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 [2]. Due to such monotonicity, subsonic phase transitions can be found in a van der Waals fluid, which is different from the well-known classical nonlinear waves such as shock waves, rarefaction waves and contact discontinuities.
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 [3] is violated for subsonic phase transitions. Hence, several admissibility criteria were introduced to select the physical admissible subsonic phase transitions, among which the viscosity capillarity criterion proposed by Slemrod [4] is an important one. Ever since, for a long time, attention has been paid to isothermal phase transitions and related problems with numerous works devoted to such topics. For problems in one dimensional spaces, see [2,4-6] and references therein. For problems in multi-dimensional spaces, see [7-10] and references therein.
Compared with isothermal phase transitions, there is much less knowledge on non-isothermal phase transitions and there are fewer papers available. Slemrod [11] and Grinfeld [12] proved the existence of traveling waves in Lagrange coordinates by Conley index theory. Hattori [13] considered certain cases of the Riemann problem by the entropy rate criterion. Recently, the author [1] proved the existence and structural stability of traveling waves by using the center manifold method, in light of which, we can expect to reveal more insights of multidimensional phase transitions.
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.
2. Viscosity Capillarity Profiles
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 [1],
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 [8], which means
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 [1].
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 [4], due to the subsonic condition (12), the Rankine-Hugoniot condition (11) is not sufficient to guarantee the wellposedness of the boundary value problem for phase transitions, which is also the situation that we encounter in the study of non-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
,
,
.
3. Linearized Problems and One Dimensional Stability
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.
3.1. Linearized Problems
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 [15], we introduce the following transformation to map the free boundary into a fixed boundary
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 [7], once we prove the continuity of. Let us show the continuity of and. Differentiating (21) with respect to j, we get
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
3.2. One Dimensional Stability
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.
4. Multidimensional Stability
First let us introduce the uniform stability in [15] and state the main result in detail. Denote by
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.
4.1. The Space <img src="1-7400374\ca4d9208-4400-43ab-ab78-3dae1bf7002f.jpg" />
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 [15], if we introduce the transformation
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 [15], for these vectors, we have Proposition 4.2 are linearly independent for and except at
.
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 [15], in the critical case, the bases of is given by Proposition 4.4 If and, then
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.
4.2. Lopatinski Determinant
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 [9], we claim that is nonzero for sufficiently small.
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.
REFERENCESNOTESReferencesS.-Y. Zhang, “Existence of Travelling Waves in NonIsothermal Phase Dynamics,” Journal of Hyperbolic Differential Equations, Vol. 3, No. 4, 2007, pp. 391-400.
doi:10.1142/S0219891607001197H. Fan and M. Slemrod, “Dynamic Flows with Liquid/ Vapor Phase Transitions,” In: H. Fan and M. Slemrod, Eds., Handbook of Mathematical Fluid Dynamics, Vol. I, North-Holland, Amsterdam, 2002, pp. 373-420. P. D. Lax, “Hyperbolic Systems of Conservation Laws. II,” Communications on Pure and Applied Mathematics, Vol. 10, 1957, pp. 537-467. doi:10.1002/cpa.3160100406M. Slemrod, “Admissibility Criteria for Propagating Phase Boundaries in a Van Der Waals Fluid,” Archive for Rational Mechanics and Analysis, Vol. 81, No. 4, 1983, pp. 301-315. M. Shearer, “Nonuniqueness of Admissible Solutions of Riemann Initial Value Problem for a System of Conservation Laws of Mixed Type,” Archive for Rational Mechanics and Analysis, Vol. 93, No. 1, 1986, pp. 45-59.
doi:10.1007/BF00250844P. G. LeFloch, “Propagating Phase Boundaries: Formulation of the Problem and Existence via the Glimm Method,” Archive for Rational Mechanics and Analysis, Vol. 123, No. 2, 1993, pp. 153-197. doi:10.1007/BF00695275S. Benzoni-Gavage, “Stability of Multi-Dimensional Phase Transitions in a Van Der Waals Fluid,” Nonlinear Analysis, Vol. 31, No. 1-2, 1998, pp. 243-263.
doi:10.1016/S0362-546X(96)00309-4S. Benzoni-Gavage, “Stability of Subsonic Planar Phase Boundaries in a Van Der Waals Fluid,” Archive for Rational Mechanics and Analysis, Vol. 150, No. 1, 1999, pp. 23-55. doi:10.1007/s002050050179Y.-G. Wang and Z. Xin, “Stability and Existence of Multidimensional Subsonic Phase Transitions,” Acta Mathematicae Applicatae Sinica. English Series, Vol. 19, No. 4, 2003, pp. 529-558. S.-Y. Zhang, “Discontinuous Solutions to the Euler Equations in a Van Der Waals Fluid,” Applied Mathematics Letters, Vol. 20, No. 2, 2007, pp. 170-176.
