By piecing together the Riemann problems, we construct weak asymptotic solution for general type initial data

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

WEAK ASYMPTOTIC SOLUTION FOR A NON-STRICTLY HYPERBOLIC SYSTEM OF CONSERVATION LAWS

HARENDRA SINGH, MANAS RANJAN SAHOO, OM PRAKASH SINGH

Abstract. In this article, we construct the weak asymptotic solution devel- oped by Panov and Shelkovich for piecewise known solutions to a prolonged system of conservation laws. This is done by introducing four singular waves along a discontinuity curve, which in turn implies the existence of weak as- ymptotic solutions for the Riemann type initial data. By piecing together the Riemann problems, we construct weak asymptotic solution for general type initial data.

1. Introduction

Systems of conservation laws arise in many physical contexts are not strictly hyperbolic. For such systems classical theories of Glimm [2] and Lax [8] do not apply. Because of the appearance of product of distributions, it is difficult to define the notion of solutions for these problems. One way to avoid this is to work with the generalized space of Colmbeau. For details see [9] and [1].

A system of this kind was introduced by Joseph and Vasudeva Murthy[5], namely, (uj)t+

j

X

i=1

(u_iu_j−i+1

2 )x= 0, j= 1,2, . . . , n. (1.1) For n = 1, system (1.1) is Burger’s equation, which is well studied by Hopf [3].

Forn= 2 case is an one dimensional model for the large scale structure formation of universe, see, [12]. Using vanshing viscosity approach it is observed by Joseph [4] that the second component contain δ measure concentrated along the line of discontinuity. The case n = 3 is studied in [6]. Solution is constructed in the Colombeau setting. Ifu1=u, u2=v, u3=w, (1.1) becomes

ut+ (u²

2 )x= 0, vt+ (uv)x= 0, wt+ (v²

2 +uw)x= 0. (1.2) A similar system,

ut+ (u²)x= 0, vt+ (2uv)x= 0, wt+ 2(v²+uw)x= 0, (1.3)

2000Mathematics Subject Classification. 35A20, 35F25, 35R05.

Key words and phrases. System of PDEs; initial conditions; weak asymptotic solutions.

c

2015 Texas State University - San Marcos.

Submitted December 5, 2014. Published January 5, 2015.

1

is studied by Panov and Shelkovich [11]. In [11] a concept of weak asymptotic solu- tion is introduced and a solution is constructed under this consideration and gener- alized integral formulation is introduced for piecewise continuous data. Note that the system (1.3) can be obtained from (1.2) using the transformation (u, v, w)→ (2u, v,^w₂). The casen= 4 is studied by joseph and Sahoo [7]. In [7], using vansh- ing viscosity approach a solution is constructed for Riemann type initial data and based on this a weak integral formulation is given.

In this paper we use the weak asymptotic method introduced by Panov and Shelkovich [11] to study the casen= 4. Putting u1 =u, u2 =v, u3=w, u4 =z and followed by a linear transformation, the system (1.1) leads to the system

u_t+ (u²)_x= 0, v_t+ (2uv)_x= 0

w_t+ 2(v²+uw)_x= 0, z_t+ 2((3vw+uz)_x) = 0. (1.4) The aim of this paper is to study the above system (1.4) with initial conditions u(x,0) =u0(x), v(x,0) =v0(x), w(x,0) =w0(x), z(x,0) =z0(x). (1.5) The content of the paper is as follows. We construct weak asymptotic solution by connecting two known solutions from the left and right. As a special case we derive weak asymptotic solution for the Riemann type initial data. Then we construct a weak asymptotic solution when the initial data for uis a monotonic increasing function and initial data forv, wandz are locally integrable functions.

2. weak asymptotic solution for Riemann type initial data In this section we connect two classical solutions by introducing a discontinuity curve in asymptote level. First of all we recall the definition of weak asymptotic solution as introduced in [11, 10].

Definition 2.1. Let us define

L1(u) =ut+ (u²)x, L2(u, v) =vt+ (2uv)x

L3(u, v, w) =wt+ 2(v²+uw)x, L4(u, v, w, z) =zt+ 2((3vw+uz)x).

