Stability and Uniqueness of Global Solutions to Euler Equations with Exothermic Reaction
Kai Hu
math.DS math.AP 35L45, 35L60, 35Q31, 35Q35





[mytitlenote]This paper is
supported by  National Natural Science Foundation of China (Grant No. 11426185),
Fundamental Research Funds for the Central Universities (XDJK2014C075 and SWU113062)
and  China Scholarship Council.



School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China
[myfootnote]E-mail address: hukaimath@swu.edu.cn













We consider the Cauchy problems of a non-strictly hyperbolic system
which describes the  compressible Euler fluid with  exothermic reaction.
 In this paper a Lyapunov-type functional is constructed
for balance laws. By analysis of the flow generated by front tracking method,
we prove the well-posedness theorems
and present the local features
of  global  solutions.



Euler equations Well-posedness Front tracking method
[2010]  35L45  35L60 35Q31 35Q35








§ INTRODUCTION






We are concerned about the Cauchy problem of the exothermically reacting compressible Euler equations
:

    v_t - u_x =0, 
       u_t + p_x =0 , 
       E_t + (pu)_x = qYϕ (T) , 
       Y_t= -Yϕ (T),

where v, u, T, p and E respectively represent specific volume, velocity, temperature, pressure and
total energy of compressible fluid.
This system arises from combustion theory in continuum physics.
Constant q>0 denotes unit binding energy released by combustion.
Y represents the mass fraction
of reactant in mixed fluid, where 0≤ Y≤ 1.
Assume that adiabatic constant γ and specific heat constant c
of the mixture are independent of Y.
 The procedure of releaseing chemical energy is controlled by reaction-rate function ϕ(T), which is a
C^1 function w.r.t. T. Suppose that ϕ(T) is positive and increasing on [0,+∞).
For instance it is of Arrhenius form
ϕ(T)=T^αe^-β/T with constants  α,β>0 .
In present paper, we prescribe the initial data

    U_0(x)=(v_0 (x), u_0 (x), E_0 (x), Y_0 (x)),
        x ∈ℝ

with
Y_0 (x) ∈ L^1(ℝ) and variation T.V.(U_0) < +∞ .







System (<ref>) is a typical hyperbolic balance law. It can be formulated by

    U_t + F(U)_x = G(U)

where U=(v,u,E,Y).
Well-posedness of solutions to balance laws was already
 established under particular circumstances.
See <cit.>
and the references therein.
Generally speaking, strictly hyperbolic balance laws has local weak solutions with bounded variation.
Their variation may blow up within finite time due to damping effect. Therefore,
in order to extend them to global solutions, one must impose appropriate
 dissipative conditions on source G.
Specifically let matrix

    G̃=R^-1(0)dG(0)R(0),

where matrix R(U) consists of linearly independent right eigenvectors of dF(U).
Dafermos and Hsiao<cit.> observe that when G̃ is strictly diagonally dominant, system (<ref>) admits a global BV solution, whose variation
decreases exponentially as t→∞.
Concerning non-autonomous source G=G(x,U), Amadori and collaborators<cit.> proposed a non-resonance condition
and a smallness condition on ω,

    |G(·,U)|+||∇_𝐮G(·,U)|| ≤ω∈ L^1 ∩ L^∞ (ℝ)

where ||·|| denotes the Euclidian norm.
Such conditions imply global well-posedness.
Whereas our system (<ref>) is ruled out of previous situations in early research,
because it is not strictly hyperbolic.
 Two eigenvalues of Jacobian dF(U) completely coincide. It is known that resonance
of different fields possibly leads to loss of uniqueness of entropy solutions.
Although one reduce (<ref>) to 3×3 strictly hyperbolic system
 by separating unknown Y from equations (<ref>), the diagonal dominance
condition still fails to hold.
So well-posedness  of our system indeed possesses new difficulties that differ
from the previous works.





The Cauchy problem (<ref>)(<ref>) has been intensively discussed in
recent research.
Main difficulty of system (<ref>) is
governing the damping induced by term Yϕ(T), which arises from exothermical
reaction and amplifies any oscillation.
 To achieve this, <cit.> demands  initial temperature satisfies

    ϕ(T_0(x))>ϕ >0 ,   ∀ x ∈ℝ,

with some constant ϕ. Then
existence of global entropy solutions  was verified in <cit.>
if U_0 is of small total variation.
Related results on Navier-Stokes equations in combustion
 are referred to <cit.>.
However, uniqueness and stability of global solutions remain unknown.
Our paper is intended to give the complete conclusions on this problem.





In this paper, we consider the global solution U(t,x) constructed  by fractional step
wave front tracking scheme, which determines a semigroup trajectory.
On stability issue, we must refer the pioneering work by Bressan, Liu
and Yang (cf. <cit.>).
They devised an elegant functional
Φ for conservation laws to analyze the L^1 distance of
distinct trajectories.
Unfortunately this functional is invalid for system (<ref>), since the exothermical damping
removes the rough monotonicity of Φ.
It is  tough to construct an appropriate functional instead of Φ.
Inspired by the argument in <cit.>,
we improve the weight included in Lyapunov functional, and then establish the
stability theorem for system (<ref>) as follows.




Assume that T.V.(U_0)
and ||Y_0||_∞ are sufficiently small,
moreover, condition (<ref>) holds. Then according to fractional step wave front tracking scheme,
 there exists a family of  domains {𝒟_t}_t≥ 0,
a unique semigroup 𝒫 : [0,+∞)×𝒟_∞→𝒟_∞ and a constant
L>0 such that 

(i) U(t,x) ≐𝒫_t (U_0) with U_0  ∈𝒟_0 is an entropy weak solution to problem (<ref>)(<ref>); 

(ii) for any U_0, V_0 ∈𝒟_0 and t, s≥ 0, there holds

    𝒫_0(U_0)=U_0,       𝒫_t(𝒫_s (U_0))= 𝒫_s+t(U_0), 
        ||𝒫_t (U_0)- 𝒫_s (V_0)||_L^1≤ L(||U_0- V_0||_L^1 + |t-s|).








We construct domain 𝒟_t by (<ref>) in a general setting.
Classic stability theory regards the constant state,especially  U≡0, as a equilibrium
point for balance laws. However, condition (<ref>) in this paper causes
the temperature  is globally  away from 0.
 We thereby replace the equilibrium state with a Riemann data U_∞ (see Subsection <ref>),
which may connect two distinct states at x=±∞. Theorem 1 demonstrates that
the flow close to U_∞ is still stable under small perturbation in BV⋂ L^1 space.



Furthermore, we find that the local characteristics of solution U(t,x)
are completely determined by either conservation laws (<ref>)
or transport equations (<ref>).
To illustrate this, we introduce
an entropy solution U_ξ^C(t,x) to  (<ref>)
and a solution U_ξ^T(t,x) to (<ref>),
then derive the following theorem.


Assume that T.V.(U_0)
and ||Y_0||_∞ are sufficiently small,
moreover, condition (<ref>) holds.
Let 𝒫 be a semigroup defined in Theorem 1,
and λ̂ be a positive constant larger than all characteristic speeds of system (<ref>).
If U(t,x)= 𝒫_t (U_0) with   U_0 ∈𝒟_0,
then for every s≥0 and a<ξ<b, there holds

    lim_θ→ 0^+1/θ∫_ξ-θλ̂^ξ+θλ̂
    ||U(s+θ,x)-U_ξ^C(θ,x)||dx =0,  
       lim_θ→ 0^+1/θ∫_a+θλ̂^b-θλ̂
    ||U(s+θ,x)-U_ξ^T(θ,x)||dx  ≤
    C_0 {T.V.(U(s);(a,b))}^2 ,

where  C_0 is a positive constant. Conversely, if mapping U: [0,+∞)
→𝒟_∞  is  Lipschitz continuous in L^1-topology and satisfies
(<ref>)(<ref>)
for almost every s≥0 and a< ξ<  b , then U(t) must coincide with the semigroup trajectory
𝒫_t (U_0).


Theorem 2 reveals that the entropy solution  possessing local features (<ref>)(<ref>)
is actually unique, no matter what approach we adopt.






§ PRELIMINARY



The author <cit.> developed the existence theorem for general mixed Euler fluid.
First of all, we directly state the algorithm and the estimates of Glimm functional for system (<ref>).
As mentioned in Section 1, assume thermodynamics parameters γ and c remain constants
throughout the paper.



 §.§ Fractional Step Wave-Front Tracking Scheme


The strategy of this scheme is  separating the effect of source term from original system,
and then transforming it into a linear problem.
Now we initiate the procedure by simplifying initial data U_0.
For each ε>0, choose suitable U^ε_0(x) as an approximation of U_0 (x) such that

  
  * U^ε_0(x) is piecewise constant, and has finite discontinuous points
        {x_α}⊂ [-1/ε, 1/ε] .
  
  * T.V.(U^ε_0) ≤ T.V.(U_0) .
  
  * ∫_-1/ε^1/ε ||U^ε_0(x)-U_0(x)|| dx < ε.




Then divide time interval [0, 1/ε] into N subintervals, such that
division points

    t_0= 0,  t_1= ε,  t_2= 2ε,  ···  t_N= Nε.

Additionally set t_N+1=+∞.



