LOCAL EXISTENCE OF SOLUTIONS OF THE MIXED PROBLEM FOR THE SYSTEM OF EQUATIONS OF IDEAL

J . R E N C L A W O W I C Z and W. M. Z A J A¸ C Z K O W S K I (Warszawa)

LOCAL EXISTENCE OF SOLUTIONS OF THE MIXED PROBLEM FOR THE SYSTEM OF EQUATIONS OF IDEAL

RELATIVISTIC HYDRODYNAMICS

Abstract. Existence and uniqueness of local solutions for the initial- boundary value problem for the equations of an ideal relativistic fluid are proved. Both barotropic and nonbarotropic motions are considered. Exis- tence for the linearized problem is shown by transforming the equations to a symmetric system and showing the existence of weak solutions; next, the appropriate regularity is obtained by applying Friedrich’s mollifiers tech- nique. Finally, existence for the nonlinear problem is proved by the method of successive approximations.

1. Introduction. In this paper we prove the local existence of solutions to the equations of ideal relativistic hydrodynamics which are the following system of conservation laws:

(1.1) T

_,x^ijj

= 0, i, j = 0, 1, 2, 3, and

(1.2) (δu

ⁱ

)

_,xⁱ

= 0,

where the summation convention over repeated indices is assumed and (1.3) T

^ij

= wu

ⁱ

u

^j

+ pg

^ij

is the energy-momentum tensor, and g

^ij

is the space-time metric tensor of

1991 Mathematics Subject Classification: 35L60, 35Q75, 35A07, 35L65.

Key words and phrases: relativistic hydrodynamics, existence, initial-boundary value problem, system of hyperbolic equations of the first order, symmetrization.

[221]

the form

(1.4) {g

^ij

} =



 

−1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1



  .

Moreover, w = e + p, where w is the density of enthalpy, e the density of the internal energy and δ the density of the fluid particles in a suitable system of coordinates in which the volume element does not move. We denote by p the pressure and by u = {u

ⁱ

}

i=0,1,2,3

the four-velocity: u

_α

= v

_α

/(cβ), α = 1, 2, 3, u

0

= −1/β, where β = p

1 − v

²

/c

²

, c is the speed of light, v

²

= v

²₁

+v

²₂

+v

₃²

, where v = (v

1

, v

2

, v

3

) is the velocity vector, and u

ⁱ

= g

^ij

u

j

, g

^ij

= g

_ij

, g

^ij

g

_jk

= δ

_kⁱ

.

In the above notation the energy-momentum tensor takes the form

(1.5)

T

αγ

= w v

α

v

γ

c

²

β

²

+ pδ

αγ

, α, γ = 1, 2, 3, T

α0

= −w v

α

cβ

²

, T

00

= w β

²

− p.

We consider problem (1.1)–(1.2) for t ∈ [0, T ] and x = (x

¹

, x

2

, x

3

) ∈ Ω ⊂ R

³

, with initial and boundary conditions

(p, u, δ) |

t=0

= (p

0

, u

0

, δ

0

), (1.6)

M z |

^∂Ω

= g(x

^′

, t), (1.7)

where z = (p, u, δ), and the matrix M is defined in Section 4.

To prove the existence of solutions to (1.1)–(1.2), we have to transform our problem to a symmetric hyperbolic system (2.2). We present this sym- metrization in Section 2. In Section 3 we introduce the necessary spaces and norms; moreover, we rewrite the symmetric system (2.2) in the form (3.1) (with the initial-boundary conditions (1.6)–(1.7) suitably transformed).

In Section 4 we consider the linearized problem (3.1); first in 4(a) we prove the existence of solutions in a half-space, in 4(b) we obtain the regu- larity of solutions and in the last part of the section, using a partition of unity and a localized problem, we transform the results of 4(a) and 4(b) to the case of a bounded domain. Using the properties of the solutions obtained, we prove the existence and uniqueness of local solutions to the nonlinear problem (3.1) by the method of successive approximations in Section 5.

Finally, in Section 6 we specify the results of Sections 4 and 5 for problem (2.2). In Section 7 the barotropic case is considered.

To prove existence of solutions to problem (1.1), (1.2), (1.6), (1.7) we

need to know that the form (6.1) is uniformly positive definite. To show it

we choose a state equation (here p = RδT ). This implies strong restrictions

on the initial velocity (see Remark 6.1). In the barotropic case we do not have such restrictions so we can also consider near light motions.