(u, v, w, z) is said to be weak asymptotic solution to problem (1.4) with initial data (1.5) if

Z

L1[u(x, t, )]ψ(x)dx=o(1), Z

L₂[u(x, t, ), v(x, t, )]ψ(x)dx=o(1), Z

L₃[u(x, t, ), v(x, t, ), w(x, t, )]ψ(x)dx=o(1), Z

L4[u(x, t, ), v(x, t, ), w(x, t, ), z(x, t, )]ψ(x)dx=o(1),

(2.1)

and initial conditions satisfy Z

u(x,0, )−u0(x)

ψ(x)dx=o(1), Z

v(x,0, )−v0(x)

ψ(x)dx=o(1), Z

w(x,0, )−w0(x)

ψ(x)dx=o(1), Z

z(x,0, )−z₀(x)

ψ(x)dx=o(1),

(2.2)

for allψ∈D(R).

To study weak asymptotic analysis first we need the following Lemma as in [11], regarding the superpositions of the singular wavesδ, δ⁰, δ⁰⁰andδ⁰⁰⁰.

Lemma 2.2. Let {w_i}_i∈I be an indexed set of Friedrich mollifiers satisfying wi(x) =wi(−x),

Z

wi = 1.

Define

Hi(x, ) =w0i(x ) =

Z ^x

−∞

wi(y)dy, δi(x, ) =1 wi(x

), δ^k_i(x, ) = 1 ^k+1w_i^k(x

).

The above assumptions implies the following asymptotic expansions, in the sense of distributions,

(Hi(x, ))^r=H(x) +OD⁰(), (Hi(x, )(Hj(x, )) =H(x) +OD⁰() (Hi(x, ))^rδj(x, ) =δ(x)

Z

w^r_0i(y)wj(y)dy+OD⁰() (δi(x, ))²=1

δ(x) Z

w²_i(y)dy+OD⁰() Hi(x, )δ⁰_j(x, ) =−1

δ(x) Z

wi(y)wj(y)dy+δ⁰(x) Z

w0i(y)wj(y)dy+OD⁰() Hi(x, )²δ⁰⁰⁰_j (x, ) = 1

δ(x) Z

w_i⁰(y)δ_j⁰(y)dy+OD⁰(), δ_i(x, ).δ_j(x, ) =1

δ(x) Z

w_i(y)w_j(y)dy+O_D⁰() δi(x, )δ_j⁰(x, ) = 1

δ⁰(x) Z

ywi(y)w_j⁰(y)dy+OD⁰(), Hi(x, )δ_j⁰⁰(x, ) = 1

δ(x) Z

w0i(y)wj(y)dy+1 2δ⁰⁰(x)

Z

y²w0i(y)wj(y)dy+OD⁰() δi(x, ))²δ_j⁰⁰⁰(x, ) =1

δ⁰(x) Z

ywi(y)w⁰⁰⁰_j (y)dy+OD⁰() Hi(x, )²δ⁰⁰⁰⁰_j (x, ) =1

δ⁰(x) Z

yw0i(y)w⁰⁰⁰⁰_j (y)dy+OD⁰() wherehOD⁰(), ψ(x)i →0 for every test functionψ.

Proof. Letψ∈D(R) be any test function. The first six relations can be found in [11]; so wee prove from the seventh onward.

Now we prove seventh asymptotic expansion. Using change of variable for- mula (x=y), employing third order Taylor expansion ψ(y) =ψ(0) +yψ⁰(0) +

1

2²y²ψ⁰⁰(0) +³y³O(1), and the fact thatR

ywi(y)wj(y)dy= 0, we have hδi(x, )δj(x, ), ψ(x)i=

Z 1 wi(x

)1 wj(x

)ψ(x)dx

=1

Z

wi(y)wj(y)ψ(y)dy

=1 ψ(0)

Z

wi(y)wj(y)dy+ψ⁰(0) Z

ywi(y)wj(y)dy+O()

=1 δ(x)

Z

w_i(y)w_j(y)dy+O().

