ON THE STRUCTURE Of IONIZING SHOCK WAVES IN MAGNETOFLUIDDYNAMICS

Ionizing shock waves in magnetofluiddynamics occur when the coefficient of electrical conductivity is very small ahead of the shock and very large behind it. For planner motion of plasma, the structure of such shock waves are stated in terms of a system of fourdimensional equations. In this paper, we show that for the above electrical conductivity as well as for limiting cases, that is, when this coefficient is zero ahead of the shock and/or is infinity behind it, ionizing fast, slow, switch-on and switch-off shocks admit structure. This means that physically these shocks occur.


1.
Introduction.An ionizing shock is defined as a compressive wave which propagates into a nonionized, nonconducting gas, ionizes it, and thus makes the post-shock gas electrically conducting and capable of interacting with an electromagnetic field.Thus in this type of shock wave the pre-shock of the gas is nonconducting and the post-shock state is ionized and a good electrical conductor.Thus, this type of shock wave is considered as a system of magnetohydrodynamics (MHD) or magnetofluiddynamics (MFD).
From the mathematical point of view, shock waves are discontinuous weak solutions of conservation laws.In order to distinguish physical shock wave solutions of conservation laws, among many of them, one has to apply some criteria.The most widely accepted one is the structure or viscous profile criterion [1,19,24,28].
The question of existence of structure for different types of MFD shock waves in planar motion has been considered by Germain [10], Kulikovskiȋ and Lyubimov [19], Cabannes [1], Conley and Smoller [4,5], and Mischaikow and Hatori [20].According to their work, the shock layer equations in MFD is stated in terms of the following four-dimensional system of ordinary differential equations, which is taken from [1] (1.1) Here µ > 0 is an electrical constant, (u, v) is the velocity vector of the fluid, ε is the internal energy, p and T are the pressure and the temperature, respectively.The vector (H x ,H y ) is the magnetic field in the xy-plane, and E z is the electric field in the direction of z-axis, where H x and E z are nonnegative constants.This system of equations contains four dissipation coefficients; the two coefficients of viscosities λ 1 and µ 1 , the coefficient of thermal conductivity λ and the coefficient of electrical conductivity σ .These coefficients are nonnegative functions of absolute temperature T .Finally, m, P , P 1 , and C denote constants of integration.For more details and derivation of these equations the reader is referred to [1].Now the structure problem for MFD shocks is the above four simultaneous firstorder nonlinear differential equations must be integrated between equilibrium points.This problem has been studied before by many authors, when the dissipation coefficients are continuous functions of T [4,5,7,10,11,13,14,20].We will describe these works in more details in Section 2. However, in the case of ionizing shock, the electrical conductivity of the gas is assumed to be zero (or very small) in the pre-shock gas and it continues to zero (or very small) until a value T is reached by the temperature.At this point in the shock structure the electrical conductivity jumps to infinity (or a high value) which remains the same through the remainder of the shock wave.The analogy with the ignition temperature in flame and detonation problems is evident [9,12,26,27,29].In other words, we have where 0 ≤ σ 1 (T ) 1 σ 2 (T ) ≤ ∞, and temperature T is given and is assumed to have a value between its upstream and downstream value.We may call T the ionizing temperature [1,2,12,18,25].
The structure problem for ionizing shock wave for the case σ 1 = 0 and σ 2 > 0 has been studied by Kulikovskiȋ and Lyubimov, when the gas is perfect and H x = µ 1 = λ 1 = 0 or H x = µ 1 = λ = 0 [18].Also this problem for perfect gas and σ 1 = 0, σ 2 > 0 is studied by Chu, when H x = λ = 0 (see [2]).This leaves open the question of the existence of structure of ionizing shock waves in planar motion when H x ≠ 0 and none of the viscosity parameters is zero.This general case is studied in this paper.
This paper is organized as follows, in Section 2 we make some observations related to the rest (equilibrium) points of system (1.1) and introduce the problem in detail.In Section 3, we find some general results on the orbits of autonomous system of ordinary differential equations related to the problem.In Section 4, we show the existence of structures when σ 1 (T ) ≡ 0 and σ 2 (T ) ≡ ∞.The existence of structures in the case σ 1 (T ) ≡ 0 and σ 2 (T ) is very high, will be considered in Section 5.In Section 6, we consider the problem for the case 0 < σ 1 (T ) 1 σ 2 (T ) < ∞.For an excellent description, experimental, and applications of ionizing shocks the reader is referred to [20,Chapter 4] and [22,Section 5.15].

