An Existence Result for a Model of Granular Material with Non-constant Density

Piotr Gwiazda ^{∗}

University of Technology Darmstadt (Germany)

e-mali: gwiazda@mathematik.tu-darmstadt.de

Abstract

We consider a model developed by Savage and Hutter [12], [13] which describes the flow of granular avalanches down a smoothly varying slope. The system consists of two nonstrictly hyperbolic equations for height and momentum. The existence of entropy solutions to this model is proved using the vanishing viscosity method, where we make extensive use of a generalised version of the invariant region theorem in order to prove a priori estimates. Since the model has a discontinuous source term, a new definition of entropy solution must be introduced.

AMS classification: 35L65, 35Q72, 73Q05

1 Introduction

The problem which we study can be written in the following form: For an avalanche down an inclined slope find the height h : ℝ_{+} × ℝ ⟶ ℝ, the density ρ : ℝ_{+} × ℝ ⟶ ℝ and the velocity v : ℝ_{+} × ℝ ⟶ ℝ satisfying the following system of equations in the form of conservation laws

\begin{aligned}\frac{∂}{∂t}(ρh) + \frac{∂}{∂x}(ρhv) & = & 0 \\ \frac{∂}{∂t}(ρhv) + \frac{∂}{∂x} (ρhv^{2} + \frac{1}{2}βρh^{2}) & = & ρhg, \end{aligned}  (SH)

where β := β(x) and g := g(v, x) are given functions. Since these two conservation laws are not enough for an investigation of the evolution containing three variables (ρ, h, v), we need an additional constitutive relation. One of the methods is to assume that ρ is a function of h and v. In this case we obtain the system of two differential equations for two independent variables:

∗Supported by Sonderforschungsbereich 298.

\begin{aligned}\frac{∂}{∂t}(ρ(h, v)h) + \frac{∂}{∂x}(ρ(h, v)hv) & = & 0 \\ \frac{∂}{∂t}(ρ(h, v)hv) + \frac{∂}{∂x} (ρ(h, v)hv^{2} + \frac{1}{2}βρ(h, v)h^{2}) & = & ρ(h, v)hg. \end{aligned}

Since it is very difficult to examine the last system with the general constitutive relation ρ := ρ(h, v), we will start with the very special (from the mathematical point of view) relation ρ := h^{α} with 0 ≤ α, which will simplify our considerations very much^{1}. Moreover, we assume that the functions β(x), g(v, x) are defined by

β(x) = k cos(γ(x)),
g(v, x) = sin(γ(x)) − sig_{0}(v)cos(γ(x)) tan δ_{F} (x),

where tan δ_{F} (x) is the dynamic friction angle at the position x and

sig_{0}(v) = \{ \begin{matrix}0 & for v = 0 \\ \frac{v}{|v|} & for v ≠ 0. \end{matrix}

Here −\frac{π}{2} < γ(x) < \frac{π}{2} is an angle between the horizontal direction and the tangent to the base curve. In general the Savage-Hutter model defined in [12], [13] allows for a function β with a constant k switching between two positive states depending on whether \frac{∂}{∂x}v is positive or negative. The first problem is, the discontinuity of the function g at v = 0. In fact we do not need the continuity of g, but the maximal monotonicity of −g with respect to the variable v. Thus we have to extend the function g to a multifunction

g̃(v, x) = sin(γ(x)) − sig(v)cos(γ(x))δ_{F} (x),

where

sig(v) = \{ \begin{matrix}[−1, 1] & for v = 0 \\ \frac{v}{|v|} & for v ≠ 0. \end{matrix}

Now we introduce a new variable H = ρh = h^{1+α} and rewrite the system (SH) as

\begin{aligned}\frac{∂}{∂t}(H) + \frac{∂}{∂x}(Hv) & = & 0 \\ \frac{∂}{∂t}(Hv) + \frac{∂}{∂x} (Hv^{2} + \frac{1}{2}βH^{\frac{2+α}{1+α}} ) & ∈ & Hg̃(v, x). \end{aligned}

A further problem is the flux term

F (x, H, v) = ( \begin{matrix}Hv \\ Hv^{2} + \frac{1}{2}β(x)H^{\frac{2+α}{1+α}} \end{matrix} )

1Up to now this system has been examined only with the very restrictive constitutive relation ρ := ρ_{0} where ρ_{0} is constant [1], [2].

containing β which is a function of x. For future considerations we need a flux term in homogeneous form, i.e., without x-dependence. Therefore we choose the new variable u defined by u = (u_{1}, u_{2}) = ((\frac{β}{2κ})^{1+α}H, (\frac{β}{2κ})^{1+α}Hv). Here the constant κ = \frac{1}{4} \frac{1}{(1+α)(2+α)} is introduced in consistency with the standard notation from [6], [7]. We obtain:

\frac{∂}{∂t}u + \frac{∂}{∂x}F (u) ∈ G̃(u, x),  (CL)

where

F (u_{1}, u_{2}) = ( \begin{matrix}u_{2} \\ \frac{(u_{2})^{2}}{u_{1}} + κ(u_{1})^{\frac{2+α}{1+α}} \end{matrix} )

and

G̃(u_{1}, u_{2}, x) = ( \begin{matrix}\frac{β′(2κ)^{α−1}}{β^{α−1}} u_{2} \\ \frac{β′(2κ)^{α−1}}{β^{α−1}} (\frac{(u_{2})^{2}}{u_{1}} + κ(u_{1})^{\frac{2+α}{1+α}} ) + u_{1}g̃ (\frac{u_{2}}{u_{1}} , x) \end{matrix} ) .

It is well known that some classes of “very weak” solutions for conservation laws can not have physical meaning, for example the class of “measure valued solutions” [4]. First we must define a proper class of solutions we are looking for. This class contains “the weak entropy solutions” defined below.

Definition 1.1 Suppose that η = η(u_{1}, u_{2}), q = q(u_{1}, u_{2}) are scalar ℂ^{1}-functions satisfying

∇_{(u_{1},u_{2})}η(u_{1}, u_{2}) · ∇_{(u_{1},u_{2})}F (u_{1}, u_{2}) = ∇_{(u_{1},u_{2})}q(u_{1}, u_{2}).

Such functions (η, q) are called Entropy-Flux Pairs. If η is convex, then (η, q) is called a Convex Entropy-Flux Pair.