In each strip domain
Ω_k= { (t,x) | t_k≤ t<t_k+1,  x ∈ℝ} (k=0,1,···, N),
 we approximately solve the initial value problem of conservation law

    U_t + F(U)_x   =0,      (t,x)∈Ω_k ,
       U(t_k,x)  = U^ε (t_k,x)

by standard wave-front tracking method. See <cit.> for this method in detail.
Define the solution to (<ref>) by

    U^ε (t,x)≐𝒮_t ^ε (U^ε (t_k,x)),    (t,x)∈Ω_k,

where 𝒮_t ^ε is the approximate solution operator of (<ref>) .
Notice that Y(t,x)≡ Y(t_k,x) for every point (t,x)∈Ω_k.
When t=t_k, the chemical reaction will be activated.
Therefore consider an ordinary differential equations,

    U_t= G(U),      t ∈ [0,ε],
       U(0,x)= U^ε(t_k-,x)

where U^ε(t_k-,x)≐lim_t → t_k^- U^ε(t,x).
We define the linearly approximate solution by

    𝒯_t (U(0,x))  ≐ U(0,x) + G(U(0,x))t,

where 𝒯_t denotes the approximate solution operator of (<ref>).
Then assign the value of U^ε (t,x) after reaction by

    U^ε(t_k,x)    ≐𝒯_ε (U^ε(t_k-,x)) 
        = U^ε(t_k-,x) + G(U^ε(t_k-,x))ε.


In summary, we formulate the ε-approximate solution to system (<ref>) as follow,

    U^ε (t,x) = 𝒫_t ^ε (U^ε_0(x))≐𝒮_t-kε ^ε(𝒯_ε𝒮_ε ^ε )^k(U^ε _0(x)),

where integer k=min{[t/ε], N}.
 𝒫_t^ε is actually a composition of mappings 𝒯_ε,
𝒮_ε^ε, etc..
It was proved in <cit.> that as ε→0,
the sequence of U^ε (t,x) converges to an entropy weak solution U(t,x)=𝒫_t  (U_0) of system (<ref>).




























 §.§ Evolution of Approximate Solutions in 𝒟_∞



In this subsection we will explain how the flow of  𝒫_t ^ε (U^ε _0(x))
evolves as t increases.
It is necessary to understand some fundamental properties of system (<ref>) in advance.
The Jacobian dF(U) has four eigenvalues, i.e.

    λ_1 = -√(-γ p_v) ,   λ_2 =0,   λ_3 = √(-γ p_v),   λ_4 =0,

where p=p(v,u,E). So we say that system (<ref>) is not strictly hyperbolic.
The waves of the first three fields, called elementary waves, contain shock, rarefaction wave and contact discontinuity;
the wave of the 4th field, called Y-wave, is of speed λ_4 =0.
To ε-approximate solutions, we add a family of non-physical waves (abbr. NP) moving at fixed speed  λ̂.
By construction, total strength of NP waves is less than ε. See <cit.>.

Next introduce some necessary functionals.
They  are developed from  the classic functionals in <cit.>.
For each U(t,x), let functional

    𝒱(U(t))≐∑ (|α| + |σ| +M|δ|),

where |α|, |σ|
and |δ|  respectively denote the strengths of elementary waves,  NP wave
and Y-wave  in U(t).
Positive constant M is used to remove bad terms arising from estimates of Glimm functional;
see Lemma 8 in <cit.>.
In order to distinguish Y-waves which locate in different positions at time t,
we write |δ_α| with subscript α.
It means that this Y-wave front of strength |δ_α| intersects with a front of strength |α|
 at some point (t̅, x̅),  where t̅≤ t.
We define
|δ_β|, |δ_σ|, |δ̅_α|, |δ̅_β| etc. in analogous way.
Furthermore, write

    Q(U(t))≐   ∑_App |α|·| β| +
     ∑_App|α| · M|δ _β|+
    ∑ M|δ_α|· M|δ_β|  
        +  ∑_App |σ|·| β | +
        ∑_App|σ| · M|δ_β|+
    ∑ M|δ_σ|· M|δ_β|

to measure the interaction of any two approaching waves except that of NP
 wave and Y-wave. We require
the term  M|δ_α|· M|δ_β| is included in Q(U(t))
if corresponding |α |·|β|  exists in  ∑_App |α|·| β|.
Analogously whether M|δ_σ|· M|δ_β| is included in Q(U(t)) depends on
|σ|·| β |.



Define modified Glimm functional

    ℱ(U(t))≐𝒱(U(t))+C· Q(U(t)),

where positive constant C is so large that estimate (<ref>) holds.
To generate a Lipschitz continuous semigroup, we require that Y_0 ∈ L^1⋂ L^∞(ℝ).
However, ||Y_0||_L^1 may be arbitrarily large.




Suppose there exist constants ϕ and ψ such that
0< ϕ < ϕ(T_0) < ψ.
Thus it is reasonable to assert that

    0< ϕ≤ϕ(T(t,x)) ≤ψ

holds for all t>0 provided U (t,x) is sufficiently close to initial U_0(x) in L^∞(ℝ).
It was proved
that the  Glimm functional ℱ designed for system (<ref>) is possibly increasing but eventually bounded, provided initial
data is suitably small. Here we directly state the following estimates of ℱ
in Lemma 1.
Also see conclusions (41)(42) in <cit.>.


Assume that T.V.(U_0) and ||Y_0||_∞ are small enough, moreover, condition (<ref>) holds.
 Let U^ε
be an approximate solution defined in (<ref>), and integer k=[t/ε].
Then

    ℱ(U^ε(t))
           ≤ℱ(U^ε(t_k-)) exp{B ||Y_0||_∞e^-ϕ kεε}
       ≤ℱ(U_0) exp{B ||Y_0||_∞∑_iε≤ te^-ϕ iεε}
       ≤ℱ(U_0) exp{B||Y_0||_∞/ϕ}
       ≤ℱ̅ ,
    
      Y(t)    ≤ Y(0)e^ -ϕ kε


for small ε>0, where i∈ℕ_+, both B and ℱ̅
are constants .



Comparing (<ref>) with original result in <cit.> ,
 we only maintain the first order terms of ε,
because all higher order ones can be absorbed if ε is small.
Since ℱ(U) is equivalent to T.V.(U), there exists a constant
B^*>0 such that

    ℱ(U_0) < ϵ   implies    
    ||Y_0||_∞<B^*ϵ

for each ϵ>0.
Given ϵ,  define

    ϵ(t)≐ϵ·exp{ BB^* ϵ∫_0^te^-ϕ sds }
        =ϵ·exp{BB^*ϵ/ϕ
        (1-e^-ϕ t) },   
       ϵ_∞≐ϵ·exp{BB^*ϵ/ϕ}.





Set vector 𝒰≐ (v,u,E).
State U_∞=(𝒰_∞(x),0) denotes a Riemann initial data with two pieces of constant states, i.e.

    𝒰_∞(x)={   𝒰_r,    x>0, 
       𝒰_l,    x<0.
        .

Evidentally G(U_∞)=0 in (<ref>) and (<ref>).
Now we can explicitly define the domains where
the flow 𝒫_t^ε(U) evolves.
In fact, for every  t≥0, set

    ℳ_t≐cl{U̅∈ L^1(ℝ;ℝ^4) . |U̅ is piecewise constant, 
                              .  ℱ(U̅+U_∞) < ϵ (t) ,
      ||Y̅||_∞<B^*ϵe^-ϕ t}, 
       ℳ_∞≐cl{U̅∈ L^1(ℝ;ℝ^4) . |U̅ is piecewise constant, 
                              .  ℱ(U̅+U_∞) < ϵ _∞ ,
      ||Y̅||_∞=0 },

where cl denotes closure in L^1 topology. Then define domain


    𝒟_t≐{ U | U= U̅+U_∞, U̅∈⋃_s≤ tℳ_s
    . }.



These domains obviously satisfy
𝒟_s⊂𝒟_t⊂𝒟_∞
for 0≤ s<t.







We complete the section by the dynamics of 𝒫^ε_t (U) in 𝒟_∞.



Assume that ϵ is sufficiently small.
If U ∈𝒟_s is a piecewise constant function, then there exists a
 positive t̅, which depends on U, ϵ and s, satisfying that 

(1) 𝒯_t (U) ∈𝒟_s+t for t ∈ (0, t̅ );

(2) 𝒮^ε_t (U) ∈𝒟_s,
𝒫^ε_t (U) ∈𝒟_s+t for any t∈ (0,+∞) , provided ε≪t̅.





1. If U ∈𝒟_s,  there exists
some r<s such that U ∈ℳ_r. This gives

    ℱ(U) < ϵ (r)=ϵ·exp{BB^*ϵ/ϕ
        (1-e^-ϕ r) }, 
        ||Y||_∞<B^*ϵe^-ϕ r.



Thanks to  Lemma 10 in <cit.> and the definition of 𝒯_t,
 we have the following estimates.

    ℱ(𝒯_t(U))
         ≤ℱ(U) exp{B ||Y||_∞e^-ϕ tt }
                  ≤ℱ(U) exp{B ||Y||_∞t }, 
       𝒯_t Y =Y-tYϕ(T) ≤ Ye^ -ϕ t ,

where 𝒯_t Y denotes the Y-component of 𝒯_t(U).
Immediately, they imply

    ℱ(𝒯_t(U))
         ≤ϵ·exp{BB^*ϵ/ϕ
        (1-e^-ϕ r)+ B ||Y||_∞t  }, 
         ||𝒯_t Y||_∞≤ ||Y||_∞e^ -ϕ t 
    <B^*ϵe^-ϕ (r+t)    if  tϕ<1.



Set function

    φ(t)=ϕ t/1- e^ -ϕ t .

