Multidimensional Stability of Subsonic Phase Transitions in a Non-Isothermal Van Der Waals Fluid *

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 multidimensional spaces are proved.


Introduction
The motion of a 2-dimensional non-isothermal van der Waals fluid is governed by the following Euler equations where , and i is the specific enthalpy given by .i e p   (4) Otherwise, according to the second law of thermodynamics, the specific entropy s and the specific free energy f of the fluid is defined by

 
where is the sound speed, we can rewrite (1) as or When  4 ,27 8 a bR a bR    , the state Equation ( 2) is not monotonic with respect to  , which means that there exist The fluid is in liquid phase in the region while it is in vapor phase in the region highly unstable region (spinodal region) where no state can be found in experiments is a onic phase transition is a discontinuous solution to t t [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 subs 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 and the subsonic condition where    tinu denotes the difference of a function across the discon ity    are c and on the Mach number and the sound speed on each side of the disc tinuity   x t   respectively.Due to the s pr ty (12) Lax ub er , the well-known veral ad a were introdu wer papers available.Slemrod [11] an w that the corresp sonic op entropy inequality [3] is violated for subsonic phase transitions.Hence, se missibility criteri ced 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][5][6] and references therein.For problems in multi-dimensional spaces, see [7][8][9][10] and references therein.
Compared with isothermal phase transitions, there is much less knowledge on non-isothermal phase transitions and there are fe d 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 sho onding 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 1 0   and K 1 > 0 depending on the bounds of U  and  given in (10) and  given in (18), such that for The definitions of the paramete

es. In Se
The paper is arranged as follows Secti n 2 is a f recall of the viscosity capillarity criterion for phase transitions and related existence results of traveling wav ction 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 Then, the R (11) and the s bsonic condition (12) can be rewritten as ankine-Hugoniot condition u and ively.

Viscosity Capillarity Profiles
Analogue to the traveling wave method for viscous sity capillarity criterion is applied to wave (10) which admits the existence of respect shocks, the visco find the planar the following traveling wave and the Navier-Stokes equations 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 assu in [1], mption   which was later simplified as Therefore, the admissibility of subsonic phase transitions can be defined by Definition 2.1 The planar phase transition (1 missible if and only if the problem (19) has a solution.The solution


-profile for simplicity.The pair To state the existence result of   , K  -profile, we will need the following quantities.As usual for fixed     4 ,8 27 bR a bR , the Maxwell equilibrium is defined by the equal area rule satisfying the first equation of (20) by the qual area rule as in [8], which means Moreover, for every can be connected by the -profile with the parameters j   , there exist 0 0   , -admissible with the parameters j and   .
Moreover, an additional jump c ndition can be derived for (10) where where − denotes the value of a function for  

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 pro 1-dimensional stability for the linear problem.

Linearized Problems
Endow the Euler Equation ( 1) with the following initial data is the initial discontinuity and 0   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 ve the where the third equation is a reformulation of the jump conditi , , .
Following [15], we introduce the following transformation to map the free boundary , , , , Then the problem (26) becomes , where we have dropped the tildes for simplicity of notations.
Consider the perturbation,   , satisfies the problem of the planar phase transition (10), which and earized problem for the unk .Then, the following linnowns   , , where , , ,a n d , .
Noticing that the boundary conditions of (29) involve the quantities j a  , j b  , and , we will need the following lemma to deal with these quantities.π .

Th
ely, e one dimensional stability concerns the stability of the problem (29) without terms of y-derivatives, nam 0,0,1,0 , ,0,0, respectively.Since the mass transfer flux is nonzero, we assume 0 j  .Then the subsonic condition ( 14) becomes Accordingly, we rewrite the boundary condition of (32) as to sepa free bo rate the outgoing characteristics together with the undary from the incoming characteristics.The necessary and sufficient condition for the w dness of the problem (32) is that the determinant ell-pose does not vanish.Direct c tation yields where   for with 0  given in Lemma 3.1.

Multidimensional Stability
First let us introduce the uniform stability in [15] and state the main result in detail.Denote by and Then, from (29) we kn satisfies where the space of boundary values of all bounded solutions of the special form as follows Theorem 4.1 There exist 1 0   and 1 0 K  depending on the bounds of U  ,  given in (10) and th e constant given in (1 at fo fixed 8), such th r any the viscosity-capillarity admissible phase transition (10) is uniformly stable, i.e. there is 0 holds for all and    .

The Space
For simplicity, we shall only consider the c ase and the s  Thus, we can state the uniform stability result in detail 0 j  other case can be studied similarly.Taking the Laplace-Fourier transform on the equation of (29) with here . where and ly.The eigenvalue of and the corresponding eigenvector is .
In the above cases, the following proposition help us to find the bases of and the corresponding eigenvectors are linearly independent.
Combining the above propositions, if we naturally expand the eigenvectors as , then the bases of To achieve the result, we need to verify the determinant  (45) being nonzero.
Noticing that the eigenvector remains the same in all the cases mentioned in n 4.1, the following simplification can be made to where is a bounded term depending on the   For sufficiently small , the determinant is nonzero as long as the determinant 2) In this case, we get     which is also nonzero for sufficiently small 0   and .Therefore, combining the above arguments, we draw the conclusion of the Theorem 4.1.
volume,  being the temperature, R being the perfect gas constant and a, b being positive constants, e is the specific internal energy given by phase transition(10) is stabl tions in the x-direction, which means the problem 2) is well-posed.Proof.The main idea of the proof is to at the urba-bounda utgoing characteristics and the free boundary can be determined by the boundary conditions, for which we need to investigate the eigenval ry can find 0 K depending on the bounds of U  ,  given in(10) and the constant  given in (18) such that for

etermin
Now we can show the uniform stability of the phase transition.Proof of Theorem 4.1.Taking the Laplace-Fourier ition in , which plays an essential role in the study of the )