Definition 1.2 We call u ∈ 𝕃^{∞}([0, T) × ℝ; ℝ_{+} × ℝ) a Weak Entropy Solution to the system (CL) with the initial data u^{0} ∈ ℂ^{0}(ℝ; ℝ_{+} × ℝ) iff:

1. there exists G(t, x) ∈ G̃(u(t, x), x) for a.a. (t, x) ∈ [0, T) × ℝ with G ∈ 𝕃_{loc}^{∞}([0, T) × ℝ; ℝ^{2}),

2. u is a weak solution, i.e.;

\begin{matrix}∫_{ℝ×[0,T)} [u(t, x) · \frac{∂}{∂t}ψ(t, x) + F (u(t, x)) · \frac{∂}{∂x}ψ(t, x) \\ +G(t, x) · ψ(t, x)] dtdx & = & ∫_{ℝ} u^{0}(x) · ψ(0, x)dx, \end{matrix}

for all test functions ψ ∈ ℂ_{0}^{1}([0, T) × ℝ; ℝ^{2}),

3. the entropy inequality

\begin{matrix}∫_{ℝ×[0,T)} [η(u(t, x))\frac{∂}{∂t}ϕ(t, x) + q(u(t, x))\frac{∂}{∂x}ϕ(t, x) \\ +∇_{u}η(u(t, x)) · G(t, x)ϕ(t, x)] dtdx & ≥ & ∫_{ℝ} η(u^{0}(x))ϕ(0, x)dx \end{matrix}

holds for all non-negative test functions ϕ ∈ ℂ_{0}^{1}([0, T) × ℝ; ℝ) and all convex weak entropy-flux pairs (η, q). Here the family “convex weak entropy” is the class where ∇_{u}η(u) ∈ 𝕃_{loc}^{∞}([0, T) × ℝ; ℝ), η(·, ·) is ℂ^{2}-function for (u_{1}, u_{2}) ∈ (0, ∞) × ℝ and η(0, ·) = 0.

Remark The last definition is not standard because of the differential inclusion appearing in the problem. More about the definition of the proper class of “convex weak entropies” see Chapter 4, step 2 below.

Main results:

Theorem 1.1.a (Local in time existence) Assume an initial data u^{0} = (u_{1}^{0}, u_{2}^{0}) ∈ ℂ^{2}(ℝ; ℝ^{2}), (u_{1}^{0} − \overline{u}_{1}, u_{2}^{0}) ∈ 𝕃^{2}(ℝ; ℝ^{2}) such that

1. inf_{x∈ℝ}u_{1}^{0}(x) > 0, lim_{|x|→∞} u_{1}^{0}(x) = \overline{u}_{1}

2. lim_{|x|→∞} u_{2}^{0}(x) = 0

where \overline{u}_{1} is a positive constant, and β ∈ ℂ^{3}(ℝ) with β(x) ≥ β_{0} > 0. Then the equation (CL) possesses a local in time (i.e. T = T_{max}) weak entropy solution in sense of Definition 1.2.

Theorem 1.1.b (Global in time existence) Assume an initial data u^{0} = (u_{1}^{0}, u_{2}^{0}) ∈ ℂ^{2}(ℝ; ℝ^{2}), (u_{1}^{0} − \overline{u}_{1}, u_{2}^{0}) ∈ 𝕃^{2}(ℝ; ℝ^{2}) such that

1. inf_{x∈ℝ}u_{1}^{0}(x) > 0, lim_{|x|→∞} u_{1}^{0}(x) = \overline{u}_{1}

2. lim_{|x|→∞} u_{2}^{0}(x) = 0

where \overline{u}_{1} is a positive constant and β = β_{const} > 0. Then the equation (CL) possesses a global in time weak entropy solution (i.e. T = ∞) in sense of Definition 1.2.

2 The Viscous Approximation - Existence and Estimates

Our idea is to apply the vanishing viscosity method by adding a second order elliptic term to the first order conservation law. For ε, λ > 0 let us define the viscous, regularized system

\frac{∂}{∂t}u + \frac{∂}{∂x}F (u) = G_{λ}(u, x) + ε\frac{∂^{2}}{∂x^{2}} u,  (V P)

with initial data

u^{0} = (u_{1}^{0}, u_{2}^{0}) .

Here we introduced a smooth function G_{λ} defined by

G_{λ}(u_{1}, u_{2}, x) = Λ (λx) ( \begin{matrix}\frac{β′(2κ)^{α−1}}{β^{α−1}} u_{2} \\ \frac{β′(2κ)^{α−1}}{β^{α−1}} (\frac{(u_{2})^{2}}{u_{1}} + κu1^{\frac{2+α}{1+α}} ) + u_{1}Λ (\frac{u_{2}}{λu_{1}} ) g (\frac{u_{2}}{u_{1}} , x) \end{matrix} ) .

Furthermore Λ is a symmetric, smooth, monotonically decreasing function for positive arguments with Λ(0) = 1, Λ(2) = 0, and suppΛ′ ⊂ [−2, −1] ∪ [1, 2].

Theorem 2.1 Assume an initial data u^{0} = (u_{1}^{0}, u_{2}^{0}) ∈ ℂ^{2}(ℝ) such that

1. inf_{x∈ℝ}u_{1}^{0}(x) > 0, lim_{|x|→∞} u_{1}^{0}(x) = \overline{u}_{1}

2. lim_{|x|→∞} u_{2}^{0}(x) = 0

where \overline{u}_{1} is a positive constant. Moreover let λ, ε > 0 and β ∈ ℂ^{3}(ℝ) with β(x) ≥ β_{0} > 0. Then the equation (VP) possesses a local in time classical solution, i.e., u ∈ ℂ^{0}([0, T_{loc}); ℂ^{2}(ℝ)), \frac{∂}{∂t}u ∈ C^{0}([0, T_{loc}) × ℝ) where T_{loc} = T_{loc}(ε, λ); moreover for all t ∈ [0, T_{loc})

1. inf_{x∈ℝ}u_{1}(t, x) > 0, lim_{|x|→∞} u_{1}(t, x) = \overline{u}_{1}

2. lim_{|x|→∞} u_{2}(t, x) = 0

holds.

Proof Using the Gaussian kernel

Θ(t, x) = \frac{1}{\sqrt{4πεt}}exp (\frac{−x^{2}}{4εt} )

of the heat equation we rewrite (VP) in the integral form

\begin{aligned}u(t, x) & = & ∫_{ℝ} Θ(x − y, t)u^{0}(y)dy + ∫_{0}^{t} ∫_{ℝ} \frac{∂}{∂x}Θ(x − y, t − τ)F (u(y, τ))dy dτ \\ & & + ∫_{0}^{t} ∫_{ℝ} Θ(x − y, t − τ)G_{λ}(u(y, τ), y)dy dτ. \end{aligned} (IE)

Note that

|\frac{∂}{∂x}Θ(t, x)| ≤ \frac{c}{εt}exp (\frac{−x^{2}}{2εt} )

and

∫_{0}^{t} ∫_{ℝ} |\frac{∂}{∂x}Θ(x − y, t − τ)| dydτ ≤ c \sqrt{\frac{t}{ε}}.

Defining the bounded set

U = conv{u^{0}(x)|x ∈ ℝ},

and the compact set

U_{δ} = {y ∈ ℝ^{2} | dist(y, U) ≤ δ},

we can choose δ > 0 small enough such that U ⊂ U_{δ} ⊂ (0, ∞) × ℝ. The functions F and G_{λ}, are locally lipschitz on the set (0, ∞) × ℝ and global lipschitz on U_{δ}. To solve (IE) we define the sequence u^{(1)}, u^{(2)}, ... by

u^{(1)}(t, x) = ∫_{ℝ} Θ(x − y, t)u^{0}(y)dy

and

\begin{aligned}u^{(n+1)}(t, x) & = & ∫_{ℝ} Θ(x − y, t)u^{0}(y)dy + ∫_{0}^{t} ∫_{ℝ} \frac{∂}{∂x}Θ(x − y, t − τ)F (u^{(n)}(y, τ))dy dτ \\ & & + ∫_{0}^{t} ∫_{ℝ} Θ(x − y, t − τ)G_{λ}(u^{(n)}(y, τ), y)dy dτ. \end{aligned}

Obviously u^{(1)}(t, x) ∈ U for all (t, x) ∈ ℝ_{+} × ℝ. It is also easy to observe that there exists a sufficiently small time T_{loc} such that

∑_{n=1}^{∞} ∥u^{(n+1)} − u^{(n)}∥_{ℂ^{0}([0,T_{loc})×ℝ)} ≤ δ.