Now we prove eighth asymptotic expansion. Using change of variable formula (x = y), employing third order Taylor expansion, ψ(y) = ψ(0) +yψ⁰(0) +

1

2²y²ψ⁰⁰(0) +³y³O(1), and the fact thatR

ywi(y)wj(y)dy= 0, we have hδ_i(x, )δ_j⁰(x, ), ψ(x)i= 1

² Z

w_i(y)w_j⁰(y)ψ(y)dy

= 1 ²ψ(0)

Z

w_i(y)w⁰_j(y)dy+1 ψ⁰(0)

Z

yw_i(y)w_j⁰(y)dy +1

2ψ⁰⁰(0) Z

y²wi(y)w⁰_j(y)dy+O()

= 1 δ⁰(x)

Z

ywi(y)w_j⁰(y)dy+O().

In the above calculation we also used the identity Z

wi(y)w⁰_j(y)dy= Z

y²wi(y)w⁰_j(y)dy= 0 .

Following an analysis similar as above, we prove the remaining identities. Details are as follows:

hHi(x, )δ_j⁰⁰(x, ), ψ(x)i

= Z

w_0i(y)1

²w⁰⁰_j(y)ψ(y)dy

= Z

w_0i(y)1

²w⁰⁰_j(y)(ψ(0) +yψ⁰(0) + ²y²

2 ψ⁰⁰(0))dy+O()

=1 δ⁰(x)

Z

yw0i(y)w⁰⁰_j(y)dy+1 2δ⁰⁰(x)

Z

y²w0i(y)w_j⁰⁰(y)dy+O(),

hδi(x, ))²w⁰⁰⁰_j (x, ), ψ(x)i

= 1 ²

Z

w_i(y)w_j⁰⁰⁰(y)ψ(y)dy

= 1 ²

Z

w_i(y)w_j⁰⁰⁰(y)(ψ(0) +yψ⁰(0) +²y²

2 ψ⁰⁰(0))dy+O()

= 1 ²δ(x)

Z

wi(y)w⁰⁰⁰_j (y)dy+1 δ⁰(x)

Z

ywi(y)w⁰⁰⁰_j (y)dy

+1 2δ⁰⁰(x)

Z

y²wi(y)w⁰⁰⁰_j (y)dy+O()

= 1 δ⁰(x)

Z

yw_i(y)w⁰⁰⁰_j (y)dy+O()

hH_i(x, )²δ_j⁰⁰⁰⁰(x, ), ψ(x)i

= Z

w0i(y)1

²w_j⁰⁰⁰⁰(y)ψ(y)(ψ(0) +yψ⁰(0) +²y²

2 ψ⁰⁰(0))dy+O()

= 1 δ⁰(x)

Z

yw0i(y)w⁰⁰⁰⁰_j (y)dy+O()

It is observed in [7], that the vanishing viscosity limit for the componentzadmits combinations ofδ, δ⁰, δ⁰⁰waves. So we choose ansatz as the combination of the above singular waves along the discontinuity curve. But this is not enough as it is clear in the construction ofw, see [11]. In [11], a correction term is added in the component w to construct weak asymptotic solution. As the solution for the component is more complicated, extra care has to be taken to accomplish this.This is done by choosing the correction term carefully in the componentz.

Theorem 2.3. The following ansatz

u(x, t, ) =u2(x, t) + [u]Hu(−x+φ(t), ),

v(x, t, ) =v₂(x, t) + [v]H_v(−x+φ(t), ) +e(t)δ_e(−x+φ(t), ), w(x, t, ) =w2(x, t) + [w]Hw(−x+φ(t), ) +g(t)δg(−x+φ(t), )

+h(t)δ_h⁰(−x+φ(t), ) +Rw(−x+φ(t), ),

z(x, t, ) =z2(x, t) + [z]Hz(−x+φ(t), ) +l(t)δl(−x+φ(t), ) +m(t)δ⁰_m(−x+φ(t), ) +n(t)δ⁰⁰_n(−x+φ(t), ) +R_z(−x+φ(t), ),

