigenvalue of with a negative real part for is
and the corresponding eigenvector is
Remark 4.1 The above eigenvalues and eigenvectors can be continuously extended to the case. With a little abuse of the notation, we still use it to denote those extensions of square roots appearing in the case.
As in [
.
In the above cases, the following proposition help us to find the bases of.
Proposition 4.3 1) If and, then and the vectors (41), (44) together with the following eigenvectors
are linearly independent.
2) If and, then and the vectors (41), (44) together with the following eigenvectors
are linearly independent.
As in [
and the corresponding eigenvectors
are linearly independent.
Combining the above propositions, if we naturally expand the eigenvectors as
then the bases of are given for and.
Now we can show the uniform stability of the phase transition.
Proof of Theorem 4.1. Taking the Laplace-Fourier transformation on the boundary condition in (29) with, multiplying it with the invertible matrix
with and introducing the transformation (39), we get
where
with
and
To achieve the result, we need to verify the determinant
being nonzero.
Noticing that the eigenvector remains the same in all the cases mentioned in Section 4.1, the following simplification can be made to,
where is a bounded term depending on the bounds of, given in (10) and given in (19),
For sufficiently small, the determinant is nonzero as long as the determinant
doesn’t vanish. Considering, one can find that it is similar to the Lopatinski determinant for the corresponding problem in the isothermal case [7,9]. Noticing Proposition 4.2-4.4, we need to consider the following three cases:
1).
We obtain
where
and
Following [
In fact, when, if, then we have
which implies
with being the Mach numbers. From the subsonic property (12) of the phase transition, we have. Due to, we deduce that one should take the plus sign in (51), which is not the root of I obviously. Thus, I is always nonzero, which gives that there exist constants, and, such that for any, we have
When with and, we know that if does not equal to the right hand side of (51) with the minus sign, then the inequality (52) holds for any with sufficiently small. If satisfies (51) with the minus sign, then the term I vanishes. However, the imaginary part of the term II given in (49) is nonzero, which implies that for any fixed, there is a constant such that the inequality (52) holds as well.
Therefore, we can find and depending on and given in (10) and the constant given in (18) such that does not vanish for and.
2).
In this case, we get
which is nonzero for sufficiently small and.
3).
In this case, we have
which is also nonzero for sufficiently small and.
Therefore, combining the above arguments, we draw the conclusion of the Theorem 4.1.