Then lim_{n→∞} u^{(n)}(t, x) = u(t, x) for all (t, x) ∈ [0, T_{loc}) × ℝ, where u(t, x) is a solution to (IE), and thus a “mild solution” to (VP). Moreover u(t, x) ∈ U_{δ} for all (t, x) ∈ ℝ_{+} × ℝ. Note that F, G_{λ} ∈ ℂ^{2}(U_{δ}). Then regularity of the solution (it means u ∈ ℂ^{0}((0, T_{loc}); ℂ^{2}(ℝ))) will be obtained by examining difference quotients of first and second order (see also [11]).

Remark It is easy to observe that the norm sup_{x∈ℝ}|u(t, x)| + sup_{x∈ℝ}|\frac{∂}{∂x}u(t, x)| + sup_{x∈ℝ}|\frac{∂^{2}}{∂x^{2}} u(t, x)| can

blow-up at time T if and only if lim_{t→T} inf_{x∈ℝ}u_{1}(t, x) = 0 or the norm ∥u_{1}(t)∥_{𝕃^{∞}(ℝ)} +

∥\frac{u_{2}}{u_{1}} (t)∥_{𝕃^{∞}(ℝ)} will blow-up at time T.

The last remark gives us a hint to prove an apriori estimate of the solution u(t).

Definition 2.1 Let R_{i} : ℝ^{2} → ℝ, i = 1, ..., k, and M_{i} : [0, T) → ℝ be smooth functions. We define the region

Σ(t) = ⋂_{i=1}^{k}{u ∈ ℝ^{2} : R_{i}(u) ≤ M_{i}(t)}.  (IR)

If u : [0, T) × ℝ → ℝ^{2}, we call Σ(t) a one parameter family of invariant regions for u iff u(x, 0) ∈ Σ(0) implies that u(t, x) ∈ Σ(t) for t > 0.

Lemma 2.1 Let u : [0, T) × ℝ → (0, ∞) × ℝ be a classical solution of the system

\frac{∂}{∂t}u + ∇_{u}F (u)\frac{∂}{∂x}u = G_{λ}(u, x) + ε\frac{∂^{2}}{∂x^{2}} u

and let Σ(t) be defined by (IR). Suppose that the following conditions hold:

1. R_{i}(u(0, x)) ≤ M_{i}(0)

2. ∇R_{i}(a) is a left eigenvector of ∇F (a) for all a ∈ ∂Σ_{i}(t)

3. Σ(t) is a convex set for all t ∈ [0, T)

4. sup_{x∈ℝ} sup_{a∈∂Σ_{i}(t)} ∇R_{i}(a) · G_{λ}(a, x) ≤ M′_{i}(t)

where ∂Σ_{i}(t) = ∂Σ(t) ∩ {a ∈ ℝ^{2} | R_{i}(a) = M_{i}(t)}.

Then Σ(t) is a one parameter family of invariant regions for u.

Proof cf. [1], [10].

Theorem 2.2.a (Local in time estimate) Assume that u(t) is the local in time classical solution to the problem (VP) with ∥u_{1}^{0}∥_{𝕃^{∞}(ℝ)} + ∥\frac{u_{2}^{0}}{u_{1}^{0}} ∥_{𝕃^{∞}(ℝ)} ≤ C_{1}, ∥β′∥_{𝕃^{∞}(ℝ)} ≤ C_{2} and β(x) ≥ β_{0} > 0. Then there exists a time T_{max} and a function C : [0, T_{max}) → ℝ_{+} depending only on C_{1}, C_{2} such that ∥u_{1}∥_{𝕃^{∞}([0,T)×ℝ)} + ∥\frac{u_{2}}{u_{1}} ∥_{𝕃^{∞}([0,T)×ℝ)} ≤ C(T) for all T ∈ [0, T_{max}).

Proof We apply Lemma 2.1 to the so-called Riemann invariants R_{1} = \frac{u_{2}}{u_{1}} + (u_{1})^{\frac{1}{2(1+α)}} ,

R_{2} = −\frac{u_{2}}{u_{1}} + (u_{1})^{\frac{1}{2(1+α)}} ; moreover let

M_{1}(0) = M_{2}(0) = max \{sup_{x∈ℝ}[\frac{u_{2}^{0}(x)}{u_{1}^{0}(x)} + (u_{1}^{0}(x))^{\frac{1}{2(1+α)}} ]; sup_{x∈ℝ}[−\frac{u_{2}^{0}(x)}{u_{1}^{0}(x)} + (u_{1}^{0}(x))^{\frac{1}{2(1+α)}} ]\} .

Then

\frac{∂R_{1}}{∂u_{1}} = −\frac{u_{2}}{(u_{1})^{2}} + \frac{1}{2(1 + α)}(u_{1})^{\frac{−(1+2α)}{2(1+α)}} , \frac{∂R_{1}}{∂u_{2}} = \frac{1}{u_{1}}

and

\frac{∂R_{2}}{∂u_{1}} = \frac{u_{2}}{(u_{1})^{2}} + \frac{1}{2(1 + α)}(u_{1})^{\frac{−(1+2α)}{2(1+α)}} , \frac{∂R_{1}}{∂u_{2}} = −\frac{1}{u_{1}} .

Hence for u_{2} ≥ 0

∇R_{1}(u_{1}, u_{2}) · G_{λ}(u_{1}, u_{2}, x) ≤ \frac{β′(2κ)^{α−1}}{β^{α−1}} [−\frac{(u_{2})^{2}}{(u_{1})^{2}} + \frac{u_{2}}{2(1 + α)}(u_{1})^{\frac{−(1+2α)}{2(1+α)}} ]

+\frac{β′(2κ)^{α−1}}{β^{α−1}} (\frac{(u_{2})^{2}}{(u_{1})^{2}} + κu1^{\frac{1}{1+α}} ) + sin γ((x))

≤ \frac{β′(2κ)^{α−1}}{β^{α−1}} [κu1^{\frac{1}{1+α}} + \frac{u_{2}}{2(1 + α)}(u_{1})^{\frac{−(1+2α)}{2(1+α)}} ] + sin γ((x)).

Similarly for u_{2} ≤ 0

∇R_{1}(u_{1}, u_{2}) · G_{λ}(u_{1}, u_{2}, x) ≤ \frac{β′(2κ)^{α−1}}{β^{α−1}} [−κu1^{\frac{1}{1+α}} + \frac{u_{2}}{2(1 + α)}(u_{1})^{\frac{−(1+2α)}{2(1+α)}} ] − sin γ((x)).

Now can observe that the fourth condition of Lemma 2.1 can be satisfied if M(t) = M_{1}(t) = M_{2}(t) is a function such that

\frac{dM(t)}{dt} ≥ C[M(t)]^{2} + sup_{x∈ℝ}| sin(γ(x))|.