(2.3)

where

R_w(x, t, ) =²P(t)δ⁰⁰⁰_P(−x+φ(t), ),

R_z(x, t, ) =²(Q(t)δ⁰⁰⁰_R(−x+φ(t), ) +R(t)δ_R⁰⁰⁰⁰(−x+φ(t), ).

is weak asymptotic solution to the problem (1.4)if the following relations hold:

L₁[u₁] = 0, L₁[u₂] = 0, L2[u1, v1] = 0, L2[u2, v2] = 0, L3[u1, v1, w1] = 0, L3[u2, v2, w2] = 0, φ(t) = (u˙ 1+u2)

_x=φ(t), e(t) = [u](v˙ 1+v2) _x=φ(t)

˙

g(t) = (2[v](v1+v2) + [u](w1+w2)

_x=φ(t), d

dt(h(t)[u(φ(t), t)]) = d dte²(t), Z

w0u(y)wj(y)dy= Z

y²w0v(y)we(y)dy= 1

2, j=e, g, h,

Z

wu(y)wh(y)dy= Z

w²_e(y)dy, P(t) = A

u₁(φ(t), t), where A is a constant, L₄[u₁, v₁, w₁, z₁] = 0, L₄[u₂, v₂, w₂, z₂] = 0,

l(t) =˙ −[z] ˙φ(t) + 2[3vw+uz], Z

w0u(y)wl(y)dy=1 2

Z

y²w0u(y)wn(y)dy= Z

w0u(y)wm(y)dy= 1 2, m(t) = 2[3{(v˙ 2+ [v]

Z

w0v(y)wg(y)dy)g(t) + (w2+ [w]

Z

w0w(y)we(y)dy)e(t)}

+ 3{(v2x+ [vx] Z

w0v(y)wh(y)dy)h(t)}

+ (u_2x+[ux]

2 )m(t) + (u_1xx+[uxx] 2 )n(t)], n(t) = 2[3{(v˙ 2+ [v]

Z

w0v(y)wh(y)dy)h(t)} −(2u2x+ [ux])n(t)],

R(t) = 1

[u]R

w_0u(y)w⁰⁰⁰⁰_R (y)dy h

3e(t)h(t) Z

ywe(y)w⁰_h(y)dy + 3e(t)p(t)

Z

yw_e(y)w⁰⁰⁰_P(y)dyi

Q(t) = 1

[u]R

w⁰_u(y)w⁰_Q(y)dy

h3e(t)g(t) Z

w_e(y)w_g(y)dy

−3[v]h(t) Z

wv(y)wh(y)dy−[u]m(t) Z

wu(y)wm(y)dy +[u]n(t)

2 + [ux]R(t) Z

w0u(y)w_R⁰⁰⁰⁰(y)dyi

Proof. If the first thirteen relations above hold, then the expression foru, v andw in (2.3) is a weak asymptotic solution, is shown in [11]. So, we only prove that the expression for the componentz in equation (2.3) is a weak asymptotic solution.

Multiplying the ansatz given forvandwin the equation (2.3) and using lemma 2.2, we obtain

v(x, t, )w(x, t, )

=v2w2+ [vw]H(−x+φ(t)) +n

(v2+ [v]

Z

w0v(y)wg(y)dy)g(t) + (w₂+ [w]

Z

w_0w(y)w_e(y)dy)e(t)o

δ(−x+φ(t)) + (v₂+ [v]

Z

w_0v(y)w_h(y)dy)h(t)δ⁰(−x+φ(t)) + (e(t)g(t)

Z

we(y)wg(y)dy−[v]h(t) Z

wv(y)wh(y)dy)1

δ(−x+φ(t)) + (e(t)h(t)

Z

ywe(y)w⁰_h(y)dy+e(t)p(t) Z

ywe(y)w⁰⁰⁰_P(y)dy)1

δ⁰(−x+φ(t)) +OD⁰().