Since φ(0)→ 1 as t→ 0, we select a positive t̅
such that

    ||Y||_∞φ(t) < B^*ϵe^-ϕ r,


    i.e.    B||Y||_∞ t <BB^*ϵ/ϕe^-ϕ r
    (1-e^-ϕ t)

for every t ∈ (0,t̅).
Thus it follows from (<ref>) that

    ℱ(𝒯_t(U)) < ϵ (r)exp{B ||Y||_∞
            t}
            <ϵ·exp{BB^*ϵ/ϕ
        (1-e^-ϕ (r+t)) }.

(<ref>) and (<ref>) guarantee  𝒯_t(U) ∈ℳ_r+t⊂𝒟_s+t.


2. Since 0<ε≪t̅, it is deduced from standard argument of front tracking scheme that
the Glimm functional ℱ(𝒮_t^ε (U)) is decreasing in t,
provided T.V.(U)
is small enough. Therefore, if U ∈𝒟_s, the fact
ℱ(𝒮_t^ε (U)) ≤ℱ(U)
and Y(t)≡ Y(0) yields 𝒮_t^ε (U) ∈𝒟_s for any t>0.
Recall that 𝒫_t^ε (U)= 𝒮_t-kε^ε
(𝒯_ε𝒮_ε^ε)^k (U)
where k=[t/ε]. The preceding results of 𝒯_t and 𝒮_t
directly yield 𝒫^ε_t (U) ∈𝒟_s+t.
□









§ STABILITY OF ENTROPY SOLUTION


We will construct a functional in this section to measure the distance of two approximate solutions.
This functional is almost decreasing in time t, which implies the  continuous dependence of entropy
solutions on initial data.



 §.§ Lyapunov Functional



Assume initial data U_0 and V_0 belong to 𝒟_0.
U_0^ε and V_0^ε are corresponding ε-approximations.
Let

    𝒫_t^ε(U_0^ε) = U^ε(t,x)=(𝒰_1, Y_1),


    𝒫_t^ε(V_0^ε) = V^ε(t,x)=(𝒰_2, Y_2).

Here every state U (or V) is decomposed into components 𝒰≐ (v,u,E) and Y.

Consider equations (<ref>) in phase space.
The i-th family of Hugoniot curves is determined by the mapping H_i.
For given states 𝒰_1 and 𝒰_2, there exists unique vector
𝐪= (q_1, q_2, q_3) such that

    𝒰_2 = H(𝐪)(𝒰_1)
    ≐ H_3(q_3)∘ H_2(q_2)∘ H_1(q_1)(𝒰_1)

at every point (t,x).
It is known that total strength ∑_i=1^3 |q_i| is equivalent to ||𝒰_1-𝒰_2||.


We introduce the  Lyapunov functional

    Φ (U^ε(t), V^ε(t)) ≐∑_i=1^3∫_-∞^+∞(|q_i(t,x)|+ κ_1 |Y_1(t,x) - Y_2(t,x)| )W_i(t,x) dx.


This functional is developed from the one in <cit.>. It is worth pointing out that,
in contrast with Φ in  <cit.>, we add the term of Y and modify the weight W_i
in order to overcome the damping effect.
Specifically, let α be a jump of the k_α-th  family (1≤ k_α≤ 3),
whose location is (t,x_α)  and strength |α| .
 Let δ be a jump of Y-wave, whose  strength is |δ|.
Notation 𝒥(U) denotes a set of all jumps of U for given time t.
Set 𝒥≐𝒥(U^ε)⋃𝒥(V^ε).
Then the weight

    W_i(t,x)≐ 1+ κ_2 A_i(t,x) + κ_3  {Q(U^ε)+Q(V^ε)} +κ_4 B(t),



where for i=1,3,

    A_i(t,x)=    [∑_c
                             x_α <x, i<k_α≤ 3, 
    α∈𝒥
                    +∑_cx_α >x, 1≤ k_α <i, 
    α∈𝒥] |α|
    +  ∑_δ∈𝒥 M|δ|
        + {   [∑_c
                      k_α =i ,  x_α <x,
    α∈𝒥(U^ε)
    
           + ∑_c
                  k_α =i ,  x_α >x 
    α∈𝒥(V^ε)
                ] |α|
                 q_i(x)<0 ,
       [∑_c
                      k_α =i ,  x_α <x,
    α∈𝒥(V^ε)
    
           + ∑_c
                  k_α =i ,  x_α >x 
    α∈𝒥(U^ε)
                ] |α|
              q_i(x) >0,
        . 
    
      A_2(t,x)=    [∑_c
                             x_α <x, k_α = 3, 
    α∈𝒥
                    +∑_cx_α >x,  k_α=1 , 
    α∈𝒥] |α|
    +  ∑_δ∈𝒥 M|δ|,
    
    B(t)=   ∑_jε > t
            (||Y_1 (0,·)||_∞+||Y_2 (0,·)||_∞) e^-ϕ jεε,





and Q(U) defined in (<ref>). The constants κ_i  ( 1≤ i ≤ 4) will be specified in the sequel.
Especially κ_3 = C ·κ_2. Recall that C is a large constant in the definition of ℱ(U).

We claim  1≤ W_i ≤ 2 if ϵ is small enough.
By the boundness of W_i,
one can verify that L^1 distance of U^ε(t) and V^ε(t) is equivalent to Φ (U^ε(t), V^ε(t)),
i.e.

    1/C_1||U^ε - V^ε||_L^1≤Φ (U^ε, V^ε) ≤ C_1||U^ε - V^ε||_L^1

for some constant C_1>1.
To understand the property of Φ, let us prove the monotonicity of W_i at first.



For every x ∈ℝ, W_i(x,t) is piecewise constant and decreasing in t.



When t ∈ (t_k, t_k+1), all the terms w.r.t. Y are unchanged. Thus W_i is
equivalent to the weight W_i(x) defined for conservation laws in <cit.>.
This is certainly decreasing in t. It suffices to verify the case  t=t_k.
Set

    W_i^+≐ W_i(t_k,x),    W_i^-≐ W_i(t_k-,x).

Using (<ref>), we calculate that

    W_i^+ - W_i^-    ≤κ_2 ℱ̅B (||Y_1 (0,·)||_∞ + ||Y_2 (0,·)||_∞)  ·e^-ϕ kεε
           - κ_4  (||Y_1 (0,·)||_∞ + ||Y_2 (0,·)||_∞)
           ·e^-ϕ kεε
       ≤ - 1/2κ_4 (||Y_1 (0,·)||_∞ + ||Y_2 (0,·)||_∞)
        ·e^-ϕ kεε
       <0

if κ_4 is so large that κ_2 ℱ̅B < 1/4κ_4.
The proof is completed. □


The modified weight W_i is  absolutely a key factor in the whole paper. Using its
monotonicity, we successfully prove in Lemma <ref>
that Φ is almost deceasing. This fact implies the
stability and uniqueness of entropy  solution.



According to the scheme stated in
Subsection <ref>, time t can be categorized into three cases as follows.
The time when approaching waves interact is denoted by t^*;
The time when chemical reaction is activated is denoted by t_k ;
The other time is denoted by t^∘ .
Obviously Lyapunov functional Φ(U^ε(t),V^ε(t)) is decreasing at time t^*,
because the main factor

    ∫_-∞^+∞(|q_i(t,x)|+ κ_1 |Y_1(t,x) - Y_2(t,x)| )dx

 in definition (<ref>)
is  continuous  at t^*, meanwhile, W_i is decreasing steeply.
Hence it suffices to discuss the remaining cases t=t_k or t^∘ .





 §.§ Estimate of Φ(t) at t_k





Define  the components of U^ε
on two banks of discontinuity t=t_k by

    𝒰_1^+ ≐𝒰_1 (t_k , x),      𝒰_1^- ≐𝒰_1 (t_k -, x),
        Y_1^+ ≐  Y_1(t_k , x),         Y_1^- ≐ Y_1 (t_k -, x).


And 𝒰_2^±,  Y_2^± represent the similar quantities of V^ε.
Accordingly define 𝐪^+ and 𝐪^- by expression
(<ref>).





At every point (t_k,x), assume 𝐪^+ and 𝐪^- satisfy

    𝒰_2^- = H(𝐪^-)(𝒰_1^-),  𝒰_2^+ = H(𝐪^+)(𝒰_1^+).

 Then

    q_i^+ - q_i ^- = 𝒪(1)
      {|Y_1^-  - Y_2^-|ε +
       ∑_j=1^3 |q_j^-| (Y_1^- + Y_2^-)ε} .

holds for i=1,2,3.

The Landau symbol 𝒪(1) denotes a quantity whose absolute value is bounded by a constant dependent
of System (<ref>) and the domain 𝒟_∞.




Set the differences

    Δ_1  ≐ Y^+_1 - Y^- _1= -Y^- _1ϕ(T^-_1)ε,
       Δ_2  ≐ Y^+_2 - Y^-_2 = -Y^-_2 ϕ(T^-_2)ε,



and vector 𝐰≐ (0,0,-q).
Then it follows that

    𝒰_1^+= 𝒰_1 ^- + 𝐰·Δ_1,  𝒰_2^+= 𝒰_2^- + 𝐰·Δ_2 .


Equations (<ref>)
and (<ref>) determine an implicit C^2 function

    𝐪^+ ≐𝐪(𝐪^-, Δ_1,Δ_2).