Here C is a positive constant depending only on ∥β′∥_{𝕃^{∞}(ℝ)}, β_{0} and α.

Theorem 2.2.b (Global in time estimate) Assume that u(t) is the local in time solution to the problem (VP) with ∥u_{1}^{0}∥_{𝕃^{∞}(ℝ)} + ∥\frac{u_{2}^{0}}{u_{1}^{0}} ∥_{𝕃^{∞}(ℝ)} ≤ C_{1}, ∥ sin(γ)∥_{𝕃^{∞}(ℝ)} ≤ C_{2} and β(x) = β_{const.}. Then there exists a constant C such that ∥u_{1}∥_{𝕃^{∞}([0,T)×ℝ)} + ∥\frac{u_{2}}{u_{1}} ∥_{𝕃^{∞}([0,T)×ℝ)} ≤ C(C_{1} + C_{2}T), for all positive T.

Proof Proof is analogues to the proof of Theorem 2.2.a.

Lemma 2.2 Assume that the equation (VP) possesses a local in time solution u ∈ ℂ([0, T_{loc}); ℂ^{2}(ℝ)) such that

1. lim_{|x|→∞} u_{1}(t, x) = \overline{u}_{1},

2. lim_{|x|→∞} u_{2}(t, x) = 0.

Moreover let us suppose that u_{1}^{0} − \overline{u}_{1}, u_{2}^{0} ∈ 𝕃^{2}(ℝ) and u_{1}^{0} ≥ δ, where δ, \overline{u}_{1} are positive constants. Then the following estimates hold

∥u_{1}(t) − \overline{u}_{1}∥_{𝕃^{2}(ℝ)} + ∥u_{2}(t)∥_{𝕃^{2}(ℝ)} ≤ C_{1}(ε, T_{loc}) for all t ∈ [0, T_{loc}),

∥\frac{∂}{∂x}u∥_{𝕃^{2}([0,T_{loc})×ℝ)} ≤ C_{2}(ε, T_{loc}).

Proof Define w = (w_{1}, w_{2}) = (u_{1} − \overline{u}_{1}, u_{2}). Therefore

\frac{∂}{∂t}w + \frac{∂}{∂x}F (w_{1} + \overline{u}_{1}, w_{2}) = G_{λ}(w_{1} + \overline{u}_{1}, w_{2}, x) + ε\frac{∂^{2}}{∂x^{2}} w.

Multiplying by w and integrating on [0, t] × [−l, l] we obtain

\begin{aligned}\frac{1}{2} [∥w(t)∥_{𝕃^{2}([−l,l])}^{2}]_{0}^{t} − ∫_{0}^{t} ∫_{−l}^{l} ε\frac{∂^{2}}{∂x^{2}} w · w dxdτ & = & ∫_{0}^{t} ∫_{−l}^{l} G_{λ}(w_{1} + \overline{u}_{1}, w_{2}, x) · w dxdτ \\ & & − ∫_{0}^{t} ∫_{−l}^{l} \frac{∂}{∂x}F (w_{1} + \overline{u}_{1}, w_{2}) · w dxdτ. \end{aligned}

Integrating by parts gives

\begin{matrix}\frac{1}{2} [∥w(t)∥_{𝕃^{2}([−l,l])}^{2}]_{0}^{t} & +ε ∥\frac{∂}{∂x}w∥_{𝕃^{2}([0,t)×[−l,l])}^{2} = ∫_{0}^{t} ∫_{−l}^{l} G_{λ}(w_{1} + \overline{u}_{1}, w_{2}, x) · w dxdτ \\ + & ∫_{0}^{t} ∫_{−l}^{l} F (w_{1} + \overline{u}_{1}, w_{2}) · \frac{∂}{∂x}w dxdτ+ ∫_{0}^{t} [\frac{∂}{∂x}w(x, τ) · w(x, τ)]_{−l}^{l} dτ \\ − & ∫_{0}^{t} [F (w_{1}(x, τ) + \overline{u}_{1}, w_{2}(x, τ)) · w(x, τ)]_{−l}^{l} dτ. \end{matrix}

Considering the second term of the right hand side recall that

F (w_{1} + \overline{u}_{1}, w_{2}) · \frac{∂}{∂x}w = w_{2} \frac{∂}{∂x}w_{1} + \frac{∂}{∂x}w_{2} (\frac{(w_{2})^{2}}{w_{1} + \overline{u}} + κ(w_{1} + \overline{u}_{1})^{\frac{2+α}{1+α}} ) .

However the term |κ(w_{1} + \overline{u}_{1})^{\frac{2+α}{1+α}} | can not directly be estimated by C_{1}|w_{1}|. But by the boundedness of w_{1} we obtain

−C|w_{1}| |\frac{∂}{∂x}w_{2}| + (\overline{u}_{1})^{\frac{2+α}{1+α}} \frac{∂}{∂x}w_{2} ≤ (w_{1} + \overline{u}_{1})^{\frac{2+α}{1+α}} \frac{∂}{∂x}w_{2} ≤ C|w_{1}| |\frac{∂}{∂x}w_{2}| + (\overline{u}_{1})^{\frac{2+α}{1+α}} \frac{∂}{∂x}w_{2}.

Hence

\begin{aligned}|∫_{0}^{t} ∫_{−l}^{l} F (w_{1} + \overline{u}_{1}, w_{2}) · \frac{∂}{∂x}w dxdτ| & ≤ & C (∥w_{1}∥_{𝕃^{∞}}, ∥\frac{w_{2}}{w_{1}+\overline{u}_{1}} ∥_{𝕃^{∞}}) ∫_{0}^{t} ∫_{−l}^{l} |\frac{∂}{∂x}w| |w| dxdτ \\ & & + |∫_{0}^{t} [κ\overline{u}1^{\frac{2+α}{1+α}} w_{2}(x, τ)]_{−l}^{l}| . \end{aligned}

By the properties of G_{λ} and the fact that lim_{|x|→∞}(w_{1}(x, τ), w_{2}(x, τ)) = (0, 0) for all

τ ∈ [0, T_{loc}) we can pass to the limit l → ∞ getting rid of the boundary terms. We complete the proof using Gronwall’s lemma.

Lemma 2.3 Assume that u_{1} ∈ ℂ^{0}([0, T_{loc}); ℂ^{2}(ℝ)), u_{2} ∈ ℂ^{0}([0, T_{loc}); ℂ^{1}(ℝ)) is a solution to the equation

\frac{∂}{∂t}u_{1} + \frac{∂}{∂x}u_{2} = Λ(λx)\frac{β′(2κ)^{α−1}}{β^{α−1}} u_{2} + ε\frac{∂^{2}}{∂x^{2}} u_{1},

moreover let inf_{x∈ℝ}u_{1}^{0}(x) > 0, ∥\frac{u_{2}}{u_{1}} ∥_{𝕃^{∞}([0,T_{loc})×ℝ)} + sup_{t∈[0,T_{loc})} ∥u_{1}(t) − \overline{u}_{1}∥_{𝕃^{2}(ℝ)} ≤ C_{1} and

∥Λ(λx)\frac{β′(2κ)^{α−1}}{β^{α−1}} ∥_{𝕃^{∞}(ℝ)} ≤ C_{2}. Then there exists a constant b(ε, T_{loc}) such that

u_{1}(t, x) > b(ε, T_{loc}) > 0

for all t ∈ [0, T_{loc}).