Hypotheses, rest points, and the problem.
As we pointed out before, a heteroclinic orbit of system (1.1) is called a structure for an MFD shock wave.In other words, a structure for an MFD shock wave is a complete orbit of (1.1) connecting two rest points.Thus in the first step we must know the rest points of (1.1).In order to take advantage of some results from previous works in [4,5,13], we replace λ 1 + 2µ 1 , x, H y , P , µ 1 , v −P 1 , H x , λ, ε, C, E z , and σ −1 by µ 1 , t, x 2 , J, µ, x 1 , δ, k, e, E, ε, and ν.Also, without loss of generality, we may assume µ = m = 1, as m is the mass quantity and µ is an electrical quantity.Then from u = mV [1], we obtain u = v, where V = ρ −1 and ρ is the density of the fluid.In this way, (1.1) can be written as where u = (V , x 1 ,T ,x 2 ) T and µ 1 ,µ,k, and ν are functions of T .Notice that for J ≤ 0, this system has no rest point.Hence we assume that J > 0. Also note that δ ≥ 0 and ε ≥ 0. Let S(V , T ) be the entropy of the system.Following the previous works in [4,5,11,13,14,15,16,21], we consider a general form for thermodynamic state functions (instead of giving a specific expression) and we assume that the functions p(V , T ), e(V , T ), and S(V , T ) satisfy the following hypotheses.
(H 4 ) On any interval 0 < V ≤ V 0 , S(V , T ) → 0 uniformly in V as T → 0. (H 5 ) If we consider p as a function of V and S, then p v < 0, p vv > 0 and p s > 0. As an alternative to H 4 , we could assume (H 4 ) The quantities S V (= p T ) and S T are positive whenever V ,T > 0, and for any fixed V , S(V , T ) converges to a limit independent of V as T → 0.
We will not use all of these hypotheses directly, but we will take advantage of some results based on them from previous works in [4,5,13], specially we will assume the existence of the rest points which is based on the above hypotheses as follows.
Let 0 ≤ µ, µ 1 ,k,ν < ∞.For fixed J > 0, δ > 0, and ε > 0, there are two numbers E 0 ≥ E 1 such that for E > E 0 system (2.1) admits no rest point at all.For E < E 1 it admits precisely four rest points, two of these rest points are located in the region V > δ 2 , and the other two are located in the region 0 < V < δ 2 .For E 1 < E < E 0 , this system admits two rest points, and either both of them lie in V < δ 2 or both of them lie in V > δ 2 [4,5,11,13,20].Hereafter, we assume that system (2.1) admits four rest points.We denote them by 0 ≤ i ≤ 3, which are ordered by increasing density.Here (V i ,T i ), for 0 and and δ > 0, then for negative large value of E system (2.1) admits four rest points (2.4) Again these rest points are ordered by increasing of density.For i = 0, 3, ūi = ( Vi , 0, Ti , 0), where ( Vi , Ti ) is a solution of (2.2) corresponding to ε = 0. Moreover for i = 1, 2, we have ūi = (δ 2 ,δx 2i , Ti , x2i ) T , where ( Ti , x2i ), i = 1, 2, is a solution of the system of equations, Finally, if δ = 0, then u 2 and u 3 do not exist, and u 0 and u 1 may exist.For more details about the rest points, the reader is referred to [13].
As we mentioned before, the existence of a heteroclinic orbit u i → u j corresponds to the existence of structure for the shock wave between the two states u i and u j .This means that, this shock occurs physically.It is shown that if such a heteroclinic orbit exists, then i < j.In the following, we explain the concept of different shock waves which may occur.
Shock wave between u 0 and u 1 (u 2 and u 3 ) is called fast (resp., slow) shock.Physically, this means upstream and downstream states of the shock is super-Alfvénic (resp., sub-Alfvénic).Shock wave between u i and u j , i = 0, 1 and j = 2, 3, is called intermediate shock.The downstream state of this shock is sub-Alfvénic and its upstream is super-Alfvénic.
Shock wave between ū0 and ū1 (as well as ū0 and ū2 ) is called switch-on shock, and shock wave between ū2 and ū3 (as well as ū1 and ū3 ) is called switch-off shock.This means that ū0 is super-Alfvénic, ū1 and ū2 are Alfvénic and ū3 is sub-Alfvénic [19,28].
It is known that for bounded and continuous functions of µ 1 (T ), µ(T ), k(T ), and ν(T ) fast, slow, switch-on, and switch-off shocks always exist [4,5,9,10,12].However, existence of intermediate shocks depend on the values of the above four viscosities [5,13,21].In the present paper, we are concerned with the existence of the ionizing shocks of the above shocks except intermediate shocks, which may be considered in a future work.