Notice that 𝐪(0,0,0)= 𝐪(0,Δ_1,Δ_1)=0, and 𝐪(𝐪^-,0,0)=𝐪^-.
Using Proposition 1 in <cit.> and Taylor's formula, we figure out

    𝐪^+
          = 𝐪(0,Δ_1,Δ_2) +𝐪(𝐪^-,0,Δ_2)
            -𝐪(0,0,Δ_2) +𝒪(1)||𝐪^-||·|Δ_1|·𝐈
       = 𝐪(0,Δ_1,Δ_2) + 𝐪^-
             +𝒪(1)||𝐪^-||·(|Δ_1|+|Δ_2|)·𝐈
       =  𝐪^-  +𝒪(1)|Δ_1 - Δ_2|·𝐈
             +𝒪(1)||𝐪^-||·(|Δ_1|+|Δ_2|)·𝐈

where 𝐈=(1,1,1)^T is a 3×1 vector.
By Lipschitz continuity of ϕ(T), we have

    |ϕ(T_1^-)-ϕ(T_2^-)| ≤𝒪(1) ||𝒰_1^-  -  𝒰_2^-||
    ≤𝒪(1) ∑_j=1^3 |q_j^-|.



It follows from condition (<ref>) and (<ref>) that

    |Δ_1 - Δ_2|
           ≤ |Y_1^-  - Y_2^-|ϕ(T^-_1)ε +Y_2^-|ϕ(T^-_1)-ϕ(T^-_2)|ε
       ≤𝒪(1)|Y_1^-  - Y_2^-| ε +𝒪(1) Y_2^-||𝒰^-_1 -𝒰^-_2||ε
       ≤𝒪(1){|Y_1^-  - Y_2^-|ε +
            ∑_j=1^3 |q_j^-|  Y_2^- ε} , 
    
       |Δ_1 |+| Δ_2|
            ≤𝒪(1) (Y_1^- + Y_2^-)ε.




Finally (<ref>)(<ref>)
and (<ref>) yield estimate (<ref>).
□




At every point (t_k,x), components Y of U^ε and V^ε satisfy that

    |Y_1^+ - Y_2^+| - |Y_1^- - Y_2^-| ≤
      -|Y_1^-  - Y_2^-|ϕε + 𝒪(1)  ∑_j=1^3 |q_j^-|  Y_2^-ε.





Note that ϕ(T) is bounded below by ϕ. It is easy to check that

    |Y_1^+ - Y_2^+| =    |Y_1^-(1-ϕ(T_1^-)ε) - Y_2^-(1-ϕ(T_2^-)ε)| 
    ≤    |Y_1^- - Y_2^-|·|1-ϕ(T_1^-)ε| + Y_2^-|ϕ(T_1^-)-ϕ(T_2^-)|ε
    ≤    |Y_1^- - Y_2^-|·(1-ϕε) + 𝒪(1)
       ∑_j=1^3 |q_j^-|Y_2^-ε

for small ε. This implies the estimate (<ref>).
□






At every time t_k=kε, Lyapunov functional is decreasing, i.e.

    Φ(U^ε(t_k ),V^ε(t_k )) ≤Φ(U^ε(t_k -),V^ε(t_k -)).








By Lemmas <ref>-<ref> and
estimate (<ref>)
we have

    (|q_i^+| +κ_1 |Y_1^+ - Y_2^+|)- (|q_i ^-|  + κ_1|Y_1^- - Y_2^-|)
    
      =     (|q_i^+| - |q_i ^-|) + κ_1 (|Y_1^+ - Y_2^+| - |Y_1^- - Y_2^-|)
    ≤    (-κ_1 ϕ + 𝒪(1)) |Y_1^- - Y_2^-|ε + 𝒪(1)  ∑_j=1^3 |q_j^-| (Y_1^- + Y_2^-)ε
    ≤    - 1/2κ_1 ϕ  |Y_1^- - Y_2^-|ε +𝒪(1)  ∑_j=1^3 |q_j^-| (Y_1^- + Y_2^-)ε
    ≤   𝒪(1)  ∑_j=1^3 |q_j^-| ·
       (||Y_1 (0,·)||_∞ + ||Y_2 (0,·)||_∞)
       e^-ϕ kεε

provided κ_1 is large enough.
Then  estimates (<ref>) and (<ref>)
give

    (|q_i^+|+ κ_1 |Y_1^+ - Y_2^+|)W_i^+ - (|q_i^-|+ κ_1  |Y_1^- - Y_2^-|)W_i^- 
        = { (|q_i^+| - |q_i ^-|) + κ_1 (|Y_1^+ - Y_2^+| - |Y_1^- - Y_2^-|)}W_i^+  
          + (|q_i^-|+ κ_1 |Y_1^- - Y_2^-|) (W_i^+ -W_i^-)
       ≤𝒪(1) ∑_j=1^3 |q_j^-| · (||Y_1 (0,·)||_∞ + ||Y_2 (0,·)||_∞) e^-ϕ kεε· W_i^+  
          - 1/2κ_4 (|q_i^-|+ κ_1 |Y_1^- - Y_2^-|)(||Y_1 (0,·)||_∞ + ||Y_2 (0,·)||_∞)·e^-ϕ kεε
       ≤𝒪(1) ∑_j=1^3 |q_j^-| · (||Y_1 (0,·)||_∞ + ||Y_2 (0,·)||_∞) e^-ϕ kεε
            - 1/2κ_4 |q_i^-|· (||Y_1 (0,·)||_∞ + ||Y_2 (0,·)||_∞) ·e^-ϕ kεε  .




Furthermore

    ∑_i=1^3 (|q_i^+|+ κ_1 |Y_1^+ - Y_2^+|)W_i^+ -
      ∑_i=1^3 (|q_i^-|+ κ_1  |Y_1^- - Y_2^-|)W_i^- 
       ≤𝒪(1) ∑_j=1^3 |q_j^-| · (||Y_1 (0,·)||_∞ + ||Y_2 (0,·)||_∞)·e^-ϕ kεε
            - 1/2κ_4 ∑_i=1^3 |q_i^-|·
      (||Y_1 (0,·)||_∞ + ||Y_2 (0,·)||_∞) ·e^-ϕ kεε
       ≤ 0

provided κ_4 is large enough.
Consequently

    Φ(U^ε(t_k ),V^ε(t_k )) -  Φ(U^ε(t_k -),V^ε(t_k -)) 
    
      =    ∫_-∞^+∞{∑_i=1^3 (|q_i^+|+ κ_1 |Y_1^+ - Y_2^+|)W_i^+ -
      ∑_i=1^3 (|q_i^-|+ κ_1  |Y_1^- - Y_2^-|)W_i^- } dx 
    ≤    0  .

□








 §.§ Estimate of Φ(t) at t^∘



At any time t^∘, jump α∈𝒥
locates at x_α and moves at the speed of ẋ_α.
We use the notations

    W_i^α+≐  W_i(t^∘,x_α+),       W_i^α-≐  W_i(t^∘,x_α-),  
        q_i^α+≐  q_i(t^∘,x_α+),      q_i^α-≐  q_i(t^∘,x_α-),

etc. to represent the quantities on two banks of x_α.
Then we find Φ is almost decreasing at t^∘.




At every t=t^∘, Lyapunov functional satisfies

    d/dtΦ(U^ε(t),V^ε(t))
    ≤𝒪(1) ε.





We can calculate the derivative of Φ by means of the technique in <cit.>.
Set w_0=𝒰_1,  w_i=H_i(q_i)∘⋯∘ H_1(q_1)( 𝒰_1),  w_3=𝒰_2.
Then define λ_i ≐λ_i(w_i-1,w_i) as the speed of the i-shock connecting states
w_i-1 and w_i. We derive that

    d/dtΦ(U^ε(t),V^ε(t))
        =   ∑_α∈𝒥∑_i=1^3{(|q_i^α-|+κ_1|Y_1-Y_2|^α-)W_i ^α-.
             -.(|q_i^α+|+κ_1|Y_1-Y_2|^α+)
        W_i ^α+}·ẋ_α
    
    =   ∑_α∈𝒥∑_i=1^3{(|q_i^α+|+κ_1|Y_1-Y_2|^α+)W_i ^α+(λ_i^α+-ẋ_α ).
             -.(|q_i^α-|+κ_1|Y_1-Y_2|^α-)
        W_i ^α-(λ_i^α--ẋ_α )}
    
    =   ∑_α∈𝒥∑_i=1^3 E_i,α

where

    E_i,α≐   (|q_i^α+|+κ_1|Y_1-Y_2|^α+)W_i ^α+(λ_i^α+-ẋ_α )
       -(|q_i^α-|+κ_1|Y_1-Y_2|^α-)
    W_i ^α-(λ_i^α--ẋ_α ).



Observe |Y_1-Y_2|^α-=|Y_1-Y_2|^α+ at t^∘.
One compute that

    E_i,α
    =   (|q_i^α+|-|q_i^α-)W_i ^α+(λ_i^α+-ẋ_α )
       + (|q_i^α-|+κ_1|Y_1-Y_2|^α-)(W_i ^α+-W_i ^α-)(λ_i^α+-ẋ_α) 
       + (|q_i^α-|+κ_1|Y_1-Y_2|^α-)W_i ^α-(λ_i^α+-λ_i^α-).


Let S, R and NP respectively denote the front of shock, rarefaction and non-physical wave.
Since |q_i^α-|+κ_1|Y_1-Y_2|^α- is a bounded quantity, we claim that

    ∑_i=1^3 E_i,α≤𝒪(1)·|σ_α|,      α∈ NP 
       ∑_i=1^3 E_i,α≤𝒪(1)·ε|α|,      α∈ S∪ R

following from the conclusions (3.5)(3.6) in <cit.>.
Recall that ∑_α∈ NP|σ_α| ≤ε. Then it yields

    d/dtΦ(U(t),V(t))=
    ∑_α∈ NP∑_i=1^3 E_i,α+
    ∑_α∈ S∪ R∑_i=1^3 E_i,α≤𝒪(1) ε.