2. Symmetrization. To symmetrize equations (1.1)–(1.2) we use con- siderations from [1], [2]. We have a system of conservation laws; now we write a new conservation law, which is a consequence of the old ones. (1.1) implies

u

_i

∂(wu

^k

)

∂x

^k

+ wu

^k

∂u

i

∂x

^k

+ ∂p

∂x

i

= 0.

Multiplying by u

ⁱ

, summing over i and using (2.0) u

ⁱ

u

i

= −1, u

ⁱ

∂u

i

∂x

^k

= 0 we get

− ∂(wu

^k

)

∂x

^k

+ u

ⁱ

∂p

∂x

i

= 0, which is equivalent to

∂

∂x

^k

 w δ δu

^k



− 1 δ

∂p

∂x

^k

δu

^k

= 0.

From this and (1.2) we obtain δu

^k

 ∂

∂x

^k

 w δ



− 1 δ

∂p

∂x

^k



= 0 so using the thermodynamical identity, we can write

T δu

^k

∂

∂x

^k

 s δ



= 0 where s is the entropy.

Finally, because u

^k

s −

¹_δ

_∂δ

∂x^k

= s

^∂x_∂x^kk

from (1.2), we get T

_∂x^∂k

(su

^k

) = 0, so

∂

∂x

^k

(su

^k

) = 0 is a new conservation law.

We have shown that equations (1.1)–(1.3) are linearly dependent, that is, there exist functions λ

^m

such that

λ

ⁱ

∂T

_i^k

∂x

^k

+ λ

⁴

∂(δu

^k

)

∂x

^k

+ λ

⁵

∂(su

^k

)

∂x

^k

= 0

for arbitrary functions z

^j

(p, u

¹

, u

²

, u

³

, δ), where λ

ⁱ

= u

ⁱ

, i = 0, 1, 2, 3; λ

⁴

=

(w − sT )/δ, λ

⁵

= T and T = T (δ, p), s = s(δ, p).

Equations (1.1)–(1.3) can be written in the form

∂

_z^j

q

_m^k

(z) ∂z

^j

∂x

^k

= 0, m = 0, 1, . . . , 5, and multiplying by ∂

z^τ

λ

^m

we obtain

∂

_z^τ

λ

^m

∂

_zj

q

^k_m

(z) ∂z

^j

∂x

^k

= 0 ⇔ A

^kτ j

∂z

^j

∂x

^k

= 0 where the matrices A

^k

(z) are symmetric (see [1], [2]).

The matrices ∂

z

q

^k

(z) take the form

∂

z

q

⁰

=



 

 

 

 

 

1 −_β2¹ −_β2^{1 ∂e}_∂p −2u¹w −2u²w −2u³w −_β2^{1 ∂e}_∂δ

1

βu₁+_β¹u₁^∂e_∂p βu²₁w+^w_β βu₁u²w βu₁u³w ^u1_β ^∂e_∂δ

1

βu₂+_β¹u₂^∂e_∂p βu₂u¹w βu²₂w+^w_β βu₂u³w ^u2_β ^∂e_∂δ

1

βu₃+_β¹u₃^∂e_∂p βu₃u¹w βu₃u²w βu²₃w+^w_β ^u3_β ^∂e_∂δ

0 βu¹δ βu²δ βu³δ _β¹

−¹_β

∂s

∂p βu¹s βu²s βu³s _β^1∂s_∂δ



 

 

 

 

  ,

∂

z

q

¹

=



 

 

 

 

 

−^u1_β −^u1_β ^∂e_∂p −βu²1w −^w_β −βu¹u³w −βu¹u²w −^u1_β ^∂e_∂δ 1 + u²₁+ u²_1∂p^∂e 2u1w 0 0 u²₁^∂e_∂δ u¹u₂+ u¹u₂^∂e_∂p u₂w u¹w 0 u¹u₂^∂e_∂δ u¹u₃+ u¹u₃^∂e_∂p u₃w 0 u¹w u¹u₃^∂e_∂δ

0 δ 0 0 u¹

u^{1 ∂s}_∂p s 0 0 u^{1 ∂s}_∂δ



 

 

 

 

  ,

∂

z

q

²

=



 

 

 

 

 