Some theorems in ODE.
In this section, we present some existence theorems which will be used as main tools in the next section.

Consider the autonomous system of ODEs
on R n , where f is smooth.We denote by x • t the value of the solution of (3.1) at time t which is x initially.Thus this solution is uniquely defined on an open interval of t containing origin and assumed to be maximal.The set S ⊂ R n is called invariant with respect to (3.1) if x • t ∈ S, for all t ∈ R and x ∈ S. By an orbit we mean a solution of (3.1), and by a complete orbit we mean an orbit which is defined for all values of t ∈ R. We say that the orbit γ(t) is running from x 0 (or running to x 1 ) if γ(t) is defined for t ≤ 0 (or t ≥ 0) and lim t→−∞ γ(t) = x 0 (or lim t→∞ γ(t) = x 1 ).If γ(t) is running from x 0 to x 1 and x 1 ≠ x 0 , then this orbit is called a heteroclinic orbit.System (3.1) is called gradient-like in the open set U ⊂ R n , if there is a continuous real-valued function h on U which is strictly increasing on each nonconstant solution of (3.1), lying in U , [3,6].
The ω-limit (α-limit) set of the orbit x • t is the set of limit points of sequences x • t n , where t n goes to +∞(−∞).If x • t is a complete bounded orbit, then its ω-limit set and α-limit set are nonempty, closed, connected, and invariant.In the case of a gradient-like system, the restriction of h (the gradient-like function) to each of this sets is constant.Therefore each of them consists of rest points [3,24].
It is known that, the ω-limit set and α-limit set are nonempty and connected if x •t is bounded.In the case of gradient-like flows, the restriction of h to any of these sets is constant.Therefore each of these sets consists of a rest point.For more details the reader is referred to [3,6,22].
The (C 1 ) The set {x ∈ D : The single point {x ∈ D : h(x) = 1} which is denoted by x, is a rest point, and this is the only rest point of (3.1) From continuous dependence of the solution with respect to initial conditions and uniqueness of solution it follows that ψ is a homomorphism from Ē to F \{ x}.This is impossible as Ē is closed and F \{ x} is not closed.Thus Ẽ ≠ ∅.Let x ∈ Ẽ.Then x •t is defined for all t > 0 and is lying in D. Since system (3.1) is gradient-like, the ω-limit set of x • t must be { x}.This completes the proof.
For more details of the above proof the reader is referred to the proof of [13, Theorem 2.1.1].Theorem 3.2.Suppose that f and D are the same as in Theorem 3.1, and system (3.1) is gradient-like with respect to a real-valued function g in D.Moreover, the following conditions hold.
(C 1 ) The set {x ∈ D : The set {x ∈ D : g(x) = 0} which consists of a single point, say x, is a rest point of (3.1), and x is the only rest point of (3.1) Proof.Since f is Lipschitz on D and the flow cannot leave D as t decreases, p • t must be defined for all p ∈ ∂D\ F and t < 0 and lying in D.
As a modification of [13, Theorem 2.1.2]we have the following theorem.Then there is Proof.We claim that the set The next theorem can be considered as an extension of the continuous dependence of solution of ordinary differential equations on parameters.Theorem 3.4.Let {f m } be a sequence of vector fields in R n , and let D ⊂ R n be a bounded domain.Suppose for each m, f m ∈ C 1 ( D), and there is a constant which is defined on the interval (a, b) and lies in D for all m.Then {γ m (t)} has a uniformly convergent subsequence on compact subsets of (a, b), which converges to a continuous of bounded variations function, say γ(t).Moreover, suppose there is a t 0 ∈ (a, b) and a C 1 vector field, say f , on a neighborhood of γ(t 0 ) such that f m converges to f uniformly, as m → ∞, in this neighborhood.Then γ(t) is a solution of the initial-value problem on some neighborhood of t 0 .
Proof.First of all notice that by assumptions {f m } and {γ m (t)} are uniformly bounded.It then follows that {γ m (t)} is uniformly of bounded variations on (a, b).On the other hand, for a < t 1 < t 2 < b, we have By Arzela-Ascoli theorem there is a subsequence of this sequence which is convergent uniformly on [α, β], to a continuous function, say γ(t).Hence we may assume γ(t) is defined on (a, b) and {γ m (t)} converges uniformly to γ(t) on each compact subset of (a, b).Moreover, by Helly's first theorem [17], γ(t) is of bounded variations too.Now, suppose for large values of m, in a neighborhood of γ(t 0 ), f m converges to f uniformly.Since f has continuous first derivatives on the closure of this neighborhood, we may assume that f is uniformly Lipschitz with Lipschitz constant λ > 0, in this neighborhood.Since {γ m (t)} converges uniformly on compact subsets of (a, b), there is ε > 0 such that γ m (t) lies in the above neighborhood for t ∈ [t 0 −ε, t 0 +ε] and all m.Now for t ∈ [t 0 ,t 0 + ε) and n, m, we have where " " means d/dt.Therefore by Gronwall's inequality [6], we must have Thus γ(t) is the solution of (3.3) in a neighborhood of t 0 .Now, in this section, we have the following theorem which can be considered as a continuous dependence of solutions to the parameter for singular situations.
For x ∈ R n and y ∈ R n , we consider the following system of equations in where "• = d/dt," as before, f : R n+m → R n and g : R n+m → R m are continuous functions and ε ∈ R is a parameter.
Theorem 3.5.Let D 1 ⊂ R n and D 2 ⊂ R m be bounded domains and f and g on D 1 × D 2 have continuous second derivatives and (x 0 ,y 0 ) ∈ D 1 × D 2 be hyperbolic rest point of system (3.9).Moreover, assume that the following conditions hold.
(C 1 ) g(x, y) ≡ 0 if and only if y = G(x), where G : D 1 → R m has continuous second derivatives.
(C 2 ) There exist nonnegative constant integers k s and k u , with k s + k u = m, such that for all x ∈ D 1 the m×m matrix ∂g(x, G(x))/∂y has k s eigenvalues with negative real part and k u eigenvalues with positive real part, uniformly bounded away from zero for all x ∈ D 1 .
(C 3 ) For ε = 0, there is an orbit of system (3.9) say γ 0 (t) which is running to the rest point (x 0 ,y 0 ) and intersects a hypersurface, say Q, at the point γ 0 (t 0 ), transversely.
Thus γ 0 (t) is lying on this manifold.For each q ∈ Γ s (p), let F s (q) be the stable manifold of system ẋ = 0, ẏ = g(x, y), (3.10) at the rest point q of this system.Now define By [8, Theorem 12.2(ii)], W s (p) perturbes smoothly to the stable manifold W s ε (p) of the hyperbolic rest point p of system (3.9) for small ε.This means that for γ 0 (t 0 ) ∈ W s (p) and given δ > 0 there is an ε 0 > 0 such that for every ε ∈ (−ε 0 ,ε 0 ) there is a is the solution of (3.9) corresponding to this ε initiating at p ε , then γ ε (t) is running to p, moreover, for all t ≥ t 0 .Thus for δ small, γ ε (t) intersects Q, transversely.