□





In summary, we draw the following conclusion from Lemmas <ref>-
<ref>.




Let U^ε(t),V^ε(t) be ε-approximate solutions of (<ref>) constructed by fractional step front tracking
method. Choose suitable constants ϵ, κ_1, κ_2, κ_3, κ_4. If
U^ε(0), V^ε(0) ∈𝒟_0,
 then there exists a constant C_2 such that

    Φ(U^ε(t),V^ε(t)) - Φ(U^ε(s),V^ε(s)) ≤ C_2 ε (t-s), 
        ||U^ε(t)-V^ε(t)||_L^1≤ C_1^2||U^ε(s)-V^ε(s)||_L^1+C_1 C_2ε (t-s)

for all t>s≥ 0, where constant C_1 is specified in (<ref>).




Letting G≡0 or Y_0=0, one can derive the stability estimate for 𝒮_t^ε (U); that is,

    ||𝒮_t^ε (U) - 𝒮_t^ε (V)||_L^1≤ C_1^2||𝒮_s^ε (U)- 𝒮_s^ε (V) ||_L^1+C_1 C_2ε (t-s)

for all t>s≥ 0. This coincides with the result of conservation laws in <cit.>.
Besides,  Lipschitz continuity of 𝒮_t^ε (U) w.r.t time t actually holds.
Therefore we assert that there exists a Lipschitz constant C_3 such that

    ||𝒮_t^ε (U) - 𝒮_s^ε (U)||_L^1≤ C_3(t-s)

for any t>s≥ 0. See <cit.>.






Lemma <ref> and Lemma <ref> on convergence
give rise to the stability  of entropy solution.









§ UNIQUENESS OF LIMIT SOLUTION


In the section we attempt to prove that by means of the scheme in Subsection <ref>,
 every sequence of approximate solutions must converge to a unique limit. And then give a rigorous proof
of Theorem 1.












 §.§ Estimates of Operator 𝒫^ε_t





Note that 𝒮_t^ε is identity mapping w.r.t. component Y.
We claim  the following Lemma developed from <cit.> still holds
for non-strictly hyperbolic system (<ref>).
Its proof in detail is referred to Theorem 4.1 in <cit.>.



Assume that U and V are piecewise constant functions in 𝒟_s.
ω(x) ∈ L^1(ℝ^4) is a piecewise constant function with small total variation,
and has finitely many discontinuities. Then

    ||𝒮_t^ε  (U)-𝒮_t^ε  (V)-ω||_L^1≤𝒪(1)|| U- V -ω||_L^1 +𝒪(1) t{ε+ T.V.(ω)}.





Additionally we need some information about source G.
Since G(U) in (<ref>)(<ref>) is locally Lipschitz continuous, there exists a constant
L'>0 such that

    ||G(U)-G(V)||≤ L' ||U-V||, 
        T.V.(G(U)) ≤ L'· T.V.(U) , 
        ||G(U)||_L^1 =||G(U)-G(U_∞)||_L^1≤ L'||U-U_∞||_L^1

for any U, V ∈𝒟_∞.
Next we can calculate the error caused by commutation of 𝒯_t
 and 𝒮_t^ε.



(Commutation estimate)
Assume that U is a piecewise constant function in 𝒟_s.
Then

    ||𝒮_t^ε𝒯_t (U)-𝒯_t𝒮_t^ε  (U)||_L^1≤𝒪(1)t^2 +𝒪(1) t ε

holds for ε≤ t <t̅.



The Lipschitz continuity of G implies

    T.V.( G(𝒮_t^ε  (U)) ≤ L'· T.V.(𝒮_t^ε(U))
    ≤𝒪(1) ϵ_∞.

Combining this with Lemma <ref> and Remark <ref>,
we deduce

    ||𝒮_t^ε𝒯_t (U)-𝒯_t𝒮_t^ε  (U)||_L^1
    
        =    ||𝒮_t^ε  (U+tG(U))-𝒮_t^ε  (U)-t G(𝒮_t^ε  (U))||_L^1
    ≤   𝒪(1) || (U+tG(U))-U-t G(𝒮_t^ε (U))||_L^1
            + 𝒪(1) t {ε+ T.V.(t G(𝒮_t^ε  (U))}
    ≤   𝒪(1)t·  ||𝒮_t^ε  (U)-U||_L^1
             + 𝒪(1) t·{ε+ t· T.V.( G(𝒮_t^ε  (U))}
    ≤   𝒪(1)t^2 + 𝒪(1)tε .

□



Similar property on strictly hyperbolic system was also observed in <cit.>.
Lemma <ref> is indeed the basis to establish uniqueness of solutions generated by semigroup 𝒫.
It shows that commutation of 𝒮_ε^ε and 𝒯_ε once gives rise to a small error of ε^2.
Since composition 𝒮_kε^ε(𝒯_ε)^k can be transformed to 𝒫_kε^ε by k(k-1)/2 commutations, the total errors accumulated in such process is less than k^2ε^2. The fact guarantees the conclusion in Lemma <ref>.


To compute the error of 𝒫_t^ε (U)- 𝒮_t^ε𝒯_t (U),
 we establish the estimates of operator (𝒯_ε)^k in advance.



For any piecewise constant functions U, V∈𝒟_s, the following relations hold.

    ||(𝒯_ε)^k (U)-U_∞||_L^1≤e^kε L'||U-U_∞||_L^1, 
         ||(𝒯_ε)^k (U)-U||_L^1≤
       kε L' e^kε L'||U-U_∞||_L^1, 
         ||(𝒯_ε)^k (U)- (𝒯_ε)^k (V) ||_L^1≤e^kε L'||U-V||_L^1.







The last inequality is obvious according to Lipschitz continuity of G.
It suffices to prove the first two inequalities by induction.
Firstly, for  k=1, one deduce from (<ref>) that

    ||𝒯_ε (U)-U_∞||_L^1
        =    ||U-U_∞+ ε{G(U)-G(U_∞)}||_L^1
    ≤    ||U-U_∞||_L^1 + ε L' ||U-U_∞||_L^1
    ≤   e^ε L' ||U-U_∞||_L^1, 
    
          ||𝒯_ε (U)-U||_L^1
        =   ε ||G(U)-G(U_∞)||_L^1
    ≤   ε L' ||U-U_∞||_L^1
    ≤   ε L' e^ε L' ||U-U_∞||_L^1 .




Secondly,  for integer k>1 , we have

    ||(𝒯_ε)^k (U)-U_∞||_L^1
        =    ||(𝒯_ε)^k-1 (U)-U_∞+ ε{G((𝒯_ε)^k-1 (U))-G(U_∞)}||_L^1
    ≤    (1+ε L')||𝒯_ε)^k-1 (U)-U_∞||_L^1
    ≤   e^ε L' ||𝒯_ε)^k-1 (U)-U_∞||_L^1
    ≤   e^k ε L' ||U-U_∞||_L^1

and

    ||(𝒯_ε)^k (U)-U||_L^1
        =    ||(𝒯_ε)^k-1 (U)-U+ ε{G((𝒯_ε)^k-1 (U))-G(U_∞)}||_L^1
    ≤    ||(𝒯_ε)^k-1 (U)-U ||_L^1+
            ε L' ||(𝒯_ε)^k-1 (U)-U_∞||_L^1
    ≤    (k-1)ε L'e^(k-1)ε L' ||U -U_∞||_L^1
            +  ε L'e^(k-1)ε L' ||U -U_∞||_L^1
    ≤    kε L' e^k ε L' ||U-U_∞||_L^1

by induction. □








Assume that U ∈𝒟_s is piecewise constant.
For any time t<t̅ and positive ε≪min{t,t^2}, we have

    ||𝒫_t^ε (U)- 𝒮_t^ε𝒯_t (U)||_L^1≤𝒪(1)(1+||U-U_∞||_L^1) t^2 .




Set integer n=[t/ε] and k=1,2,⋯,n.
Define diameters

    d_k≐sup_V∈𝒟_s+kε
                ||(𝒯_ε)^k  𝒮_ε^ε (V)
                - 𝒮_ε^ε  (𝒯_ε )^k (V)||_L^1,
       d̅_k  ≐sup_V∈𝒟_s+kε
                 || 𝒫_kε^ε (V)
                - 𝒮_ε^ε  (𝒯_ε )^k (V)||_L^1.



Applying Lemma <ref> and definition of d_k, we deduce that

    ||(𝒯_ε)^k  𝒮_ε^ε (U)
                - 𝒮_ε^ε  (𝒯_ε )^k (U)||_L^1≤    ||(𝒯_ε)^k-1𝒯_ε𝒮_ε^ε (U)
                - (𝒯_ε )^k-1𝒮_ε^ε𝒯_ε(U)||_L^1
         + || (𝒯_ε )^k-1𝒮_ε^ε𝒯_ε(U)
            -  𝒮_ε^ε(𝒯_ε)^k-1𝒯_ε (U)||_L^1
    ≤   e^(k-1)ε L'|| 𝒯_ε𝒮_ε^ε (U)
                - 𝒮_ε ^ε𝒯_ε(U)||_L^1
        +|| (𝒯_ε )^k-1𝒮_ε ^ε𝒯_ε(U)
            -  𝒮_ε^ε(𝒯_ε)^k-1𝒯_ε (U)||_L^1
    ≤   e^tL' d_1 + d_k-1 .