−

u2 β −

u2 β

∂e

∂p −βu2u₁w −βu²2w −^wβ −βu3u₂w −

u2 β

∂e

∂δ

u²u₁+ u²u₁^∂e_∂p u²w u₁w 0 u²u₁^∂e_∂δ 1 + u²₂+ u²_2∂p^∂e 0 2u₂w 0 u²₂^∂e_∂δ u²u₃+ u²u₃^∂e_∂p 0 u₃w u²w u²u₃^∂e_∂δ

0 0 δ 0 u²

u^{2 ∂s}_∂p 0 s 0 u^{2 ∂s}_∂δ



 

 

 

 

  ,

∂

z

q

³

=



 

 

 

 

 

−^u3_β −^u3_β ^∂e_∂p −βu3u₁w −βu³u₂w −βu²3w −^w_β −^u3_β ^∂e_∂δ u³u₁+ u³u₁^∂e_∂p u³w 0 u₁w u³u₁^∂e_∂δ u³u₂+ u³u₂^∂e_∂p 0 u³w u₂w u³u₂^∂e_∂δ 1 + u²₃+ u²_3∂p^∂e 0 0 2u3w u²₃^∂e_∂δ

0 0 0 δ u³

u^{3 ∂s}_∂p 0 0 s u^{3 ∂s}_∂δ



 

 

 

 

 

.

The matrix ∂

z^τ

λ

^m

has the form



 

 

 

0 0 0 0

_∂p^{∂ w−sT}_δ

_∂

∂p

T

βu

₁

1 0 0 0 0

βu

₂

0 1 0 0 0

βu

₃

0 0 1 0 0

0 0 0 0

_∂δ^{∂ w−sT}_δ

_∂

∂δ

T



 

 

 

so we get

A

⁰

=



 

 

 



1 β

∂s

∂p

∂T

∂p

βu

1

βu

2

βu

3 1

β

∂s

∂p

∂T

∂δ

βu

1

−βu

²1

w +

^w_β

−βu

¹

u

²

w −βu

¹

u

³

w 0 βu

₂

−βu

2

u

¹

w −βu

²2

w +

^w_β

−βu

2

u

³

w 0 βu

₃

−βu

3

u

¹

w −βu

3

u

²

w −βu

²3

w +

^w_β

0

1 β

∂s

∂p

∂T

∂δ

0 0 0

_β^1∂T_∂δ ^∂s_∂δ

−

^s_δ



 

 

 

 ,

A

¹

=



 

 

 

u

^{1 ∂s}_∂p^∂T_∂p

1 0

1 −β

²

u

³₁

w + u

¹

w −β

²

u

²₁

u

²

w 0 −β

²

u

²₁

u

2

w −β

²

u

¹

u

²₂

w + u

¹

w 0 −β

²

u

²₁

u

3

w −β

²

u

¹

u

²

u

3

w

u

^{1 ∂s}_∂p^∂T_∂δ

0 0

0 u

^{1 ∂s}_∂p^∂T_∂δ

−β

²

u

²₁

u

³

w 0

−β

²

u

¹

u

2

u

³

w 0

−β

²

u

¹

u

²₃

w + u

¹

w 0 0 u

^{1 ∂T}_∂δ ^∂s_∂δ

−

^s_δ



 

 

  ,

(2.1)

A

^k₀₀

= u

^k

∂s

∂p

∂T

∂p , A

⁰_α0

= βu

α

, A

^k_α0^′

= δ

_α^k′

, α, k

^′

, γ = 1, 2, 3, A

^k₄₀

= u

^k

∂s

∂p

∂T

∂δ , A

^k_αγ

= −β

²

u

^k

u

α

u

^γ

w + u

^k

wδ

^γ_α

, A

^k_4α

= 0, A

^k₄₄

= u

^k

∂T

∂δ

 ∂s

∂δ − s δ

 . Now we consider the following symmetric system:

A

^k

(z)



 

 

 p

_,x^k

u

¹_,xk

u

²_,xk

u

³_,xk

δ

_,xk



 

 

 = 0; k = 0, 1, 2, 3,

z = (p, u

¹

, u

²

, u

³

, δ),

or, because x

0

= ct,

(2.2) A

⁰

(z)z

t

+ c

X

3 i=1

A

ⁱ

(z)z

xⁱ

= 0.