4.
Existence of structure when σ = 0 for T ≤ T and σ = ∞ for T > T .In this section, we discuss the existence of structure for fast, slow, switch-on, and switch-off shocks when the coefficient of electrical conductivity is zero ahead of the shock and very large behind it.That is in this case we have Thus for T ≤ T , ẋ2 ≡ 0 which means that x 2 is constant.Let x2 be the fourth component of u at the downstream state.Then at this state from 2 , then this system of algebraic equations is the same as [13, If the point ( Ṽ , T ) exists as above, then downstream state of the shock is for some 0 < T < T .Notice that also for 0 < T < T , system (2.1) reduces to the following system: where û = (V , x 1 ,T ).
. If we substitute this value of x 2 into the other equations of system (2.1), then this system reduces to the system where û = (V , x 1 ,T ).
Let T i be the third component of u i , 0 ≤ i ≤ 3, which is given by (2.2).If T 0 < T < T 1 , then consider systems (4.5) and (4.6) in the region V > δ 2 and the downstream and upstream states of the shock are ũ0 = ( Ṽ ,δx 2 , T , x2 ) and u 1 = (V 1 ,x 11 ,T 1 ,x 21 ), respectively.This case corresponds to the fast ionizing shock.For T 2 < T < T 3 , these systems must be considered in 0 < V < δ 2 .In this case the downstream and upstream states of the shocks are ũ2 = ( Ṽ ,δx 2 , T , x2 ), and u 3 = (V 3 ,x 13 ,T 3 ,x 23 ), respectively.In this case the shock between ũ2 and u 3 corresponds to slow shock.In the following we prove the existence of structures.