It implies  d_k  ≤e^tL' d_1 + d_k-1, furthermore,

    d_k ≤e^tL'k d_1 .

From stability result in Lemma <ref> , Remark <ref> and inequality (<ref>), we derive

    || 𝒫_kε^ε (U) - 𝒮_kε^ε  (𝒯_ε )^k (U)||_L^1
    ≤    || 𝒫_(k-1)ε^ε𝒯_ε𝒮_ε^ε (U)
         - 𝒫_(k-1)ε^ε𝒮_ε^ε𝒯_ε(U)||_L^1
              + || 𝒫_(k-1)ε^ε𝒮_ε^ε𝒯_ε (U)
         - 𝒮_(k-1)ε^ε(𝒯_ε )^k-1𝒮_ε^ε𝒯_ε(U)||_L^1
                 + || 𝒮_(k-1)ε^ε(𝒯_ε )^k-1𝒮_ε^ε𝒯_ε (U)
         - 𝒮_(k-1)ε^ε𝒮_ε^ε  (𝒯_ε )^k-1𝒯_ε(U)||_L^1
    ≤    C_1^2|| 𝒯_ε𝒮_ε^ε (U)
         - 𝒮_ε^ε𝒯_ε(U)||_L^1 +  C_1 C_2(k-1)ε^2 
              + || 𝒫_(k-1)ε^ε𝒮_ε^ε𝒯_ε (U)
         - 𝒮_(k-1)ε^ε(𝒯_ε )^k-1𝒮_ε^ε𝒯_ε(U)||_L^1
              + C_1^2|| (𝒯_ε )^k-1𝒮_ε^ε𝒯_ε (U)
         - 𝒮_ε^ε  (𝒯_ε )^k-1𝒯_ε(U)||_L^1
          + C_1 C_2 (k-1)ε^2 
    ≤    C_1^2 d_1 + d̅_k-1 + C_1^2 d_k-1 + 2C_1 C_2 kε^2  
    ≤    C_1^2(1+e^tL'k) d_1 + d̅_k-1  + 2C_1 C_2 kε^2 .






Hence there holds

    d̅_k≤ C_1^2(1+e^tL'k)  d_1 + d̅_k-1+ 2C_1 C_2 kε^2,

and then

    d̅_n   ≤ C_1^2(1+e^tL'n)n  d_1 + 2C_1 C_2 n^2ε^2 
        = C_1^2{nε^2+e^tL'(nε)^2}d_1/ε^2+
            2C_1 C_2 n^2ε^2 
       ≤ C_1^2(tε +e^tL't^2)  d_1/ε^2+ 2C_1 C_2 t^2 
       ≤𝒪(1) t^2,




since ε≤ t^2 and d_1 / ε^2 is bounded
in light of Lemma <ref>.
Observe that Lemma <ref> implies

    ||(𝒯_ε)^k  (U)
                -  𝒯_kε (U)||_L^1
    
        =     ||𝒯_ε (𝒯_ε)^k-1   (U)
                - 𝒯_(k-1)ε (U) - ε G(U)||_L^1
    
        =      ||(𝒯_ε)^k-1 (U)
                - 𝒯_(k-1)ε (U)
                + ε G((𝒯_ε)^k-1 (U))- ε G(U)||_L^1
    ≤    ||(𝒯_ε)^k-1 (U)- 𝒯_(k-1)ε (U) ||_L^1
                + (k-1)ε^2 L'^2 e^(k-1)ε L' || U-  U_∞||_L^1.

Hence

    ||(𝒯_ε)^k  (U) -  𝒯_kε (U)||_L^1≤ k^2ε^2 L'^2 e^kε L' || U-  U_∞||_L^1 ,

moreover,

    ||(𝒯_ε)^n  (U)  -  𝒯_t (U)||_L^1
    ≤    ||(𝒯_ε)^n  (U)  -  𝒯_nε (U)||_L^1
            + ||𝒯_t (U) -  𝒯_nε (U)||_L^1
    ≤    n^2ε^2 L'^2 e^nε L' || U-  U_∞||_L^1
            +(t-nε) L' || U-  U_∞||_L^1
    ≤    2 t^2 L'^2 e^t L'|| U-  U_∞||_L^1

provided L'>1.
Consequently, we see that

    ||𝒫_t^ε (U)- 𝒮_t^ε𝒯_t (U)||_L^1≤    ||𝒫_t^ε (U)-   𝒫_nε^ε (U)||_L^1
            + ||𝒫_nε^ε (U)- 𝒮_nε^ε𝒯_nε (U)||_L^1
            + ||𝒮_nε^ε𝒯_nε (U) - 𝒮_t^ε𝒯_t (U)||_L^1
    ≤   𝒪(1)ε^2 + d̅_n + 𝒪(1)||𝒯_nε (U) - 𝒯_t (U)||_L^1
                + 𝒪(1)(t-nε) ε
    ≤   𝒪(1)ε^2 +𝒪(1) (1+ || U-  U_∞||_L^1)t^2  
    ≤   𝒪(1) (1+ || U-  U_∞||_L^1) t^2

by Lemma <ref>, Remark <ref>
and estimates (<ref>)(<ref>).
□


In the end we state an approximate semigroup property of 𝒫_t^ε.


(Semigroup estimate)
For any t_1, t_2 >0 and U∈𝒟_s, the inequality

    ||𝒫_t_2^ε𝒫_t_1^ε (U)- 𝒫_t_1 +t_2^ε (U)||_L^1≤𝒪(1)(1+ε +t_1 + t_2+||U-U_∞||_L^1) ε

holds if 0<ε≪t̅.



Fix ε≪t̅. Then there exists two integers
k and K such that

    (k-1)ε < t_1 ≤ kε,     Kε≤ t_1 +t_2 < (K+1)ε.

They yield

    [t_2/ε] = K-k   or  K-k+1.

Since the arguments of these two cases are analogous, we only discuss the case  [t_2/ε] = K-k.

Let V≐𝒫_t_1^ε (U) ,   V_∞:= 𝒫_t_1^ε (U_∞). Then
Lemma <ref> and Remark <ref>
give that

    ||𝒫_t_2^ε𝒫_t_1^ε (U)- 𝒫_t_1 +t_2^ε (U)||_L^1
    ≤    ||𝒮_t_1+t_2-Kε^ε𝒮_kε- t_1^ε𝒫_(K-k)ε^ε (V) -
            𝒮_t_1+t_2-Kε^ε𝒫_(K-k)ε^ε𝒯_ε𝒮_kε- t_1^ε (V)
        ||_L^1
    ≤   𝒪(1)|| 𝒮_kε- t_1^ε𝒫_(K-k)ε^ε (V)
            -𝒫_(K-k)ε^ε𝒯_ε𝒮_kε- t_1^ε (V)
        ||_L^1
         +  𝒪(1) (t_1+t_2-Kε)ε
    ≤   𝒪(1)|| 𝒫_(K-k)ε^ε𝒯_ε𝒮_kε- t_1^ε (V)
        -  𝒫_(K-k)ε^ε (V)||_L^1
            +  𝒪(1)|| 𝒮_kε- t_1^ε𝒫_(K-k)ε^ε (V)
        -  𝒫_(K-k)ε^ε (V)||_L^1 + 𝒪(1) ε^2 
    ≤   𝒪(1)||  𝒯_ε𝒮_kε- t_1^ε (V) -  V||_L^1
        + 𝒪(1) (K-k)ε^2 
         +  𝒪(1) (kε - t_1) + 𝒪(1) ε^2 
    ≤   𝒪(1){|| 𝒮_kε- t_1^ε (V) -  V||_L^1
        +  ε || G(𝒮_kε- t_1^ε (V)) -  G(𝒮_kε- t_1^ε (V_∞))||_L^1}
             + 𝒪(1) (t_2ε + ε+ ε^2)  
    ≤   𝒪(1) ||V-V_∞||_L^1ε
           + 𝒪(1)(kε - t_1)ε
        +    𝒪(1)( t_2ε + ε+ ε^2)  
    ≤   𝒪(1)||U-U_∞||_L^1ε + 𝒪(1) (t_1+t_2 +1+ε)ε .

We complete the proof.□





 §.§ Convergence of ε-approximate Solutions


First, we need a local estimate of different trajectories 𝒫_t^μ
and  𝒫_t^ν which originate from the same state U.



Assume that U ∈𝒟_s is piecewise constant . Then the inequality

    ||𝒫_t^μ (U)- 𝒫_t^ν (U)||_L^1≤𝒪(1)(1+||U-U_∞||_L^1) t^2

holds for any positive μ, ν≪min{t,t^2}, and time t<t̅.



Note that for any V∈𝒟_∞,

    ||𝒮_t^μ (V)- 𝒮_t^ν  (V)||_L^1≤𝒪(1)max{μ, ν}t

holds (cf. <cit.>) .
By Lemma <ref> we deduce

    ||𝒫_t^μ (U)- 𝒫_t^ν (U)||_L^1
    ≤    ||𝒫_t^μ (U)- 𝒮_t^μ𝒯_t(U)||_L^1
           + ||𝒫_t^ν (U)- 𝒮_t^ν𝒯_t(U)||_L^1
           +  ||𝒮_t^μ𝒯_t (U)- 𝒮_t^ν𝒯_t(U)||_L^1
    ≤   𝒪(1)(1+||U-U_∞||_L^1) t^2 + 𝒪(1)max{μ, ν}t 
    ≤   𝒪(1)(1+||U-U_∞||_L^1) t^2  .