3. Notations. In the next sections we will use the following norms, spaces and notations.

We will consider initial-boundary value problems in Ω

^T

= Ω × [0, T ] where Ω ⊂ R

³

, x ∈ Ω, t ∈ [0, T ]. We write

D

^γ_t,x

= ∂

^γ0

∂t

^γ0

∂

^γ1

∂x

^γ₁¹

∂

^γ2

∂x

^γ₂²

∂

^γ3

∂x

^γ₃³

, |γ| = γ

0

+ γ

1

+ γ

2

+ γ

3

, and we denote by H

^s

(Ω

^T

) the Sobolev space with the norm

kuk

H^s(Ω^T)

=  X

|γ|≤s T

\

0

\

Ω

|D

^γt,x

u |

²

dx dt 

1/2

= kuk

s,2,Ω^T

.

Similarly, we introduce H

^s

(Ω) and H

^s

(∂Ω

^T

) with norms k k

s,2,Ω

and k k

s,2,∂Ω^T

. We will use L

p

(Ω

^T

) and L

p

(Ω) with norms k k

p,Ω^T

and k k

^p,Ω

, respectively.

For α ∈ R we denote by H

α^s

(Ω

^T

) the weighted Sobolev space, the closure of C

^s

(Ω

^T

) in the norm

kuk

H_α^s(Ω^T)

= kuk

s,Ω^T,α

=  X

|γ|≤s T\

0

\

Ω

|D

^γt,x

u |

²

e

^−2αt

dx dt 

1/2

so we obtain L

2,α

(Ω

^T

) = H

_α⁰

(Ω

^T

) with k k

L2,α(Ω^T)

= k k

Ω^T,α

. Let

u ∈ L

^s∞

(0, T ; H

ⁱ

(Ω)) ⇔ ess sup

t∈[0,T ]

∂

^s

∂t

^s

u(t)

i,2,Ω

< ∞.

Then we define

Π

_k^l

(Ω

^T

) =

\

l i=k

L

^l−i_∞

(0, T ; H

ⁱ

(Ω))

with kuk

Π_k^l(Ω^T)

= kuk

l,k,∞,Ω^T

. Finally, we introduce Γ

₀^l

(Ω) by kuk

Γ₀^l(Ω)

= |u|

^l,0,Ω

=  X

|γ|≤l

\

Ω

|D

^γt,x

u |

²

dx 

1/2

.

Furthermore, Γ

^◦l₀

, H

^◦s_α

denote the sets of functions in the respective spaces

vanishing on the boundary ∂Ω; |u| is the Euclidean norm.

To simplify the following considerations, in Sections 4 and 5 we will consider the mixed problem

(3.1)

Lu ≡ E(t, x, u)u

^t

+ X

3 i=1

A

i

(t, x, u)u

x_i

= F (t, x), M (t, x

^′

, u)u |

∂Ω

= g(t, x

^′

), x

^′

∈ ∂Ω,

u |

^t=0

= u

0

(x),

where u takes values in R

^m

, x ∈ Ω ⊂ R

³

, t ∈ [0, T ], E, A

i

are real m × m matrices, the values of u lie in an open domain G and the values of the initial data u

0

belong to an open subset G

0

such that G

0

⊂ G. Next, assuming u := z ∈ R

⁵

, z = (p, u

¹

, u

²

, u

³

, δ) we will formulate results for problem (2.2) with initial and boundary conditions (1.6), (1.7).

4(a) The existence of solutions for the linearized equations in a half- space. In this part we shall consider the linearized problem (3.1) in the half-space x

1

> 0:

(4.1)

Lu = E(t, x)u

_t

+ X

3 i=1

A

_i

(t, x)u

_xi

= F (t, x), M (t, x

^′

)u |

x1=0

= g(t, x

^′

),

u |

t=0

= u

₀

,

where x = (x

1

, x

^′

) and we assume that Ω = {x ∈ R

³

: x

1

> 0 }, ∂Ω = {x ∈ R

³

: x

1

= 0 }. In part (c) we shall obtain results for a bounded domain Ω, using a partition of unity.

Lemma 4.1. (1) Let E, A

i

, i = 1, . . . , 3, be symmetric matrices and (4.2) Eu · u ≥ α

⁰

u

²

, α

0

> 0.

Let n be the unit outward vector normal to ∂Ω and assume −A

^n¯

= A

1

has eigenvalues λ

µ

, where λ

⁺_µ

, µ = 1, . . . , k, and λ

⁻_µ

, µ = k + 1, . . . , m, are respectively the positive and negative ones. Suppose that

(4.3) min

µ

min

Ω^T

|λ

µ

| ≥ c

0

> 0 and

(4.4) max

ν∈{1,...,k}

max

∂Ω^T

λ

⁺_ν

(x

^′

, t) ≤ c

¹

, where c

0

, c

1

are constants.