Fast shock.
In order to prove the existence of structure for fast shock we define Now, if we differentiate G i0 ( û), 1 ≤ i ≤ 3, along the orbits of system (4.6)we obtain where in the last equality we used the identity e V + p = T S V .In D f we have δ 2 < V.
It follows, from G 20 ( û) < 0, that δx 1 − ε < 0 in D f .Then from (4.8) we see that every orbit of system (4.6) is oriented in such a way that as t increases, it must go out of D f when crossing the boundary of D f on G i0 ( û) ≡ 0, at a point different from the rest point û1 = (V 1 ,x 11 ,T 1 ).Also system (4.6) is gradient-like with respect to h( û) = T , and the hypersurface T = constant, intersects D f on a set homomorphic to the unit disk.Finally, from we see that every orbit of system (4.6)starting at a point on { û ∈ ∂D f : T = T }, it must get into D f as t increases.Therefore, system (4.6)together with D f , the rest point û1 = (V 1 ,x 11 ,T 1 ), and the gradient-like function h( û) = T satisfy all the conditions of Theorem 3.1.Hence, by that theorem, there must be a point ûf on { û ∈ ∂D f : T = T } such that the orbit of system (4.6), initiating at this point, is lying in D f and is running to û1 = (V 1 ,x 11 ,T 1 ).We denote this orbit by û(t) = ( Ṽ (t), x1 (t), T (t)), t ∈ [0, ∞).Then û(0) = ûf and lim t→∞ û(t) = û1 .Now consider system (4.5) with x2 as x2f 2 follow F1 ( Ṽ (0), T ) < 0 and F2 ( Ṽ (0), T ) > 0. Thus from Lemma 4.1, this system admits the unique rest point say û0f = ( Ṽf , x1f , Tf ) in T < T , where x1f = δ x2f .This rest point is located on the boundary of the set along the orbits of system (4.5)yield where in the third equality we used the identity e V + p = T S V from thermodynamics.Thus the surface G 2∞ ( û) = 0 is invariant and the flow of system (4.5)goes out of D f on ∂D f \{ û : G 2∞ ( û) = 0}.Since this system is gradient-like with respect to h( û) = T , thus every orbit of this system initiating at a point on ∂D f ∩{ û : T = T } must be lying in D f for t < 0 and is running to the rest point ûof as t tends to minus infinity.Let ū(t) be the orbit initiating at the point ûf , which is defined for t ≤ 0.

Now define
where V (t), 0), and V (t), T (t), x1 (t) are the components of ū(t).Finally, notice that the first three components of u f has continuous first derivative at t = 0. Therefore, we have proved the following theorem.
Theorem 4.2.Suppose that system (2.1) admits the rest points u 0 and u 1 and T 0 < T < T 1 , where T 0 and T 1 are the temperatures at u 0 and u 1 , respectively.Moreover, assume that µ 1 (T ), µ(T ), and k(T ) are smooth, bounded and bounded away from zero, and ν(T ) is given by (4.1).Then there is a rest point of (2.1), say û0 in T < T and a complete orbit of this system corresponding to fast shock which is running from u 0f = ( Ṽf , x1 , Tf , x2f ) to u 1 and the components of this orbit are monotone.

Slow shock.
Suppose that the rest points u i = (V i ,x 1i ,T i ,x 2i ), i = 2, 3, exist and in (4.1) we have T 2 < T < T 3 .In order to show the existence of heteroclinic orbit corresponds to slow shock, we define First of all notice that in D s we have δx 1 − ε > 0.
From the second equality in (4.8) it follows that the flow gets into D s on G 20 ( û) = 0. Also from third equality of (4.8) we see that the flow goes out of D s on G 30 ( û) = 0. Finally the first equality of (4.8) implies that the flow goes out of D s on { û ∈ ∂D s : where "• = d/dt" as before.
Here we wish to show that the flow goes out of D s on In order to do this, it is sufficient to show that the second derivative of In order to see Case 4, notice that for the ideal gas we have p = nRT /V and e = nRT /(γ − 1).Thus p T T = 0 and e T − V p T = nR(2 − γ)/(γ − 1) which is positive for γ ≤ 2. On the other hand, in D s we have Therefore Case 3 implies Case 4. Now, system (4.6)together with D s , the rest point û3 = (V 3 ,x 13 ,T 3 ), and the function h( û) = T satisfy all conditions of Theorem 3.3.Thus there is an orbit of this system initiating at a point ûs ∈ {û ∈ ∂D s : T = T }, lying in D s and is running to û3 .We denote this orbit by û+

Now, consider system (4.5). If we substitute for x2 in this system by x2s =
then similar to the fast case this system admits a unique rest point say û2s = ( Ṽs , x1s , Ts ) in T < T , where x1s = δ x2s and ( Ṽ , T ) is given in (4.4).This rest point and ûs = û+ (0) are located on the boundary of Similar to the case of fast shock, the orbit of system (4.5)corresponds to the initial condition û(0) = ûs = û+ (0) is defined for t < 0, lies in D s and is running from û2s .Let u where 0), as before.Thus we have proved the following theorem.Theorem 4.3.Suppose that system (2.1) admits the rest points u 2 and u 3 and T 2 < T < T 3 , where T i is the temperature at the rest point u i , i = 2, 3, and the viscosity parameters µ 1 (T ), µ(T ), k(T ), and ν(T ) are as in Theorem 4.2.Moreover, one of the above four cases holds.Then the ionizing slow shock admits structure.Along this structure, density, temperature and the vertical component of velocity are increasing, but the vertical component of the magnetic field is nondecreasing.