□

Second, we can generalize the result in Lemma <ref> to that
of trajectories with distinct origins.
The following Lemma exhibits  that {𝒫_t^ε (U^ε)}
is a Cauchy sequence  w.r.t. ε in L^1-topology.



Assume that U^μ_0, U^ν_0 are piecewise constant functions in 𝒟_0 such that

    ||U^μ_0- U_0||_L^1 < μ,     ||U^ν_0- U_0||_L^1 < ν ,


    T.V.(U^μ_0) ≤ T.V.(U_0),     T.V.(U^ν_0) ≤ T.V.(U_0)

with some U_0 ∈𝒟_0. Then for every t≥ 0,

    ||𝒫_t^μ (U^μ_0)-𝒫_t^ν (U^ν_0)||_L^1→ 0

as μ, ν→ 0.



Divide the interval [0,t] equally into m parts such that Δ t= t/m is small.
Let the points of such division  satisfy

    0=t_0 < t_1 < ⋯ < t_m = t.

If μ, ν < min{t̅,(Δ t) ^2},  it follows from Lemma <ref>
that

    ||𝒫_t^μ (U^μ_0)-𝒫_t^ν (U^μ_0)||_L^1
    ≤   ∑_i=1^m ||𝒫_t-t_i^μ𝒫_t_i^ν(U^μ_0)
            -𝒫_t-t_i-1^μ𝒫_t_i-1^ν(U^μ_0)||_L^1
    ≤   ∑_i=1^m ||𝒫_t-t_i^μ𝒫_t_i^ν(U^μ_0)
            - 𝒫_t-t_i^μ𝒫_t_i-t_i-1^μ𝒫_t_i-1^ν(U^μ_0)||_L^1
          +  𝒪(1) ∑_i=1^m
          (1+μ+ t-t_i-1 + ||𝒫_t_i-1^ν(U^μ_0) - U_∞||_L^1)μ
    
    =:     Σ_1 + Σ_2

where Σ_1 (resp. Σ_2) denotes the first (resp. the second) sum of the
 third part in estimate (<ref>).
Thus using Lemmas <ref>,<ref>,<ref>
and Remark <ref>,
we obtain




    Σ_1
           ≤𝒪(1)  ∑_i=1^m || 𝒫_t_i^ν(U^μ_0)
            -𝒫_t_i-t_i-1^μ𝒫_t_i-1^ν(U^μ_0)||_L^1
             + 𝒪(1) ∑_i=1^m (t-t_i)μ
       ≤𝒪(1)  ∑_i=1^m || 𝒫_t_i-t_i-1^ν𝒫_t_i-1^ν(U^μ_0)
            -𝒫_t_i-t_i-1^μ𝒫_t_i-1^ν(U^μ_0)||_L^1
               + 𝒪(1)  ∑_i=1^m (1+ ν + t_i   +||U^μ_0 - U_∞||_L^1) ν
             +𝒪(1) ∑_i=1^m (t-t_i)μ
       ≤𝒪(1)  ∑_i=1^m (1+ ||𝒫_t_i-1^ν(U^μ_0) - U_∞||_L^1 )(t_i-t_i-1)^2 
               + 𝒪(1)  (1+ ν + t  +||U^μ_0 - U_∞||_L^1)m ν
             +𝒪(1) tmμ
       ≤𝒪(1)  ∑_i=1^m (1+ ||𝒫_t_i-1^ν(U^μ_0) -𝒫_t_i-1^ν(U_∞)||_L^1+
                ||𝒮_t_i-1^ν(U_∞)- U_∞||_L^1 )(Δ t)^2 
               + 𝒪(1)  (1+ ν + t +||U^μ_0 - U_∞||_L^1  )m ν
             +𝒪(1) tmμ
       ≤𝒪(1)  ∑_i=1^m (1+ ν t_i-1 +t_i-1+
                ||U^μ_0- U_∞||_L^1 )(Δ t)^2 
               + 𝒪(1)  (1+ ν +  t +||U^μ_0 - U_∞||_L^1  )m ν
             +𝒪(1) tmμ
       ≤𝒪(1)   (1+ μ + ν t +t+
                ||U_0- U_∞||_L^1 )tΔ t 
               + 𝒪(1)  (1+μ+ ν + t   +||U_0 - U_∞||_L^1)m ν
             +𝒪(1) tmμ ,



    Σ_2
           ≤𝒪(1)∑_i=1^m  (1+μ+ t-t_i-1 + ||𝒫_t_i-1^ν(U^μ_0) -
           𝒫_t_i-1^ν(U_∞)||_L^1
                                 + ||𝒮_t_i-1^ν(U_∞)- U_∞||_L^1)μ
       ≤𝒪(1) ∑_i=1^m (1+μ+ t-t_i-1 + t_i-1+ ν t_i-1  +||U^μ_0 - U_∞||_L^1)μ
       ≤𝒪(1)  (1+μ  + t + ν t  +||U_0 - U_∞||_L^1)m μ .



Estimates (<ref>)-(<ref>) yield

    ||𝒫_t^μ (U^μ_0)-𝒫_t^ν (U^μ_0)||_L^1
    ≤   𝒪(1)   (1+ μ +ν +t+ ν  t+  ||U_0- U_∞||_L^1 ){tΔ t +m (μ+ν)}.




As a result,

    ||𝒫_t^μ (U^μ_0)-𝒫_t^ν (U^ν_0)||_L^1
    ≤    ||𝒫_t^μ (U^μ_0)-𝒫_t^ν (U^μ_0)||_L^1
            + ||𝒫_t^ν (U^μ_0)-𝒫_t^ν (U^ν_0)||_L^1
    ≤   𝒪(1)    (1+ μ +ν +t+ ν t+  ||U_0- U_∞||_L^1 ){tΔ t +m (μ+ν)}
             + 𝒪(1) || U^μ_0 -U^ν_0||_L^1 + 𝒪(1)tν
    ≤   𝒪(1)    (1+ μ +ν +t+ ν t+  ||U_0- U_∞||_L^1 ){tΔ t +m (μ+ν)} .


Passing μ, ν→ 0, we have

    lim_μ, ν→ 0 ||𝒫_t^μ (U^μ_0)-𝒫_t^ν (U^ν_0)||_L^1≤𝒪(1)   (1  +t+  ||U_0- U_∞||_L^1 )tΔ t ,

which implies (<ref>) due to the arbitrariness of Δ t.
□



In the light of previous preparation, we continue to prove Theorem 1.






Proof of Theorem 1

Assume that ϵ is small enough, and U_0 (x) ∈𝒟_0.
Then according to Lemma <ref>, every sequence of ε-approximate
solution 𝒫_t^ε (U_0^ε) to Cauchy problem (<ref>)
(<ref>) converges to
a unique limit solution U(t,x) ≐𝒫_t (U_0).
It was proved in <cit.> U(t,x) is an entropy solution to problem
(<ref>)(<ref>).
Thus it suffices to verify that  the mapping 𝒫 : [0,+∞)×𝒟_∞→𝒟_∞
is a Lipschitz semigroup.


Semigroup property (<ref>) is deduced from uniqueness and
Lemmas <ref> and <ref>.
Now we only check the Lipschitz continuity of 𝒫.
At every time t_k =k ε, one see that

    ||𝒫_t_k^ε(U_0^ε) - 𝒫_t_k-^ε(U_0^ε)||_L^1   
      = 𝒪(1) ∫_-∞^+∞ Y(t_k -,x) ϕ(T(t_k-,x))ε dx  
       ≤𝒪(1) e^-ϕ kεε∫_-∞^+∞
       Y(0,x)dx

according to definition (<ref>) and decay rate in (<ref>).
By finite propagation speed of waves, we assert that

    ||𝒫_t'^ε(U_0^ε) - 𝒫_s'^ε(U_0^ε)||_L^1≤𝒪(1)
     T.V.(𝒫_t_k-1^ε(U_0^ε)) ·|t'-s'|

for any t',s' ∈ [t_k-1, t_k).
Suppose  t>s≥ 0. Choose ε small enough, then have

    ||𝒫_t^ε(U_0^ε) - 𝒫_s^ε(U_0^ε)||_L^1   ≤𝒪(1) (t-s+  ∑_s≤ kε≤ t e^-ϕ kεε) 
       ≤𝒪(1) (t-s+ ∫_s^t e^-ϕτ dτ) 
        = 𝒪(1)  |t-s|.


As ε→ 0, there holds

    ||𝒫_t (U_0)- 𝒫_s (U_0)||_L^1≤𝒪(1)  |t-s|.

Remember that Lemma <ref> implies for any U_0, V_0 ∈𝒟_0,

    ||𝒫_s (U_0)- 𝒫_s (V_0)||_L^1≤𝒪(1) ||U_0- V_0||_L^1

if passing to limits. Finally preceding two inequalities yield (<ref>).
□




§ LOCAL CHARACTERISTICS OF ENTROPY WEAK SOLUTIONS



This section is concerned with the local geometric features of entropy solutions.
U_0^ε is as usual  regarded as the ε-approximation of initial data U_0∈𝒟_0.
Based on Theorem 1 in this paper and the convergence of
𝒮_t^ε in <cit.>,
we claim that in L^1-topology sense,

    U_0^ε   → U_0,  
    𝒫_t^ε (U_0^ε)     →𝒫_t (U_0) , 
    𝒮_t^ε𝒯_t(U_0^ε)    →𝒮_t𝒯_t(U_0), 
    𝒯_t𝒮_t^ε (U_0^ε)   →𝒯_t𝒮_t (U_0),

as ε→ 0.
 Hence by Lemmas <ref> and <ref>, we draw the following conclusion.