Proof The proof is similar to the proof of Theorem 5.3 from [10], cf. also [1], though we include it for completeness. First let us define r = \frac{1}{e^{at}u_{1}} , where the constant a will be determined later. Then

\frac{∂}{∂t}r + u_{2} \frac{∂}{∂x}r − \frac{∂}{∂x}ur(a + u_{2})r = ε (\frac{∂^{2}}{∂x^{2}} r − 2 (\frac{∂}{∂x}r)^{2} r^{−1}) .  (∗)

To prove Lemma 2.3 we recall a technical lemma, cf. [10]:

Lemma 2.4 If ϕ : [c_{0}, ∞) → [0, ∞) is a nonincreasing function, and if for positive constants K, a and b

ϕ(c_{2}) ≤ Kc_{1}^{a}(c_{2} − c_{1})^{−1}ϕ(c_{1})^{1+b} for all c_{2} > c_{1} ≥ c_{0},

then there exists a c^{∗} < ∞ such that ϕ(c^{∗}) = 0.

We define ξ = max{0, r − c_{1}} for some c_{1} ≥ c_{0} large enough so that ξ(0, x) = 0, and ϕ(c) as the measure |{(t, x) ∈ [0, T_{loc}) × ℝ : r(t, x) > c}|; observe that ϕ(c) < ∞ for all c < \overline{u}_{1}. Multiply (*) by ξ^{3} and integrate over [0, t) × ℝ for t ∈ (0, T_{loc}) to get

\begin{matrix}\frac{1}{4} ∫_{ℝ} ξ(t)dx + (a + γu) ∫_{0}^{t} ∫_{ℝ} (ξ^{4} + c_{1}ξ^{3})dxdτ + \frac{3ε}{4} ∫_{0}^{t} ∫_{ℝ} |(ξ^{2})^{2}|dxdτ = \\ = 2ε ∫_{0}^{t} ∫_{ℝ} \frac{\frac{∂}{∂x}ξ^{2}}{ξ + c_{1}} dxdτ − ∫_{0}^{t} ∫_{ℝ} (5ξ^{3} + 3c_{1}ξ^{2})u_{2} \frac{∂}{∂x}ξdxdτ. \end{matrix}

Let us define C = ∥\frac{u_{2}}{u_{1}} ∥_{𝕃^{∞}([0,T_{loc})×ℝ)}^{2} . Considering the last integral Young’s inequality

yields

− ∫_{0}^{t} ∫_{ℝ} 5ξ^{3}u_{2} \frac{∂}{∂x}ξdxdτ ≤ \frac{ε}{4} ∫_{0}^{t} ∫_{ℝ} |\frac{∂}{∂x}(ξ)^{2}|^{2} dxdτ + \frac{25}{4ε}C ∫_{0}^{t} ∫_{ℝ} ξ^{4}dxdτ

and

\begin{aligned}− ∫_{0}^{t} ∫_{ℝ} 3c_{1}ξ^{2}u_{2} \frac{∂}{∂x}ξdxdτ & ≤ & \frac{ε}{4} ∫_{0}^{t} ∫_{ℝ} |\frac{∂}{∂x}(ξ)^{2}|^{2} dxdτ \\ & & +\frac{9c_{1}^{2}}{4ε} Cϕ(c_{1})^{\frac{3}{2}} ∫_{0}^{t} ∥ξ∥_{𝕃^{6}(ℝ)}^{2}dxdτ. \end{aligned}

Now choose

a = \frac{ε}{4} + \frac{25}{4ε}C + ∥\frac{β′(2κ)^{α−1}}{β^{α−1}} \frac{u_{2}}{u_{1}} ∥_{𝕃^{∞}([0,T_{loc})×ℝ)} .

We thus derive

∫_{ℝ} ξ(t)dx + ∫_{0}^{t} ∫_{ℝ} (ξ^{4} + |\frac{∂}{∂x} (ξ^{2})|^{2}) dxdτ ≤ \frac{9c_{1}^{2}}{ε} Cϕ(c_{1})^{\frac{3}{2}} ∫_{0}^{t} ∥ξ∥_{𝕃^{6}(ℝ)}^{2}dxdτ.

Combining Sobolev’s embedding theorem ∥ξ∥_{𝕃^{6}(ℝ)}^{2} ≤ C∥ξ∥_{ℍ^{1}(ℝ)}^{2} with another application of Young’s inequality we derive

sup_{t∈[0,T_{loc}]} ∫_{ℝ} ξ(t)dx ≤ \frac{αt}{ε}c_{1}^{4}ϕ(c_{1})^{\frac{4}{3}}

where α is a positive constant. Furthermore, by definition of ξ, for c_{2} > c_{1} we have

sup_{t∈[0,T_{loc}]} ∫_{ℝ} ξ(t)dx ≥ (c_{2} − c_{1})^{4}ϕ(c_{2}).

The last two inequalities show that Lemma 2.4 can be applied to complete the proof of Lemma 2.3.

Theorem 2.3.a Assume an initial data u^{0} = (u_{1}^{0}, u_{2}^{0}) ∈ ℂ^{2}(ℝ; ℝ^{2}), (u_{1}^{0} − \overline{u}_{1}, u_{2}^{0}) ∈ 𝕃^{2}(ℝ; ℝ^{2}) such that

1. inf_{x∈ℝ}u_{1}^{0}(x) > 0, lim_{|x|→∞} u_{1}^{0}(x) = \overline{u}_{1}

2. lim_{|x|→∞} u_{2}^{0}(x) = 0

where \overline{u}_{1} is a positive constant. Moreover let λ, ε > 0, and β ∈ ℂ^{3}(ℝ) with β(x) ≥ β_{0} > 0. Then the equation (VP) possesses a local in time classical solution, i.e., u ∈ ℂ^{0}([0, T_{max}); ℂ^{2}(ℝ)), \frac{∂}{∂t}u ∈ ℂ^{0}([0, T_{max}) × ℝ); moreover

1. inf_{x∈ℝ}u_{1}(t, x) > 0, lim_{|x|→∞} u_{1}(t, x) = \overline{u}_{1}

2. lim_{|x|→∞} u_{2}(t, x) = 0

for all t ∈ [0, T_{max}).

Proof Apply Theorem 2.1, Theorem 2.2.a, Lemma 2.2, Lemma 2.3.

Theorem 2.3.b Assume an initial data u^{0} = (u_{1}^{0}, u_{2}^{0}) ∈ ℂ^{2}(ℝ; ℝ^{2}), (u_{1}^{0} − \overline{u}_{1}, u_{2}^{0}) ∈ 𝕃^{2}(ℝ; ℝ^{2}) such that

1. inf_{x∈ℝ}u_{1}^{0}(x) > 0, lim_{|x|→∞} u_{1}^{0}(x) = \overline{u}_{1}

2. lim_{|x|→∞} u_{2}^{0}(x) = 0

where \overline{u}_{1} is a positive constant. Moreover λ, ε > 0, and β = β_{const}. Then the equation (VP) possesses a global in time classical solution i.e. u ∈ ℂ^{0}([0, T); ℂ^{2}(ℝ)), \frac{∂}{∂t}u ∈ ℂ^{0}([0, T) × ℝ) for all T ∈ [0, ∞); moreover

1. inf_{x∈ℝ}u_{1}(T, x) > 0, lim_{|x|→∞} u_{1}(T, x) = \overline{u}_{1}

2. lim_{|x|→∞} u_{2}(T, x) = 0

holds.