Switch-on and switch-off shocks.
As we mentioned in Section 2, switch-on and switch-off shocks occur if in system (2.1) we have ε = 0 (i.e., in the absence of electric field).Here we obtain the structure for switch-on and switch-off ionizing shocks as limits of structure for fast and slow shocks, respectively.We do this for switch-on shock, the same arguments work for switch-off too.
Suppose ūi , i = 0, 1, the rest point corresponds to the switch-on shock exist and the ionizing temperature T is between the temperature at ū0 and ū1 .Then by [13,Theorem 3.3.1],u 0 (ε) and u 1 (ε), the rest points corresponding to fast shock exist for ε > 0 and small, moreover, lim ε→0 u i (ε) = ūi .Thus T is between the temperature at u 0 (ε) and u 1 (ε).Now choose ε m → 0. By Theorem 4.2, for each ε m there is a heteroclinic orbit, say γ m (t), which is running from a rest point of system (2.1), say u 0f m to the rest point u 1 (ε m ).Since {u 0f m } is bounded, it contains a convergent subsequence.We may assume that lim m→∞ u 0f m = ū0n .
The sequence {γ m (t)} is uniformly bounded and componentwise monotone.Therefore, by Helly's theorem [17], it has a subsequence which is uniformly convergent on the compact subsets of R. We may assume that γ m (t) converges to a continuous function, say γ 0 (t).Then similar to the fast shock, we can show that γ 0 (t) is componentwise strictly monotone and intersects the surface T = T at a single point, say t 0 .Moreover, similar to the proof of Theorem 3.4 we can show that γ 0 (t) are componentwise differentiable except its last component at t 0 and satisfies system (2.1).Hence we have proved the following theorem.Theorem 4.4.If the rest point ūi , i = 0, 1, exists, then under the same conditions of Theorem 4.2, the ionizing switch-on shock admits structure, which has all of the properties of the ionizing fast shock.Similarly, if ū2 and ū3 exist, then under the same conditions of Theorem 4.3 the structure for switch-off shock exists and has the same properties as those of slow shock.

4.4.
Transverse magnetic field shock.In the case of transverse magnetic field in system (2.1), δ must be considered zero.In this case system (2.1) admits two rest points which are the limiting case of the rest points corresponding to fast shock as δ tends to zero.Thus similar to switch-on shock the structure for the related ionizing shock can be found from the structure for ionizing fast shock as δ tends to zero.Also this structure can be found directly by using the same technique which is used for the ionizing fast shock.

5.
Existence of structure for σ = 0 ahead of the shock and very large behind it.In this section, we show that the ionizing fast, slow, switch-on, switch-off admit structure when the electrical conductivity coefficient is zero ahead of the shock and is very large behind it.Thus in this case we have where ν 2 (T ) as well as µ 1 (T ), µ(T ), and k(T ) are smooth (i.e., C 1 ) positive value with ν 2 (T ) 1.As we said in the introduction, this case has been discussed in [2,18] for the case of transverse magnetic field (i.e., δ = 0) and k(T ) or µ 1 (T ) is assumed to be zero and the gas is taken to be ideal gas.In this section similar to the previous section, we consider the problem for all types of the ionizing shocks as follows.