Similarly,

u(x, t, )z(x, t, )

=u2z2+ [uz]H(−x+φ(t)) + [u2+ [u]

Z

w0u(y)wl(y)dy]l(t)δ(−x+φ(t)) + [u₂+ [u]

Z

w_0u(y)w_m(y)dy]m(t)δ⁰(−x+φ(t)) + [u₂+[u]

2 Z

y²w_0u(y)w_n(y)dy]n(t)δ⁰⁰(−x+φ(t)) + [−[u]m(t)

Z

wu(y)wm(y)dy+ [u]n(t) Z

w0u(y)wn(y)dy + [u]Q(t)

Z

w⁰_u(y)w⁰_Q(y)dy]1

δ(−x+φ(t)) + [u]R(t)

Z

w0u(y)w_R⁰⁰⁰⁰(y)dy1

δ⁰(−x+φ(t)) +OD⁰().

Arranging the coefficient ofδand the derivatives, ¹δand¹δ⁰of 3v(x, t, )w(x, t, )+

u(x, t, )z(x, t, ), we obtain

3v(x, t, )w(x, t, ) +u(x, t, )z(x, t, )

= (3v2w2+u2z2) + [3vw+uz]H(−x+φ(t)) + [3{(v2+ [v]

Z

w_0v(y)w_g(y)dy)g(t) + (w₂+ [w]

Z

w_0w(y)w_e(y)dy)e(t)}

+ (u2+ [u]

Z

w0u(y)wl(y)dy)l(t) + 3{(v2x+ [vx] Z

w0v(y)wh(y)dy)h(t)}

+ (u2x+ [ux] Z

w0u(y)wm(y)dy)m(t) + (u2xx+[uxx]

2 Z

y²w0u(y)wn(y)dy)n(t)]

_x=φ(t)δ(−x+φ(t)) + [3{(v2+ [v]

Z

w0v(y)wh(y)dy)h(t)}+ (u2+ [u]

Z

w0u(y)wm(y)dy)m(t)

−2(u_2x+[u_x] 2

Z

y²w_0u(y)w_n(y)dy)n(t)]

_x=φ(t)δ⁰(−x+φ(t)) + [u₂+[u]

2 Z

y²w_0u(y)w_n(y)dy]n(t)

x=φ(t)δ⁰⁰(−x+φ(t)) +h

3e(t)g(t) Z

we(y)wg(y)dy−3[v]h(t) Z

wv(y)wh(y)dy

−[u]m(t) Z

wu(y)wm(y)dy+ [u]n(t) Z

w0u(y)wn(y)dy + [u]Q(t)

Z

w⁰_u(y)w⁰_Q(y)dy + [u_x]R(t)

Z

w_0u(y)w⁰⁰⁰⁰_R (y)dyi _x=φ(t)

1

δ(−x+φ(t)) +h

3e(t)h(t) Z

yw_e(y)w⁰_h(y)dy+ 3e(t)p(t) Z

yw_e(y)w⁰⁰⁰_P(y)dy + [u]R(t)

Z

w0u(y)w_R⁰⁰⁰⁰(y)dyi _x=φ(t)

1

δ⁰(−x+φ(t)) +OD⁰().

(3v(x, t, )w(x, t, ) +u(x, t, )z(x, t, ))x

= (3v2w2+u2z2)x+ [(3vw+uz)x]H(−x+φ(t))−[3vw+uz]δ(−x+φ(t))

−[3{(v2+ [v]

Z

w0v(y)wg(y)dy)g(t) + (w2+ [w]

Z

w0w(y)we(y)dy)e(t)}

+ (u2+ [u]

Z

w0u(y)wl(y)dy)l(t) + 3{(v2x+ [vx] Z

w0v(y)wh(y)dy)h(t)}

+ (u2x+ [ux] Z

w0u(y)wm(y)dy)m(t) + (u2xx+[uxx]

2 Z

y²w0u(y)wn(y)dy)n(t)]

_x=φ(t)δ⁰(−x+φ(t))

−[3{(v₂+ [v]