Proof Apply Theorem 2.1, Theorem 2.2.b, Lemma 2.2, Lemma 2.3.

Lemma 2.5 Assume that u(t, x) is a strong solution of the equation

\frac{∂}{∂t}u + \frac{∂}{∂x}F (u) = G_{λ}(u, x) + ε\frac{∂^{2}}{∂^{2}x}u.

Moreover suppose that u, F (u), G_{λ}(u, x) ∈ 𝕃^{∞}([0, T) × ℝ; ℝ^{2}), u_{1} > 0 and that (η, q) is a convex entropy-flux pair with |∇η| ≤ C and η(·, ·) is ℂ^{2}-function for (u_{1}, u_{2}) ∈ (0, ∞) × ℝ. Then ε|\frac{∂}{∂x}u|^{2}, ε(\frac{∂}{∂x}u^{T} · ∇^{2}η(u) · \frac{∂}{∂x}u) are bounded independently of 0 < ε < 1 in the space 𝕃^{1}(Ω) for any bounded Ω relatively open in [0, ∞) × ℝ.

Proof If (η, q) is a convex entropy-flux pair (\frac{∂}{∂x}u^{T} · ∇^{2}η(u) · \frac{∂}{∂x}u ≥ 0) with |∇η| ≤ C, then multiplying (VP) by ∇η gives:

\frac{∂}{∂t}η + \frac{∂}{∂x}q = ε (\frac{∂^{2}}{∂^{2}x}η − \frac{∂}{∂x}u^{T} · ∇^{2}η(u) · \frac{∂}{∂x}u) + ∇η · G_{λ}.

Now we multiply by a non-negative test function θ ∈ ℂ_{0}^{∞}([0, ∞)×ℝ) with θ(t, x)_{|Ω} = 1 and integrate over [0, ∞) × ℝ. We obtain the inequality

\begin{aligned}ε ∥\frac{∂}{∂x}u^{T} · ∇^{2}η(u) · \frac{∂}{∂x}u∥_{𝕃^{1}(Ω)} & ≤ & ∫_{ℝ} ∫_{0}^{∞}\{|η \frac{∂^{2}}{∂^{2}x}θ| + q \frac{∂}{∂x}θ + η \frac{∂}{∂t}θ + θ∇η · G_{λ}\} dx dt \\ & & +∫_{ℝ} θ η dx, \end{aligned}

where the right hand side and consequently also the left hand side is bounded independently of ε. Similarly if (η, q) is an strictly convex entropy-flux pair (\frac{∂}{∂x}u^{T} · ∇^{2}η(u) · \frac{∂}{∂x}u > K|\frac{∂}{∂x}u|^{2} K > 0), for example η_{E} defined in Chapter 4 step 2, then

εK ∥\frac{∂}{∂x}u∥_{𝕃^{2}(Ω)}^{2} ≤ ∫_{ℝ} θηdx + ∫_{ℝ} ∫_{0}^{∞}\{|η \frac{∂^{2}}{∂^{2}x}θ| + q \frac{∂}{∂x}θ + η \frac{∂}{∂t}θ + θ∇η · G_{λ}\} dx dt.

3 Young Measure Limits

First we recall two important theorems:

Theorem (convergence in the sense of Young measures) Assume that the sequence {u_{k}} is bounded in 𝕃^{∞}(Ω; K), where Ω ⊂ [0, T) × ℝ, K is a compact, convex set in ℝ^{2}. Then there exists a subsequence {u_{k_{j}}} ⊂ {u_{k}} and for a.e. (t, x) ∈ Ω a Borel probability measure μ_{(t,x)} on K such that for each F ∈ ℂ(ℝ × K; ℝ^{2}) we have

lim_{k_{j}→∞} ∫_{Ω} F(x, u_{k_{j}}(t, x))φ(t, x)dxdt = ∫_{Ω} \overline{F(t, x)}φ(t, x)dxdt ∀φ ∈ 𝕃^{1}(Ω, ℝ^{2}),

where

\overline{F(t, x)} = ∫_{K} F(x, y)dμ_{(t,x)}(y)

for a.e. (t, x) ∈ Ω.

Proof The main idea is to use the Banach-Alaoglu Theorem and the fact that 𝕃_{w}^{∞}(Ω; M(K)), is the dual of the separable space 𝕃^{1}(Ω; ℂ^{0}(K)) cf. [17], and for F without x-dependence [3],[9].

Remark By Lemma 2.2 we have |\frac{u_{2}}{u_{1}} | + |u_{1}| ≤ C(T) and u_{1} ≥ 0. For such (u_{1}, u_{2}) the flux term F is continuous.

Theorem (Div-Curl lemma) Assume that (q_{k}^{1}, η_{k}^{1}) and (q_{k}^{2}, −η_{k}^{2}) are bounded sequences in 𝕃^{2}(Ω; ℝ^{2}) such that (\frac{∂}{∂t}q_{k}^{i}, \frac{∂}{∂x}η_{k}) (for i=1,2) lies in a compact subset of 𝕎^{−1,2}(Ω; ℝ^{2}). Suppose further that (q_{k}^{1}, η_{k}^{1}) ⇀ (q^{2}, η^{2}), (q_{k}^{2}, −η_{k}^{2}) ⇀ (q^{2}, −η^{2}) in 𝕃^{2}(Ω; ℝ^{2}). Then

∫_{Ω}(q_{k}^{1}η_{k}^{2} − η_{k}^{1}q_{k}^{2})φ dxdt → ∫_{Ω}(q^{1}η^{2} − η^{1}q^{2})φ dxdt ∀φ ∈ ℂ_{0}^{∞}(Ω, ℝ).

Proof For a similar statement see [9]. Note that (\frac{∂}{∂t}, \frac{∂}{∂x}) · (q, η) is the two-dimensional divergence operator and (\frac{∂}{∂t}, −\frac{∂}{∂x}) · (q, η) is the two-dimensional curl operator.