Fast shocks.
Suppose the rest points u 0 and u 1 exist and T is between the temperature at u 0 and u 1 .Here we define where u = (v, x 1 ,T ,x 2 ) and T 1 is the value of temperature at u 1 .First of all, V > δ 2 , G 2 (u) < 0, and G 4 (u) < 0 imply that x 2 < 0 and x 1 < 0 in D f .Now, if we differentiate G i (u), 0 ≤ i ≤ 4, and T along the orbits of system (2.1) we obtain where in the third equality we used the identity e V = p + T S V which is a result of second law of thermodynamics.Thus the flow goes out of D f on {u ∈ ∂D f : T > T } and gets into D f on {u ∈ ∂D f : T = T }.Moreover, system (2.1) is gradient-like with respect to h(u) = T in D f .Similar to the fast shock in the previous section, it follows from Theorem 3.1, there exists a point, say u f , on {u ∈ ∂D f : T = T } such that the orbit of system (2.1), initiating at this point, is lying in D f and running to u 1 as t tends to ∞ and its components are monotone.We denote this orbit by u + f (t), t ∈ [0, ∞).Let x2 be the value of x 2 at the point u f = u + f (0).Now, consider system (4.5) for this value of x2 .Similar to the fast and slow shocks in Section 4, we can show system (4.5)admits a unique rest point in T < T , say u 0f which is on the surface x 2 = x2 and the orbit of system (4.5)initiating at the point u f is running from u 0f ; lying on the surface x 2 = x2 , and the other its components are strictly monotone.We call this orbit u − f (t).Now define (5.4) This orbit is the structure for ionizing fast shock, in this case.Along this orbit, V , x 1 , and x 2 are decreasing and T is increasing.Hence we have the following theorem.
Theorem 5.1.Let in Theorem 4.2, ν(T ) be given by ( 5.1) and the other assumptions remain the same.Then all of its conclusions are valid.

Slow shock.
The technique we use here to show the existence of structure for slow shock, for ν(T ) is given by (5.1), is different from the previous one.The reason for using a different approach is that Theorems 3.1, 3.2, and 3.3 cannot be applied in this case.The technique we use here is to obtain the structure for ionizing slow shock, corresponding to ν(T ) which is given by (5.1), as a perturbation of the structure of ionizing slow shock which is found in Section 4.
Let T and g(x, y) = G 4 (u), where u = (V , x 1 ,T ,x 2 ) T , ν and G i , 1 ≤ i ≤ 4, are the same as in system (2.1).Then system (2.1) is of the form of system (3.9).We consider this system on the bounded domain (5.5) Thus f and g have continuous second derivative on D 1 × D 2 and u 3 is the only rest point of this system in D 1 × D 2 .It is known that this rest point is hyperbolic [10,11].
For ε ≠ 0 its linearized matrix has two positive eigenvalues and two negative.For ε = 0 it has one positive and two negative eigenvalues.For details about the eigenvalues the reader is referred to [13,Section 2.5].
In order to see conditions (C 1 ) and (C 2 ) of Theorem 3.5 hold, notice that the equation g(x, y) = 0, namely, −δx 1 +V x 2 +ε = 0, implies x 2 = V −1 (δx 1 −ε).Hence condition (C 1 ) holds too.Also the one by one matrix ∂g(x, G(x))/∂y = V has one positive eigenvalue uniformly bounded away from zero.This means that condition (C 2 ) holds.Now, consider the orbit u s (t) which is given by (4.9).For t ≥ 0 this orbit is a solution of (2.1) corresponding to ν = 0.This orbit intersects the surface T = T transversely, runs to u 3 and lies in Thus condition (C 4 ) holds too.Therefore by Theorem 3.5 there is ν 0 > 0 such that for each 0 < ν < ν 0 there exists an orbit of system (2.1) corresponding to ν, intersecting the surface T = T , transversely, running to u 3 , and lying in the region {u ∈ R 4 : G 1 (u) < 0, G 2 (u) < 0, G 3 (u) > 0, T ≤ T }.We denote this orbit by u + νs (t), t ∈ [0, ∞).Along this orbit −V , −x 1 , and T are increasing.Let, in system (4.5),x2 be the value of x 2 at u νs (0).Then the orbit u s (t) which is given by (4.9) is a solution of this system for t < 0, initiating at u νs (0) and is running from a rest point of this system, say u 2s , which exists similar to previous cases since and T are increasing.We call this orbit u − νs (t).Now define which is the structure for ionizing slow shock corresponding to ν(T ) which is given by (5.1).Therefore we have the following theorem.
Theorem 5.2.Let in Theorem 4.3, ν(T ) be given by ( 5.1) and the other assumptions be the same.Then all of its conclusions remain valid, except the x 2 component of the structure orbit may not be monotone.

Switch-on and Switch-off shocks.
In the previous section, by taking advantage from the existence of structure for fast and slow ionizing shocks and componentwise monotonicity of the related orbits, and using Helly's theorem, we were able to prove the existence of structure for switch-on and switch-off ionizing shocks as a limiting of the structure of the above shocks as ε tends to zero.Here in this section we have all of the above situations with one exception for the orbit of the slow shock.This exception is that the x 2 component of this orbit is of bounded variations instead of monotone.In the proof we used monotonicity for applying Helly's theorem.But Helly's theorem works for the class of bounded variations, too [17].Thus we have the following theorem.Theorem 5.3.Let, in Theorem 4.4, ν(T ) be given by (5.1) and the other assumptions remains the same.Then the structure for ionizing switch-on and switch-off exists and has all of the properties the same except that the x 2 -component of the structure for switch-off shock is of bounded variations instead of being monotone.