(2) Let γ

_µ⁺

, γ

_µ⁻

be orthonormal eigenvectors of the matrix −A

n¯

, corre-

sponding to the eigenvalues λ

⁺_µ

, λ

⁻_µ

. Assume that the matrix M (t, x

^′

) has

the form

M = X

k µ,ν=1

α

_µν

(t, x

^′

)γ

_µ⁺

(t, x

^′

)γ

_ν⁺

(t, x

^′

) (4.5)

+ X

k µ=1

X

m ν=k+1

β

µν

(t, x

^′

)γ

_µ⁺

(t, x

^′

)γ

_ν⁻

(t, x

^′

) where

(a) max

∂Ω^T

|α

⁻¹µν

(t, x

^′

) | ≤ δ

⁻¹0

, (b) max

∂Ω^T

|β

µν

(t, x

^′

) | ≤ β

0

, (4.6)

(c) (c

0

+ c

1

)δ

₀⁻²

β

₀²

≤

¹2

c

0

, and δ

0

, β

0

are constants.

(3) Let e L = (E, A

1

, A

2

, A

3

), e L, M ∈ Π

0³

(Ω

^T

) and suppose that α satisfies (4.7) |e L |

3,0,∞,Ω^T

< αα

0

/2

and

(4.8) sup

Ω^T

|E| ≤ c

2

, where c

₂

is a constant.

Then, for every u ∈ C

^∞

(Ω

^T

) and t ≤ T we have the estimate (4.9) α

0

e

^−2αt

\

Ω

u

²

dx + αα

0

2

\

Ω^t

u

²

e

^−2αs

dx ds + α

0

2

\

∂Ω^t

u

²

e

^−2αs

dx

^′

ds

≤ (c

0

+ c

₁

)δ

₀⁻²

\

∂Ω^t

|Mu|

²

e

^−2αs

dx

^′

ds

+ 2 αα

0

\

Ω^t

|Lu|

²

e

^−2αs

dx ds + c

₂

\

Ω

u

²

dx   

_t=0

. P r o o f. Multiplying (4.1)

1

by ue

^−2αt

and integrating by parts in Ω, we obtain

(4.10) d dt e

^−2αt

\

Ω

Eu

²

dx + 2αe

^−2αt

\

Ω

Eu

²

dx + e

^−2αt

\

∂Ω

A

_n

u

²

dx

^′

− e

^−2αt

\

Ω

 X

³

i=1

A

i,x_i

+ E

t

 u

²

dx − 2e

^−2αt

\

Ω

Lu · u dx = 0.

Integrating (4.10) from 0 to t, using (4.2) and (4.8) we get (4.11) α

0

e

^−2αt

\

Ω

u

²

dx + 2αα

0

\

Ω^t

u

²

e

^−2αs

dx ds +

\

∂Ω^t

A

n

u

²

e

^−2αs

dx

^′

ds

≤

\

Ω^t

 X

³

i=1

A

_i,xi

+ E

_s



u

²

e

^−2αs

dx ds

+ 2

\

Ω^t

Lu · ue

^−2αs

dx ds + c

₂

\

Ω

u

²

dx   

t=0

. From (4.7) we get

max

Ω^t

 |E

t

| + X

3 i=1

|A

i,xi

| 

≤ 2c|e L |

3,0,∞,Ω^T

≤ αα

0

so using the Young inequality (with ε = p

2/(αα

0

)) in (4.11) we have (4.12) α

0

e

^−2αt

\

Ω

u

²

dx + αα

0

2

\

Ω^t

u

²

e

^−2αs

dx ds +

\

∂Ω^t

A

n

u

²

e

^−2αs

dx

^′

ds

≤ 2 αα

₀

\

Ω^t

|Lu|

²

e

^−2αs

dx ds + c

2

\

Ω

u

²

dx   

_t=0

.

We have to consider the boundary term. From (2), u =

X

k µ=1

c

µ

γ

_µ⁺

+ X

m µ=k+1

c

µ

γ

⁻_µ

= u

^′

+ u

^′′

where c

µ

= uγ

µ

,

u

^′

= X

k µ=1

c

_µ

γ

_µ⁺

, so |u

^′

|

²

= X

k µ=1

c

²_µ

, |u

^′′

|

²

= X

m µ=k+1

c

²_µ

,

and

−A

n¯

u

²

= X

k µ=1

λ

⁺_µ

c

²_µ

+ X

n µ=k+1

λ

⁻_µ

c

²_µ

.