Now we prove the existence of a so-called “measure valued solution” defined in [4] by R. DiPerna to our problem (CL). We also show the very important property that for a.e. (t, x) the measures reduce to Dirac measures concentrated in points. Let us divide our analysis into three steps:

Step 1 First we fix λ = ε and write the approximate system in the weak form

\begin{aligned}∫_{[0,T)×ℝ} [u^{ε} · \frac{∂}{∂t}ψ + F (u^{ε}) · \frac{∂}{∂x}ψ + G_{ε}(u^{ε}, x) · ψ] dtdx & = & ε ∫_{[0,T)×ℝ} \frac{∂}{∂x}u^{ε} · \frac{∂}{∂x}ψ dtdx \\ & & + ∫_{ℝ} u^{0}(x) · ψ(0, x) dx \end{aligned}

for all test functions ψ ∈ ℂ_{0}^{1}([0, T) × ℝ; ℝ^{2}). Defining the new variable z^{ε} by z^{ε} = (z_{1}^{ε}, z_{2}^{ε}) = (\frac{u_{2}^{ε}}{u_{1}^{ε}} , u_{1}^{ε}) we obtain

\begin{matrix}∫_{[0,T)×ℝ} [H_{1}(z^{ε}) · \frac{∂}{∂t}ψ + H_{2}(z^{ε}) · \frac{∂}{∂x}ψ + H_{3}^{ε}(z^{ε}, x) · ψ] dtdx = \\ ε ∫_{[0,T)×ℝ} \frac{∂}{∂x}u^{ε} · \frac{∂}{∂x}ψ dtdx + ∫_{ℝ} u^{0}(x) · ψ(0, x)dx \end{matrix}  (V P)

for all test functions ψ ∈ ℂ_{0}^{1}([0, T) × ℝ; ℝ^{2}), where

H_{1}(z_{1}^{ε}, z_{2}^{ε}) = ( \begin{matrix}z_{2}^{ε} \\ z_{1}^{ε}z_{2}^{ε} \end{matrix} ) , H_{2}(z_{1}^{ε}, z_{2}^{ε}) = ( \begin{matrix}z_{1}^{ε}z_{2}^{ε} \\ (z_{1}^{ε})^{2}z_{2}^{ε} + κ(z_{2}^{ε})^{\frac{2+α}{1+α}} \end{matrix} )

and

H_{3}^{ε}(z_{1}^{ε}, z_{2}^{ε}, x) = Λ (εx) ( \begin{matrix}\frac{β′(2κ)^{α−1}}{β^{α−1}} z_{1}^{ε}z_{2}^{ε} \\ \frac{β′(2κ)^{α−1}}{β^{α−1}} ((z_{1}^{ε})^{2}z_{2}^{ε} + κ(z_{2}^{ε})^{\frac{2+α}{1+α}} ) + Λ (\frac{1}{εz_{1}^{ε}} ) z_{1}^{ε}g (z_{1}^{ε}z_{2}^{ε}, x) \end{matrix} ) .

Note that ψ_{t}, ψ_{x}, ψ are elements from the space 𝕃^{1}([0, T)×ℝ; ℝ^{2}). Then for the sequence of solutions z^{ε} of the approximate problems (for example setting ε = \frac{1}{k}, k ∈ ℕ) we can pass to the limit in the sense of Young measures. Since \frac{1}{k} \frac{∂}{∂x}u^{k} → 0 strongly in 𝕃^{2}([0, T) × ℝ; ℝ^{2}) by Lemma 2.5 and hence the first term on the right hand side of (VP) goes to zero, we obtain:

∫_{[0,T)×ℝ} [\overline{H_{1}(t, x)}ψ_{t}(t, x) + \overline{H_{2}(t, x)}ψ_{x}(t, x) + \overline{H_{3}(t, x)}ψ(t, x)] dtdx = ∫_{ℝ} u^{0}ψ(0, x)dx;

(ME)

here by the characterisation of Young measures we know that

\overline{H_{1}(t, x)} = ∫_{K} H_{1}(y_{1}, y_{2})dμ_{(t,x)}(y_{1}, y_{2}),

\overline{H_{2}(t, x)} = ∫_{K} H_{2}(y_{1}, y_{2})dμ_{(t,x)}(y_{1}, y_{2})

for a.e (t, x) ∈ Ω and that μ_{(t,x)} is the Young measure generated by a subsequence {(z_{1}, z_{2})^{k_{j}}}. However we can not characterise the “weak-star” limits of H_{3}^{ε} in the same way. We know only that the limit function H_{3} ∈ 𝕃^{∞}(ℝ × ℝ_{+}; ℝ^{2}). In the standard theory (ME) defines a weak solution if and only if Young measures μ_{(t,x)} are Dirac measures. Here the characterisation of the limit H_{3}^{ε} poses an additional difficulty.

Step 2 To prove that μ_{(t,x)} is a Dirac measure we apply the Div-Curl Lemma to the family of entropies. Multiplying the equation (VP) by ∇_{u}η(u^{k}) we obtain

\frac{∂}{∂t}u^{k}∇_{u}η(u^{k}) + \frac{∂}{∂x}F (u^{k})∇_{u}η(u^{k}) = G_{k}(u^{k}, x)∇_{u}η(u^{k}) + \frac{1}{k} \frac{∂^{2}}{∂^{2}x}u^{k}∇_{u}η(u^{k}),

and by a simple calculation

\frac{∂}{∂t}η(u^{k})+ \frac{∂}{∂x}q(u^{k}) = G_{k}(u^{k}, x)∇_{u}η(u^{k})+ \frac{1}{k} (\frac{∂^{2}}{∂^{2}x}η(u^{k}) − (\frac{∂}{∂x}u^{k})^{T} · ∇_{u}^{2}η(u^{k}) · \frac{∂}{∂x}u^{k}) .

(EE)

We want to prove that the right hand side lies in a compact subset of 𝕎^{−1,p}(Ω; ℝ^{2}) for all bounded, relatively open sets Ω in [0, T) × ℝ and p ∈ (1, ∞). It is well known that any “weak entropy” defined by R. DiPerna [4] can be expressed in the form of a convolution:

η(u_{1}, u_{2}) = ∫_{ℝ} f(ξ)χ (u_{1}, ξ − \frac{u_{2}}{u_{1}} ) dξ

where

χ(a, b) = (a^{\frac{1}{2(1+α)}} − b^{2})_{+}^{2+4α}

and

(c)_{+}^{λ} = \{ \begin{matrix}c^{λ} & for c > 0 \\ 0 & for c ≤ 0. \end{matrix}

Now rewrite the convolution as

η(u_{1}, u_{2}) = u_{1} ∫_{−1}^{1} f (\frac{u_{2}}{u_{1}} + ξu1^{\frac{1}{2(1+α)}} ) (1 + ξ^{2})_{+}^{2+4α}dξ.

Differentiation yields

\begin{aligned}\frac{∂}{∂u_{1}} η(u_{1}, u_{2}) & = & ∫_{−1}^{1} f (\frac{u_{2}}{u_{1}} + ξu1^{\frac{1}{2(1+α)}} ) (1 + ξ^{2})_{+}^{2+4α}dξ \\ & & + ∫_{−1}^{1} (\frac{1}{2(1 + α)}ξu1^{\frac{1}{2(1+α)}} − \frac{u_{2}}{u_{1}} ) f′ (\frac{u_{2}}{u_{1}} + ξu1^{\frac{1}{2(1+α)}} ) (1 + ξ^{2})_{+}^{2+4α}dξ, \end{aligned}

and

\frac{∂}{∂u_{2}} η(u_{1}, u_{2}) = ∫_{−1}^{1} f′ (\frac{u_{2}}{u_{1}} + ξu1^{\frac{1}{2(1+α)}} ) (1 + ξ^{2})_{+}^{2+4α}dξ.

Obviously the assumption ∥f∥_{𝕎^{1,∞}(ℝ)} < ∞ implies that ∥∇_{u}η(u^{k})∥_{𝕃^{∞}(Ω;ℝ^{2})} ≤ C.