Z

w_0v(y)w_h(y)dy)h(t)}+ (u₂+ [u]

Z

w_0u(y)w_m(y)dy)m(t)

−2(u_2x+[u_x] 2

Z

y²w_0u(y)w_n(y)dy)n(t)]

_x=φ(t)δ⁰⁰(−x+φ(t))

−[u2+[u]

2 Z

y²w0u(y)wn(y)dy]n(t)

_x=φ(t)δ⁰⁰⁰(−x+φ(t))

−h

3e(t)g(t) Z

we(y)wg(y)dy−3[v]h(t) Z

wv(y)wh(y)dy

−[u]m(t) Z

wu(y)wm(y)dy + [u]n(t)

Z

w0u(y)wn(y)dy+ [u]Q(t) Z

w⁰_u(y)w⁰_Q(y)dy + [u_x]R(t)

Z

w_0u(y)w_R⁰⁰⁰⁰(y)dyi _x=φ(t)

1

δ⁰(−x+φ(t))

−h

3e(t)h(t) Z

yw_e(y)w⁰_h(y)dy+ 3e(t)p(t) Z

yw_e(y)w_P⁰⁰⁰(y)dy + [u]R(t)

Z

w0u(y)w⁰⁰⁰⁰_R (y)dyi _x=φ(t)

1

δ⁰⁰(−x+φ(t)) +OD⁰().

(2.4) Differentiatingz with respect tot,

z_t(x, t, )

=z_1t+ [z_t]H(−x+φ(t)) +h

[z] ˙φ(t) + ˙l(t)i

δ(−x+φ(t)) +

l(t) ˙φ(t) + ˙m(t)

δ⁰(−x+φ(t)) +

m(t) ˙φ(t) + ˙n(t)

δ⁰⁰(−x+φ(t)) +n(t) ˙φ(t)δ⁰⁰⁰(−x+φ(t)) +OD⁰().

(2.5)

Putting the value of z_t(x, t, ) from the equations (2.5) and (3v(x, t, )w(x, t, ) + u(x, t, )z(x, t, ))_x from the equations (2.4) in the fourth equation of (1.4), we obtain

zt+ 2((3vw+uz)x)

=z1t+ 2(3v2w2+u2z2)x+h

[zt] + 2[(3vw+uz)x]i

H(−x+φ(t)) +h

[z] ˙φ(t) + ˙l(t)−2[3vw+uz]i

δ(−x+φ(t))

+h

l(t) ˙φ(t) + ˙m(t)−2[3{(v2+ [v]

Z

w0v(y)wg(y)dy)g(t) + (w₂+ [w]

Z

w_0w(y)w_e(y)dy)e(t)}

+ (u₂+ [u]

Z

w_0u(y)w_l(y)dy)l(t) + 3{(v2x+ [v_x] Z

w_0v(y)w_h(y)dy)h(t)}

+ (u2x+ [ux] Z

w0u(y)wm(y)dy)m(t) + (u2xx+[uxx]

2 Z

y²w0u(y)wn(y)dy)n(t)]i

δ⁰(−x+φ(t)) +h

m(t) ˙φ(t) + ˙n(t)−2[3{(v2+ [v]

Z

w0v(y)wh(y)dy)h(t)}

+ (u2+ [u]

Z

w0u(y)wm(y)dy)m(t)

−(2u_2x+ [u_x] Z

y²w_0u(y)w_n(y)dy)n(t)]i

δ⁰⁰(−x+φ(t)) +h

n(t) ˙φ(t)−[2u2+ [u]

Z

y²w0u(y)wn(y)dy]n(t)i

δ⁰⁰⁰(−x+φ(t))

−2h

3e(t)g(t) Z

we(y)wg(y)dy−3[v]h(t) Z

wv(y)wh(y)dy

−[u]m(t) Z

wu(y)wm(y)dy + [u]n(t)

Z

w0u(y)wn(y)dy+ [u]Q(t) Z

w_u⁰(y)w_Q⁰ (y)dy + [u_x]R(t)

Z

w_0u(y)w⁰⁰⁰⁰_R (y)dyi _x=φ(t)

1

δ⁰(−x+φ(t))

−2h

3e(t)h(t) Z

yw_e(y)w⁰_h(y)dy+ 3e(t)p(t) Z

yw_e(y)w_P⁰⁰⁰(y)dy + [u]R(t)

Z

w0u(y)w_R⁰⁰⁰⁰(y)dyi _x=φ(t)

1

δ⁰⁰(−x+φ(t)) +OD⁰().