Using this and (4.3), (4.4) we get, from (4.12), (4.13) α

0

e

^−2αt

\

Ω

u

²

dx + αα

₀

2

\

Ω^t

u

²

e

^−2αs

dx ds + c

0

\

∂Ω^t

|u

^′′

|

²

e

^−2αs

dx

^′

ds

≤ c

1

\

∂Ω^t

|u

^′

|

²

e

^−2αs

dx

^′

ds

+ 2 αα

0

\

Ω^t

|Lu|

²

e

^−2αs

dx ds + c

2

\

Ω

u

²

dx

 



t=0

.

Now, to express |u

^′

|

²

= P

k

µ=1

c

²_µ

by |Mu|

²

, we consider M u =

X

k µ,ν=1

α

µν

γ

_µ⁺

c

⁺_ν

+ X

k µ=1

X

m ν^′=k+1

β

µν^′

γ

_µ⁺

c

⁻_ν′

= X

k µ=1

g

µ

γ

_µ⁺

so

g

µ

= X

k ν=1

α

µν

c

⁺_ν

+ X

m ν^′=k+1

β

µν^′

c

⁻_ν′

and this implies c

⁺_ν

=

X

k µ=1

α

⁻¹_µν

g

_µ

− X

m ν^′=k+1

X

k µ=1

α

⁻¹_µν

β

_µν^′

c

⁻_ν′

. Adding c

0

T

∂Ω^t

|u

^′

|

²

e

^−2αs

dx

^′

ds, using (4.6) and the last expression, we ob- tain

(4.14) α

0

e

^−2αt

\

Ω

u

²

dx + αα

0

2

\

Ω^t

u

²

e

^−2αs

dx ds + c

0

\

∂Ω^t

u

²

e

^−2αs

dx

^′

ds

≤ (c

0

+ c

₁

)δ

₀⁻²

\

∂Ω^t

|Mu|

²

e

^−2αs

dx

^′

ds

+ 2 αα

0

\

Ω^t

|Lu|

²

e

^−2αs

dx ds

+ (c

₀

+ c

₁

)δ

₀⁻²

β

₀²

\

∂Ω^t

|u

^′′

|

²

e

^−2αs

dx

^′

ds + c

₂

\

Ω

u

²

dx   

t=0

. Finally, from (4.14) and (4.6)(c) we have (4.9).

To prove the existence of solutions to (4.1) we have to split it into a Cauchy problem and a boundary value problem. Let χ ∈ C

0^∞

( −δ, δ); we assume that a solution of (4.1) has the form u = χu

1

+ u

2

, where

(4.15) Lu

1

= 0, u

1

|

^t=0

= u

0

, and

(4.16) Lu

2

= F − Eu

¹

∂

∂t χ, M u

2

|

^∂Ω

= g, u

2

|

^t=0

= 0.

Further, introducing w

1

= u

1

− e u

0

(where u e

0

denotes an extension of u

0

to the half-space t > 0) we get, from (4.15),

(4.17) Lw

1

= −Le u

0

, w

1

|

t=0

= 0.

We define the formally adjoint operator L

^(∗)

by (4.18) L

^(∗)

= −E∂

t

−

X

3 i=1

A

i

∂

xi

− E

t

− X

3 i=1

A

i,xi

so we have the identity

(4.19) (Lw

1

, v

1

)

_ΩT

= (w

1

, L

^(∗)

v

1

)

_ΩT

for all w

1

, v

1

∈ C

0^∞

(Ω

^T

) with w

1

|

^t=0

= 0 and v

1

|

^t=T

= 0.

Next, for such w

1

, v

1

we obtain by (4.9) the following estimates:

(4.20) α

0

e

^−2αt

\

Ω

w

²₁

dx + αα

0

2

\

Ω^t

w

²₁

e

^−2αs

dx ds

≤ 2 αα

0

\

Ω^t

|Lw

¹

|

²

e

^−2αs

dx ds and (with time travelling backward)

(4.21) α

₀

e

^2αt

\

Ω

v

₁²

dx + αα

₀

2

\

Ω^t

v

₁²

e

^2αs

dx ds ≤ 2 αα

₀

\

Ω^t

|L

^(∗)

v

₁

|

²

e

^2αs

dx ds.