Transverse magnetic field shock.
In this case, again system (2.1) admits two rest points and by the same argument which is used in the previous section we can show this shock admits structure.Here we should mention that this is the only case which is considered in literature, where at least one of the viscosity parameters, µ 1 , µ or k is assumed to be zero.

6.
Existence of structure for σ very small ahead of the shock and very large behind it.In order to prove the existence of structure for slow as well as fast shock, [4, page 435] showed that the above complete orbits must be lying in the subspace y 1 = y 2 = 0 (y 1 and y 2 are the fifth and sixth variables in their work, the first four variables are the same as ours).Now, if the substitute y 1 = y 2 = 0 in their system (2.4), we obtain our system (6.1).
Another point about Conely and Smoller's work is that in their work they had never assumed that the viscosity parameters are constants, nor they mentioned that the viscosities are functions of T .However, their proofs are organized in such a way that they work even if the viscosities are functions of u, as long as they are smooth, bounded and bounded away from zero.Thus their Theorem 4.1 in [3] implies Theorem 6.3 above.Now we can prove the existence of structure for ionizing shocks as follows.
6.1.Fast shock.Consider system (6.1), the rest points u 0 and u 1 and assume that T 0 < T < T 1 where T i , i = 0, 1, is the temperature at u 0 and u 1 , respectively.
According to Theorem 6.2 for m = 1, 2,..., there is a unique orbit, γ m (t), which is running from u 0 to u 1 and is lying in the bounded domain D f .Thus by Theorem 6.1, {γ m (t)} contains a subsequence which is convergent to a continuous function γ(t).Let γ m (t) = (V m (t), x 1m (t), T m (t), x 2m (t)) and γ(t) = (V (t), x 1 (t), T (t), x 2 (t)).Since H 1 (u) is continuous and bounded on Df , from the first equation of system (6.1) and Lebesgue dominated convergence theorem we have H 1 γ(s) ds.(6.6)This means that for t ∈ R, we have V (t) = H 1 γ(t) . (6.7) Similarly, ẋ1 (t) = H 2 γ(t) , Ṫ (t) = H 3 γ(t) .(6.8) By the proof of [13, Theorem 2.2.2], the flow goes out of D f on H 3 (u) ≡ 0 ≡ G 3 (u).Thus H 3 (γ(t)) > 0 for all t ∈ R. Thus γ(t) intersects the surface {u ∈ R 3 : T = T } at a single point.Therefore by Theorem 3.4, γ(t) is differentiable for all t ∈ R except at a single point, and satisfies system (2.1).On the other hand, γ(t) is componentwise monotone and bounded, thus lim t→∞ γ(t) exists and is a rest point of system (2.1) in D f = {u ∈ R 4 : T > T }.But the only possibility is u 1 .That is lim t→∞ γ(t) = u 1 .Similarly lim t→−∞ γ(t) = u 0 .Finally, in [10], Germain showed that the stable manifold of system (2.1) at u 1 is one-dimensional.Therefore the above heteroclinic orbit is unique.Hence we have proved the following theorem.is not considered in the literature, but the cases of ionizing fast, switch-on and the case of transverse magnetic field can be solved with the same technique which is used in this section and the previous sections.

Mathematical Problems in Engineering
Special Issue on Time-Dependent Billiards

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
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Theorem 3 . 1 .
following three theorems are modifications of [13, Theorems 2.1.1 and 2.1.2].Suppose that f in (3.1) is locally Lipschitz in a neighborhood of the closure of an open bounded set D which is homomorphic to the semisphere {x ∈ R n : |x| < 1, x n > 0}, and (3.1) is gradient-like with respect to a real-valued function, h in D. Moreover the following conditions hold.

Theorem 3 . 3 . 1 ) 2 ) 3 ) 4 ) 5 ) 7 )
Let f , D, h, x, and F be the same as in Theorem 3.1.Moreover, the following conditions hold.(C The same as condition (C 1 ) in Theorem 3.1.(C The same as condition (C 2 ) in Theorem 3.1.(C If p ∈ F \{ x}, then p • t ∈ ∂D for |t| ≠ 0 and small.(C If p ∈ F \{ x} and p•t ∈ D for t > 0 and small, then p•t ∈ D for t <0 and |t| small.(C The set I = {p ∈ ∂D : p • t ∈ D for t > 0 and small} is disconnected.(C 6 ) Let E = ∂D\ F .If p ∈ E, then p • t ∈ D for t > 0 and small.(C The connected set E intersects at least with two components of I.