So if the relations 14-21 holds then the coefficients of δ and their derivatives, ¹δ

and ¹δ⁰ vanishes. The proof is complete.

For Riemann type data the above expression is simple, and it is described in the following corollary.

Corollary 2.4. If ui, vi, wi zi fori= 1,2are constants then expression (2.3)is a weak asymptotic solution provided the following equalities hold.

φ(t) = (u˙ 1+u2)

_x=φ(t), e(t) = [u](v˙ 1+v2) _x=φ(t),

˙

g(t) = (2[v](v1+v2) + [u](w1+w2)

_x=φ(t), quadd

dt(h(t)[u(φ(t), t)]) = d dte²(t) Z

w_0u(y)w_j(y)dy= Z

y²w_0v(y)w_e(y)dy= 1

2, j=e, g, h, Z

w_u(y)w_h(y)dy= Z

w²_e(y)dy, P(t) = A

u1(φ(t), t), whereA is a constant,

l(t) =˙ −[z] ˙φ(t) + 2[3vw+uz], Z

w0u(y)wl(y)dy=1 2

Z

y²w0u(y)wn(y)dy= Z

w0u(y)wm(y)dy= 1 2, m(t) = 2[3{(v˙ ₂+ [v]

Z

w_0v(y)w_g(y)dy)g(t) + (w₂+ [w]

Z

w_0w(y)w_e(y)dy)e(t)}, n(t) = 2[3{(v˙ 2+ [v]

Z

w_0v(y)w_h(y)dy)h(t)},

R(t) = 1

[u]R

w0u(y)w⁰⁰⁰⁰_R (y)dy[3e(t)h(t) Z

ywe(y)w_h⁰(y)dy + 3e(t)p(t)

Z

ywe(y)w_P⁰⁰⁰(y)dy],

Q(t) = 1

[u]R

w⁰_u(y)w⁰_Q(y)dy

h3e(t)g(t) Z

w_e(y)w_g(y)dy

−3[v]h(t) Z

wv(y)wh(y)dy−[u]m(t) Z

wu(y)wm(y)dy+[u]n(t) 2

i (2.6)

Piecing together the Riemann problems we construct a weak asymptotic solution for general type initial data under the assumption thatuis a monotonic increasing function.

Theorem 2.5. Ifu₀, v₀, w₀ andz₀ are locally integrable functions onR, and u₀ is monotonic increasing, then there exists weak asymptotic solution(u, v, w, z)to the system (1.4)with initial data (1.5).

Proof. Letφbe a test function onRhaving support in [−K, K]. Given >0, there exist piecewise constant functions (u0, v0, w0, z0) such that

Z

[−K,K]

|u0(x)−u0(x)|dx < , Z

[−K,K]

|v0(x)−v0(x)|dx < , Z

[−K,K]

|w0(x)−w0(x)|dx < , Z

[−K,K]

|z0(x)−z0(x)|dx < .

In addition to this we can take u₀ monotonic increasing and all functions have same points of discontinuities. (u₀, v₀, w₀, z₀) in [−K, K] can be represented as

u0=

n

X

i=1

u0i(H(x−a_i−1)−H(x−ai)),

v0=

n

X

i=1

v0i(H(x−a_i−1)−H(x−ai)),

w0=

n

X

i=1

w0i(H(x−ai−1)−H(x−ai)),

z₀=

n

X

i=1

z_0i(H(x−a_i−1)−H(x−a_i)).