We can also show (see [4],[10]) that if ∥f∥_{C^{2}(ℝ)} < ∞, then

|r^{T} · ∇_{u}^{2}η(u^{k}) · r| ≤ C(∥f∥_{C^{2}(ℝ)}) r^{T} · ∇_{u}^{2}η_{E}(u^{k}) · r for all r ∈ ℝ^{2}

and that η(·, ·) is a ℂ^{2}-function for (u_{1}, u_{2}) ∈ (0, ∞) × ℝ. Here η_{E} denotes the mechanical energy obtained for f(ξ) = \frac{ξ^{2}}{2} and is given by

η_{E}(a, b) = \frac{a^{2}}{2b} + κ(1 + α)b^{\frac{2+α}{1+α}} .

Thus for each entropy generated by f ∈ ℂ_{0}^{∞}(ℝ):

1. {G_{k}(u^{k}, x)∇_{u}η(u^{k})}_{k=1}^{∞} is bounded in 𝕃^{∞}(Ω; ℝ^{2}) and therefore precompact in 𝕎^{−1,2}(Ω; ℝ^{2}),

2. \{\frac{1}{k}(\frac{∂}{∂x}u^{k})^{T} · ∇_{u}^{2}η(u^{k}) · \frac{∂}{∂x}u^{k}\}_{k=1}^{∞} is bounded in 𝕃^{1}(Ω; ℝ^{2}) and consequently in M(Ω; ℝ^{2}). Hence this set is precompact in 𝕎^{−1,p}(Ω; ℝ^{2}) for p < 2,

3. \frac{1}{k} \frac{∂}{∂x}η(u^{k}) = \sqrt{\frac{1}{k}}∇_{u}η(u^{k})\sqrt{\frac{1}{k}} \frac{∂}{∂x}u^{k} → 0 in 𝕃^{2}(Ω; ℝ^{2}); thus \{\frac{1}{k} \frac{∂^{2}}{∂^{2}x}η(u^{k})\}_{k=1}^{∞} is a precompact set in 𝕎^{−1,2}(Ω; ℝ^{2}),

4. the left hand side of equation (EE) is bounded 𝕎^{−1,∞}(Ω; ℝ^{2}).

Then by Murat’s Lemma [9] the left hand side of (EE) is precompact in 𝕎^{−1,2}(Ω; ℝ^{2}).

Step 3 We apply the Div-Curl lemma to all two entropy-flux pairs (η^{1}, q^{1}), (η^{2}, q^{2}) generated by f^{1}, f^{2} ∈ C_{0}^{∞}(ℝ). This is the starting point to the analysis from [6], [7]. In the end this will imply that in the (z_{1}, z_{2})-coordinates the measure μ_{(t,x)} has the form

μ_{(t,x)} = \{ \begin{matrix}(δ_{z_{1}(t,x)}, δ_{z_{2}(t,x)}) & if z_{2}(t, x) > 0 \\ (ν_{(t,x)}, δ_{z_{2}(t,x)}) & if z_{2}(t, x) = 0, \end{matrix}

where supp{ν_{(t,x)}} ⊂ [lim inf_{k→∞}(z_{2}^{k}(t, x), lim sup_{k→∞}(z_{2}^{k}(t, x)]. It follows that in (u_{1}, u_{2})-

coordinates the measure is concentrated in one point.

4 Strong 𝕃^{p} Convergence and Existence of Weak Entropy Solutions

To prove that our solution is a weak entropy solution to problem (CL) we need that the limit function G(t, x) satisfies G(t, x) ∈ G̃(z(t, x), x) for a.e. (t, x) ∈ Ω, and that the entropy inequality is satisfied for all weak entropies.

First we prove u^{k_{j}} → u in the strong topology of 𝕃^{p}(Ω; ℝ^{2}) for 1 ≤ p < ∞. Note that by the Radon-Riesz theorem the assumption f^{k} ⇀ f weakly in 𝕃^{p} and ∥f^{k}∥_{𝕃^{p}} → ∥f∥_{𝕃^{p}} imply that f^{k} → f strongly in 𝕃^{p} (for each p ∈ (1, ∞)). Thus we must check that ∥u^{k_{j}}∥_{𝕃^{p}} → ∥u∥_{𝕃^{p}}. By the convergence in the sense of Young measures and by the fact that these measures are Dirac measures

∫_{ℝ×[0,T)} |u^{k_{j}}|^{p}dtdx → ∫_{ℝ×[0,T)} ∫_{K} |u|^{p}dδ_{u(t,x)}dtdx = ∫_{ℝ×[0,T)} |u|^{p}dtdx.

We can also prove the strong 𝕃^{p}-convergence for the terms (u_{2}^{k_{j}}(u_{1}^{k_{j}})^{−a})^{b}, (u_{1}^{k_{j}})^{b} and (u_{1}^{k_{j}})^{a}∇_{u}η(u^{k_{j}}) for all 0 < a < 1 and 0 < b. Just observe that (u_{1})^{a}∇_{u}η(u) is a continuous bounded function, if ∥∇_{u}η(u)∥ ≤ C, η(·, ·) is a ℂ^{1}-function for (u_{1}, u_{2}) ∈ (0, ∞) × ℝ and η(0, ·) = 0.

To complete the proof of the existence of a weak entropy solution we need that there exists G(t, x) ∈ G̃(u(t, x), x) for a.a. (t, x) ∈ [0, ∞) × ℝ such that

1. G_{k_{j}}(u^{k_{j}}) ⇀ G in 𝕃^{p}(Ω; ℝ^{2}),

2. G_{k_{j}}(u^{k_{j}})∇_{u}η(u^{k_{j}}) ⇀ G∇_{u}η(u) in 𝕃^{p}(Ω; ℝ^{2})

for all 1 < p < ∞. Now we prove the following lemma:

Lemma 4.1 Assume u^{k} → u in the strong topology 𝕃^{p}(Ω; ℝ) for 1 ≤ p < ∞, sig_{k} → sig uniformly in ℝ − [−\frac{1}{n}, \frac{1}{n}] for all n ∈ ℕ and |sig_{k}| ≤ 1. Then there exists a function S ∈ 𝕃^{p}(Ω; ℝ) and a subsequence k_{j} such that

1. sig_{k_{j}}(u^{k_{j}}) ⇀ S in the weak topology of 𝕃^{p}(Ω; ℝ)

2. S ∈ sig(u) a.e. in Ω

holds.

Proof We prove only S ∈ sig(u) a.e. in Ω. Let us observe |S(t, x)| ≤ 1 a.e. in Ω. Then we divide Ω into three sets: Ω_{−} = {(t, x) ∈ Ω | u_{2}(t, x) < 0}, Ω_{0} = {(t, x) ∈ Ω | u_{2}(t, x) = 0}, Ω_{+} = {(t, x) ∈ Ω | u_{2}(t, x) > 0}. It is obvious that S ∈ sig(u) a.e. in Ω_{0}. Now we can assume that k_{j} is a subsequence with u^{k_{j}} → u a.e. in Ω. Thus sig_{k_{j}}(u^{k_{j}}) → 1 for a.a. (t, x) ∈ Ω_{+} and sig_{k_{j}}(u^{k_{j}}) → −1 for a.a. (t, x) ∈ Ω_{−}. This completes the proof of the lemma.

By applying Lemma 4.1 we end the proof of the existence of a weak entropy solution to our problem (CL).

5 Acknowledgements

The author would like to express his gratitude to K. Chelminski and R. Farwig for remarks to this paper.

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