Now we use the following (see [3]).

Theorem 4.1. Let L denote the space of square integrable functions on Ω

^T

, D

_L

the domain of L consisting of u ∈ C

^∞

(Ω

^T

∪ ∂Ω

^T

) which satisfy the boundary (initial) condition, and D

_L(∗)

the domain of L

^(∗)

of those v ∈ C

^∞

(Ω

^T

∪ ∂Ω

^T

) which satisfy the adjoint boundary (initial) condition.

If there exists a constant c such that

c kuk ≤ kLuk, c kvk ≤ kL

^(∗)

v k,

for u ∈ D

L^¯

and v ∈ D

L^¯(∗)

, then L and L

^(∗)

map their domains one-to-one onto L.

From this theorem and inequalities (4.20), (4.21) we get:

Lemma 4.2. There exists a unique weak solution u

1

of (4.15) such that w

₁

∈ L

^◦2,α

(Ω

^T

).

Now we are looking for solutions of problem (4.16). For the adjoint L

^(∗)

we obtain the identity

(4.22) (Lu, v) = (u, L

^∗

v) + (A

n¯

u, v) = (u, L

^(∗)

v) − (A

¹

u, v) where u, v ∈ C

0¹

(Ω × R).

We can find the boundary matrix M

^∗

for the adjoint problem from (A

₁

u, v) = 0 for u ∈ ker M and v ∈ ker M

^∗

(see [10], [11]). Let F, g = 0 for t < 0 and t > T . Then we consider (4.16) in Ω × R and we can prove, similarly to (4.9),

(4.23) αα

0

2

\

Ω×R

u

²₂

e

^−2αs

dx ds + c

0

2

\

∂Ω×R

u

²₂

e

^−2αs

dx

^′

ds

≤ (c

0

+ c

1

)δ

₀⁻²

\

∂Ω×R

|Mu

2

|

²

e

^−2αs

dx

^′

ds + 2 αα

0

\

Ω×R

|Lu

2

|

²

e

^−2αt

dx ds.

We have to obtain an estimate for the adjoint problem. If we take L

^(∗)

, M

^∗

instead of L, M and we assume that the time is travelling backward, then we can prove (in the same way as Lemma 4.1):

Lemma 4.3. Assume that (1) and (3) of Lemma 4.1 hold. Let M

^∗

=

X

m µ,ν=k+1

α

^∗_µν

(t, x

^′

)γ

_µ⁻

(t, x)γ

_ν⁻

(t, x

^′

)

+ X

m µ=k+1

X

k ν=1

β

_µν^∗

(t, x

^′

)γ

_µ⁻

(t, x

^′

)γ

⁺_ν

(t, x

^′

), with

(4.24) max

∂Ω^T

|α

^∗−1µν

| ≤ δ

0⁻¹

, max

∂Ω^T

|β

µν^∗

| ≤ γ

⁰

, (c

0

+ c

4

)δ

₀⁻²

γ

₀²

≤ c

⁰

/2.

Moreover , let

(4.25) max

ν∈{1,...,m}

max

∂Ω×R

|λ

⁻ν

(x

^′

, t) | ≤ c

4

. Then for v

2

∈ C

0^∞

(Ω × R) ∩ L

2,−α

(Ω × R) we obtain (4.26) αα

₀

2

\

Ω×R

v

₂²

e

^2αs

dx ds + c

₀

2

\

∂Ω×R

v

₂²

e

^2αs

dx

^′

ds

≤ (c

0

+ c

₄

)δ

₀⁻²

\

∂Ω×R

|M

^∗

v

₂

|

²

e

^2αs

dx

^′

ds + 2 αα

0

\

Ω×R

|L

^∗

v

₂

|

²

e

^2αs

dx ds.

Now, by (4.23), (4.26) and Theorem 4.1 we have

Lemma 4.4. Let g ∈ L

^2,α

(∂Ω × R) with g|

^t<0

= g |

^t>T

= 0 and F ∈ L

2,α

(Ω × R) with F |

t<0

= F |

t>T

= 0. Let the assumptions of Lemmas 4.1 and 4.3 be satisfied. Then there exists a unique solution u

₂

∈ L

2,α

(Ω × R) of (4.16) such that u

2

|

^∂Ω

∈ L

^2,α

(∂Ω × R).