Since u₀ is a monotonic increasing function, discontinuity curve arising in the solution of (u, v, w, z) do not intersect for any time. So the following functions are

weak asymptotic solutions

u(x, t, η) =u01Hu(−x+c1t+a1, η) +

n−1

X

i=2

u0i

Hu(x−ci−1t−ai−1, η)

−Hu(x−cit−ai, η)

+u0n(Hu(x−c_n−1t−a_n−1, η), v(x, t, η) =v01Hv(−x+c1t+a1, η) +

n−1

X

i=2

v0i

Hv(x−ci−1t−ai−1, η)

−Hv(x−cit−ai, η)

+v0nHv(x−c_n−1t−a_n−1, η) +

n−1

X

i=1

e_i(t)δ_e(−x+c_it, η),

w(x, t, η) =w₀₁H_w(−x+c₁t+a₁, η) +

n−1

X

i=2

w_0i

H_w(x−c_i−1t−a_i−1, η)

−Hw(x−cit−ai, η)

+w0nHw(x−cn−1t−an−1, η) +

n−1

X

i=1

g_i(t)δ_g(−x+c_it, η) +

n−1

X

i=1

h_i(t)δ⁰_h(−x+c_it, η)

+

n−1

X

i=1

R_wi(−x+c_it, η),

z(x, t, η) =z₀₁H_z(−x+c₁t+a₁, η) +

n−1

X

i=2

z_0i

H_z(x−c_i−1t−a_i−1, η)

−Hz(x−cit−ai, η)

+z0n(Hz(x−cn−1t−an−1, η)) +

n−1

X

i=1

l_i(t)δ_l(−x+c_it, η) +

n−1

X

i=1

m_i(t)δ_m⁰ (−x+c_it, η)

+

n−1

X

i=1

n_i(t)δ_n⁰⁰(−x+c_it, η) +

n−1

X

i=1

R_zi(−x+c_it, η),

where ei, gi, hi, li, mi, ni, Rwi and Rzi satisfy (2.6) with u1, u2, v1, v2, w1, w2, z1,z2,e,g,h,l,m,n,Rw andRzreplaced byui−1,ui,vi−1,vi,wi−1,wi,zi−1,zi, ei,gi,hi,li, mi,ni,Rwi andRzi. Given >0 chooseη() small enough such that the following estimates hold.

Z

L1[u(x, t, η())]ψ(x)dx

< η(), Z

L2[u(x, t, η()), v(x, t, η())]ψ(x) < ,

Z

L₃[u(x, t, η()), v(x, t, η()), w(x, t, η())]ψ(x)dx < ,

Z

L₄[u(x, t, η()), v(x, t, η()), w(x, t, η()), z(x, t, η())]ψ(x)dx < ,

Z

u(x,0, η())−u0(x)

ψ(x)dx <2,

Z

v(x,0, η())−v0(x)

ψ(x)dx <2,

Z

w(x,0, η())−w₀(x)

ψ(x)dx <2,

Z

z(x,0, η())−z₀(x)

ψ(x)dx <2.

Define

(¯u(x, t, ),v(x, t, ),¯ w(x, t, ),¯ z(x, t, ))¯

= (u(x, t, η(), v(x, t, η(), w(x, t, η(), z(x, t, η()).

Then (¯u,v,¯ w,¯ z) is a weak asymptotic solution of system (1.4)-(1.5).¯ References

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[3] E. Hopf; The Partial differential equation ut+uux = νuxx, Comm. Pure Appl.Math., 3 (1950), 201-230.

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Anal.,7(1993), 105-120.

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[9] M. Oberguggenberger; Multipication of distributions and Applications to PDEs, Pittman Research Notes in Math,Longman, Harlow259(1992).

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Harendra Singh

Department of Mathematical Sciences, IIT (BHU), Varanasi 221005, India E-mail address:[email protected]

Manas Ranjan Sahoo

Department of Mathematical Sciences, IIT (BHU), Varanasi 221005, India E-mail address:[email protected]

Om Prakash Singh

Department of Mathematical Sciences, IIT (BHU), Varanasi 221005, India E-mail address:[email protected]