doi:10.1016/j.aml.2006.03.010M. Slemrod, “Dynamic Phase Transitions in a Van Der Waals Fluid,” Journal of Differential Equations, Vol. 52, No. 1, 1984, pp. 1-23.
doi:10.1016/0022-0396(84)90130-XM. Grinfeld, “Nonisothermal Dynamic Phase Transitions,” Quarterly of Applied Mathematics, Vol. 47, No. 1, 1989, pp. 71-84. H. Hattori, “The Riemann Problem for a Van Der Waals fluid with Entropy Rate Admissibility Criterion Nonisothermal Case,” Journal of Differential Equations, Vol. 65, No. 2, 1986, pp. 158-174.
doi:10.1016/0022-0396(86)90031-8H. O. Kreiss, “Initial Boundary Value Problems for Hyperbolic Systems,” Communications on Pure and Applied Mathematics, Vol. 23, 1970, pp. 227-298.
doi:10.1002/cpa.3160230304A. Majda, “The Stability of Multi-Dimensional Shock Fronts,” Memoirs of the American Mathematical Society, Vol. 41, No. 275, 1983, pp. 1-95.
, 
is the density, 
is the velocity with
, p is the pressure satisfying the following state equation
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
as independent variables and denoting
,
is the sound speed, we can rewrite (1) as
, the state Equation (2) is not monotonic with respect to
, which means that there exist 
and 
such that
, 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 [
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
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.
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.
, K, 
and 
- admissible will be given in Section 2, and the uniform stability will be described in detail in Section 4.
the mass transfer flux, and 
and
. Then, the Rankine-Hugoniot condition (11) and the subsonic condition (12) can be rewritten as
and the Navier-Stokes equations
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
,
.
. Employing the Rankine-Hugoniot conditions (13), the hecteroclinic problem (17) becomes
is called the viscosity capillarity profile, or 
-profile for simplicity. The pair
, 
is called 
-admissible.
-profile, we will need the following quantities. As usual for fixed 
, the Maxwell equilibrium 
is defined by the equal area rule
, which satisfies that the chord connecting the points
and 
is tangent to the graph of 
at the point 
. Denote
, the 
-profile satisfies
. Setting
, there exists 
satisfying the first equation of (20) by the generalized equal area rule as in [
and 
, a unique pair 
can be found such that 
and 
can be connected by the 
-profile with the parameters j and
.
-profile, the following theorem shows the existence of 
-profile for small 
and small K in [
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
.
and integrating from 
to 
with respect to
, we get the following jump condition on the phase boundary
, 
and 
being the 
-profile with the parameters j and
.
is a bounded function which also possesses a uniform limit as
,
. Indeed, from the second equation of (19), we have
.
, 
, 
has a uniform limit
is the 
-profile.
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
and
,
,
.
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
,
,
, 
satisfying
.
into a fixed boundary 
, 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),
, 
, 
and
, we will need the following lemma to deal with these quantities.
, 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 
. Let us show the continuity of 
and
. Differentiating (21) with respect to j, we get
and 
is an infinitesimal as r goes to zero. Substituting the above identities into (31) and employing the calculations (23), we get
and 
as the following
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.
. The eigenvalues of 
are
on the bases 
is nonzero, we assume
. Then the subsonic condition (14) becomes
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
with 
given in Lemma 3.1.
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
. 
-variables with
. Then, from (29) we know that 
satisfies
.
all the distinct eigenvalues of
with multiplicity being mj. Obviously, we have
.
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
and
.
can be studied similarly.
yields
with negative real part for 
are
denotes the positive real part square root of a complex value. The corresponding eigenvectors are
with a negative real part for 
is
. With a little abuse of the notation
, we still use it to denote those extensions of square roots appearing in the case
.
are linearly independent for 
and 
except at
.
.
and
, then 
and the vectors (41), (44) together with the following eigenvectors
and
, then 
and the vectors (41), (44) together with the following eigenvectors
, the bases of 
is given by Proposition 4.4 If 
and
, then
are given for 
and
.
, multiplying it with the invertible matrix
and introducing the transformation (39), we get
remains the same in all the cases mentioned in Section 4.1, the following simplification can be made to
,
is a bounded term depending on the bounds of
, 
given in (10) and 
given in (19),
, the determinant 
is nonzero as long as the determinant
, 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:
.
is nonzero for sufficiently small
.
, if
, then we have
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
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.
and 
depending on 
and 
given in (10) and the constant 
given in (18) such that 
does not vanish for 
and
.
.
and
.
.
and
.