In Lemmas 4.2 and 4.4 we can obtain strong solutions, using the tech- nique of mollifiers (see [6], [12]) with respect to (x

^′

, t) where x = (x

1

, x

^′

).

Then we have the sequence u

ε

= J

ε

u = j

ε

∗u (the operator J

ε

is the mollifier) and from the properties of J

ε

we have the convergences

u

ε

→ u in L

2

(Ω

^T

), Lu

ε

→ Lu = F in L

²

(Ω

^T

), M u

ε

→ Mu = g in L

²

(∂Ω

^T

), and u

ε

is continuous up to the boundary.

Now for u = χu

1

+ u

2

we formulate

Theorem 4.2. Let u

₀

∈ H

¹

(Ω), u

₀

|

∂Ω

= 0, e L ∈ H

α¹

(Ω

^T

) and F ∈

L

2,α

(Ω

^T

), g ∈ L

^2,α

(∂Ω

^T

). Let the assumptions of Lemmas 4.1 and 4.3 be

satisfied. Then there exists a unique strong solution u of problem (4.1) in the half-space Ω such that u ∈ L

^2,α

(Ω

^T

) ∩L

^2,α

(∂Ω

^T

) ∩L

∞

(0, T ; Γ

₀¹

(Ω)) and (4.9) holds.

4(b) Regularity of solutions. To prove the existence of solutions of (3.1) we have to use the method of successive approximations; so we need more regular solutions of (4.1) such that u ∈ H

³

(Ω

^T

). Since u ∈ L

2,α

(Ω

^T

) we have to use mollifiers to derive the regularity of u. Let u

_δ

= j

_δ

∗ u = J

δ

u, where j(t, x) ∈ C

0^∞

(R

¹

× R

ⁿ

), j ≥ 0,

T

j(t, x) dx dt = 1 and j

δ

(t, x) = δ

⁻ⁿ⁻¹

j(t/δ, x/δ). We consider the problems

LD

_t,x^s ′

u

δ

= D

^s_t,x′

Lu

δ

+ (LD

^s_t,x′

u

δ

− D

^st,x^′

Lu

δ

),

M D

^s_t,x′

u

δ

|

^x1=0

= D

^s_t,x′

M u

δ

+ (M D

_t,x^s ′

u

δ

− D

t,x^{s ′}

M u

δ

), (4.27)

D

_t,x^s ′

u

δ

|

t=0

= D

_t,x^s ′

u

δ

|

t=0

, for s = 1, 2, 3, where

D

_t,x^s ′

u = X

|γ|=s

∂

^γ0

∂t

^γ0

∂

^γ2

∂x

^γ₂²

∂

^γ3

∂x

^γ₃³

u, |γ| = γ

⁰

+ γ

2

+ γ

3

, Lu

δ

= (Lu)

δ

− [(Lu)

δ

− Lu

δ

]

= (Lu)

_δ

− C

δ

u (C

_δ

u is called the commutator).

Lemma 4.5. Assume that (1)–(3) of Lemma 4.1 hold, M ∈ H

α³

(∂Ω

^T

), g ∈ H

α³

(∂Ω

^T

), u

0

∈ H

³

(Ω) and F ∈ H

α³

(Ω

^T

). Set

a = |e L |

^3,0,∞,Ωt

, b = |M|

3,0,∞,Ω^T

+ kMk

^3,∂Ωt,α

for t ≤ T . Then there exist polynomials p

0

(a, b), p

s

(a, b), q

s

(a, b), 1 ≤ s ≤ 3, such that the solution of problem (4.1) satisfies the following estimate:

(4.28) α

0

|u|

²s,0,Ω

e

^−2αt

+ αα

0

4 kuk

²s,Ω^t,α

+ c

0

2 kuk

²s,∂,Ω^t,α

≤ p

s

(a, b)[ |Lu|

²s,Ω^t,α

+ |Lu|

²s−1,0,Ω

|

t=0

] + q

_s

(a, b) kMuk

²s,∂Ω^t,α

+ c

₂

|u|

²s,0,Ω

|

t=0

where

(4.29) α satisfies p

0

(a, b) ≤ αα

⁰

. Moreover , there exist polynomials r such that

(4.30) |u|

²s,0,Ω

|

^t=0

≤ r(|e L |

^s−1,0,Ω

|

^t=0

, |Lu|

^s−1,0,Ω

|

^t=0

, ku

⁰

k

^s,2,Ω

).

P r o o f. For |s| = 1 and problem (4.27) we have by (4.9),