Suppose U(s,x)=𝒫_s (U_0) with U_0∈𝒟_0. Then

    ||𝒫_θ (U)-𝒮_θ𝒯_θ(U)||_L^1≤𝒪(1)(1+||U-U_∞||_L^1) θ^2 , 
        ||𝒫_θ (U)-𝒯_θ𝒮_θ(U)||_L^1≤𝒪(1)(1+||U-U_∞||_L^1) θ^2, 
        ||𝒮_θ𝒯_θ(U)-𝒯_θ𝒮_θ(U)||_L^1≤𝒪(1) θ^2

hold for  θ∈ (0, t̅).



We will utilize this Lemma to analyze the relations between balance laws (<ref>)
and conservation laws (resp. transport system) .
Now let us focus on a small neighborhood of point (s,ξ). Introduce Riemann problem of conservation laws,

    w_t+ F(w)_x =0, 
        w(0,x) = {    U(s, ξ+)   if x<ξ, 
        U(s, ξ-)   if x>ξ.
                    .



and Cauchy problem of transport equations.

    w_t+ dF(U(s, ξ))w_x = G(U(s,x)), 
        w(0,x) = U(s,x).



Then regard U_ξ^C as the entropy solution to (<ref>),
and U_ξ^T as the solution to (<ref>).
One can figure out that

    U_ξ^T (t,x)= U_ξ^1 + U_ξ^2, 
        U_ξ^1 (t,x)= ∑_i=1^3  (l_i^ξ· U(s,x - λ_i^ξ t))  r_i^ξ,  
        U_ξ^2 (t,x)= ∑_i=1^3  ∫_0^t (l_i^ξ· G(U(s,x - λ_i^ξτ)) r_i^ξ dτ,

where l_i^ξ (resp. r_i^ξ) denotes the left (resp. right) eigenvector of dF(U(s, ξ)).
Now we proceed to prove Theorem 2.






Proof of Theorem 2

Suppose U(t,x)= 𝒫_t (U_0) with U_0 ∈𝒟_0. It is an entropy solution to (<ref>)(<ref>).
Fix s≥0 and ξ∈ℝ.
Then we compute the limit (<ref>).
Observe that

    1/θ∫_ξ-θλ̂^ξ+θλ̂
        ||U(s+θ, x)-U_ξ^C (θ, x)||dx  
    ≤   1/θ∫_ξ-θλ̂^ξ+θλ̂
                ||𝒫_θ (U(s, x))-S_θ (U(s, x)) - θ G(U(s, x))||dx 
         + 1/θ∫_ξ-θλ̂^ξ+θλ̂
                ||S_θ ( U(s, x)) - U_ξ^C (θ, x)||dx   +
                1/θ∫_ξ-θλ̂^ξ+θλ̂
                ||θ G(U(s, x))||dx  
    ≤   1/θ ||𝒫_θ (U(s, x))-S_θ (U(s, x)) - θ G(U(s, x))||_L^1
         + 1/θ∫_ξ-θλ̂^ξ+θλ̂
                ||S_θ ( U(s, x)) - U_ξ^C (θ, x)||dx   +
                ∫_ξ-θλ̂^ξ+θλ̂ ||G(U(s, x))||dx  .



Since

    ||𝒫_θ (U)-𝒯_t 𝒮_θ(U)||_L^1≤𝒪(1)(1+||U-U_∞||_L^1) θ^2

for  θ∈ (0, t̅), we have

    lim_θ→ 0^+1/θ ||𝒫_θ (U(s, x))-S_θ (U(s, x)) - θ G(U(s, x))||_L^1
    
    =    lim_θ→ 0^+1/θ ||𝒫_θ (U(s, x))-𝒯_θ𝒮_ θ(U(s, x)) ||_L^1
    
    =    0




Additionally

    lim_θ→ 0^+1/θ∫_ξ-θλ̂^ξ+θλ̂
     ||S_θ ( U(s, x)) - U_ξ^C (θ, x)||dx =0

is proved by Bressan <cit.>(see also <cit.>);

    lim_θ→ 0^+∫_ξ-θλ̂^ξ+θλ̂ ||G(U(s, x))||dx  =0

is deduced from absolute continuity of Lebesgue integral.
Therefore conclusion (<ref>) holds.


Next we calculate the limit (<ref>). For every a<ξ<b,

    1/θ∫_a+θλ̂^b-θλ̂
            ||U(s+θ,x)-U_ξ^T(θ,x)||dx    
    ≤   1/θ∫_a+θλ̂^b-θλ̂
            ||𝒫_θ (U(s))(x)-𝒮_θ (U(s, x)) - θ G(U(s, x))||dx  
       + 1/θ∫_a+θλ̂^b-θλ̂
            ||𝒮_θ (U(s, x)) - U_ξ^1 (θ,x)||dx  
       + 1/θ∫_a+θλ̂^b-θλ̂
            ||θ G(U(s, x)) - U_ξ^2 (θ,x)||dx  
    ≤   1/θ
            ||𝒫_θ (U(s))(x)-𝒯_θ𝒮_θ (U(s, x))||_L^1  + 𝒪(1) {T.V.(U(s); (a,b))}^2  
       + 1/θ∫_a+θλ̂^b-θλ̂
            ||θ G(U(s, x)) - U_ξ^2 (θ,x)||dx.

The first term  of the sum above tends to 0 as θ→ 0^+.
The second term was derived by <cit.> etc..
So it suffices to compute the third term .
In fact

    1/θ∫_a+θλ̂^b-θλ̂
            ||θ G(U(s, x)) - U_ξ^2 (θ,x)||dx    
    ≤   1/θ∫_a+θλ̂^b-θλ̂∫_0^θ∑_i=1^3 l_i^ξ· G(U(s,x )) r_i^ξ dτ -
            ∫_0^θ∑_i=1^3 l_i^ξ· G(U(s,x - λ_i^ξτ)) r_i^ξ dτdx  
    ≤   𝒪(1) 1/θ∑_i=1^3
            ∫_0^θ∫_a+θλ̂^b-θλ̂   G(U(s,x ))  -   G(U(s,x - λ_i^ξτ)) dx   dτ
    ≤   𝒪(1) 1/θ∑_i=1^3
            ∫_0^θ  T.V. (G(U(s));(a,b)) ·λ_i^ξτ  dτ
    ≤   𝒪(1) θ· T.V. (G(U(s));(a,b)).



Consequently we have

    lim_θ→ 0^+1/θ∫_a+θλ̂^b-θλ̂
            ||θ G(U(s, x)) - U_ξ^2 (θ,x)||dx =0.

Then (<ref>) holds for some constant C_0.




Conversely, suppose mapping U: [0,+∞)
→𝒟_∞  is  Lipschitz continuous in L^1-topology and satisfies
(<ref>)(<ref>) for s ∈ [0,+∞)∖𝒩 and ξ∈ℝ,
where 𝒩 is a set  with measure 0. Given s ∈ [0,+∞)∖𝒩 and ε>0,
we can cover the interval [a,b] with finitely many  intervals [a_i,b_i] such that

  
  * ⋃_i [a_i,b_i]=[a,b]  and (a_i,b_i)⋂(a_j,b_j)=∅ with i≠ j;
  
  * T.V.(U(s);(a_i,b_i))<ε ;
  
  * lim_θ→ 0^+1/θ∫_a_i+θλ̂^b_i-θλ̂
        ||U(s+θ,x)-U_ζ_i^T(θ,x)||dx  ≤
            C_0 {T.V.(U(s);(a_i,b_i))}^2
        where ζ_i ≐ (a_i+b_i)/2.




Set ξ_i=a_i. Then construct a function

    U^*(t,x)≐{    U_ξ_i^C(t,x)           if  |x-ξ_i|≤ tλ̂ , 
        U_ζ_i^T(t,x)           if  a_i+tλ̂<x<b_i-tλ̂ ,
            .

for t small enough. Hence

    lim_θ→ 0^+1/θ∫_a+θλ̂^b-θλ̂
    ||U(s+θ)(x)-U^*(θ,x)||dx  
    ≤   ∑_i lim_θ→ 0^+1/θ∫_ξ_i-θλ̂^ξ_i+θλ̂
            ||U(s+θ)(x)-U_ξ_i^C(θ,x)||dx  
         + ∑_i lim_θ→ 0^+1/θ∫_a_i+θλ̂^b_i-θλ̂
            ||U(s+θ)(x)-U_ζ_i^T(θ,x)||dx   
    ≤   ∑_i
             C_0 {T.V.(U(s);(a_i,b_i))}^2        
    ≤     C_0  T.V.(U(s);(a,b)) ε .


Similarly we have

    lim_θ→ 0^+1/θ∫_a+θλ̂^b-θλ̂
    ||𝒫_s+θ (U_0)-U^*(θ,x)||dx
    ≤    C_0 T.V.(𝒫_s (U_0);(a,b))ε   .

Therefore,

    lim_θ→ 0^+1/θ∫_a+θλ̂^b-θλ̂
    ||U(s+θ)(x) - 𝒫_s+θ (U_0)||dx  
    ≤    C_0  {T.V.(U(s);(a,b)) +  T.V.(𝒫_s (U_0);(a,b))}ε.

We find U(s )= 𝒫_s (U_0) according to the arbitrariness of (a,b) and ε in the last estimate.
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