����������� ������ 56 (2004), 23–46

UDK 517.988.6

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NEW APPROACH TO SHOCKS GENERATION FOR CONSERVATION LAWS. EXAMPLE: GLOBAL SOLUTION TO HOPF EQUATION

V. G. Danilov and D. Mitrović

Abstract. We present a new method for constructing global solution to Hopf equation. Propagation and formation of nonlinear waves are described.

1. Introduction

In general, the use of asymptotic method for solving a differential equation means the construction of function which depends on some small parameter, say ε, and satisfy a differential equation up to O_{S}(ε^{α}) when ε → 0 and α > 0. Here, O_{S}(ε^{α}) means that expression \frac{O_{S}(ε^{α})}{ε^{α}} stays bounded in S sense when ε → 0, where S is suitable functional space, usually C^{k}. In our new approach, S = D′(R). That means that for a given differential equation we look for a function which satisfies this equation in D′(R) or in a weak sense. Therefore, the name of this method is the weak asymptotic method (see [3]).

In what follows, we use our method on the following problem:

Lu = u_{t} + (u^{2})_{x} = 0, x ∈ R, t > 0,  (1)
u|_{t=0} = u_{0}(x),  (2)

where u_{0} is decreasing, Lipschitz continuous and takes values in some compact interval. Asymptotic solutions for this problem has already been searched by several authors as J.Witham and A.M.Il’in. These authors were interested exactly in the scalar conservation law. On the other hand, more general approach (analysis of the system of conservation laws) was proceeded by J.Glimm (see [1]) and Bressan (see [8]). (We mentioned only the authors which completed appropriate approaches.)

Witham uses Florin-Hopf-Cole linearization of equation u_{t} + (u^{2})_{x} = εu_{xx}. In that manner he obtains its exact solution. Moreover, he analyzes it as ε → 0 for different periods of time.

AMS Subject Classification: 35 F 25

Keywords and phrases: Nonlinear waves, shocks generation, Hopf equation.

This method cannot be used for more general situation of the scalar conservation law since in general it is not known how to linearize corresponding problem with vanishing viscosity. Successful attempt to generalize Whitham approach was made by Il’in. He introduced “the matching method”. In his work he considers Cauchy problem u_{t} +(f(u))_{x} = εu_{xx}, u(0, x) = φ(x) where φ is bounded, piecewise smooth function. For fixed ε one knows that the last problem has bounded infinitely differentiable solution everywhere for t > 0 with the exception of the discontinuity points of the initial function. On the contrary, when ε → 0 the solution stops to be even continuous and the discontinuity line appears. So, the solution of the problem he searches in the form of two asymptotical series. He chooses the first one to be the solution of the problem in an arbitrary domain which does not comprise the discontinuity line while the second series is the solution of the problem near the discontinuity line. He shows that those two series coincides in the vicinity of the discontinuity line. The solution obtained in a such way has complicated form.

The drawback of mentioned approaches lies in the fact that they cannot be applied on the systems of hyperbolic conservation laws. More comprehensive approach was offered by Glimm (“random choice method”) and Bressan (“front tracking method”). (Front tracking method was first proposed by DiPerna for 2 × 2 systems.)

Our idea in this paper is the closest to Glimm’s ideas. He breaks x-axis as well as t-axis creating a mesh in xt-plane. He assigns recursively, (beginning from the time t_{0} = 0) some constant value on each of intervals (x_{i−1}, x_{i+1}) × {t_{s}}, i ∈ Z, s ∈ N, i + s even. In that manner, he obtains a sequence of Riemann problems for equation (1) on disjoint intervals (x_{2i−1}, x_{2i+3}) (or (x_{2i}, x_{2i+2})) with the initial moment t = t_{s} where i ∈ Z_{2}, s ∈ N_{2} + 1 (or i ∈ Z_{2}, s ∈ N_{2}). He breaks t-axis in such a way that no two shocks created from mentioned Riemann problems interact till t = t_{s+1}. In t = t_{s+1} he reapproximates the solution with new step function and so he obtains the Cauchy problem as in the beginning (with the difference that initial moment is not t = 0 but t = t_{s+1}). Finally, he proves that solution constructed in this way tends to the admissible weak solution of the original problem. Shortcoming of this approach lies in the fact that it does not give any explicit formula which would represent the asymptotic solution of problem (1), (2) but it gives the proof of existence of the Cauchy problem for the initial data with small variation. Method can be applied on Hopf type equation with an arbitrary nonlinearity. (Since Bressan approach is quite different from ours we will not mention it. One can find more information about it in [8].)

We will search the weak asymptotic solution of problem (1), (2) with the accuracy ε^{1−μ}, ε → 0, 0 < μ < 1. That means that we look for a net of smooth functions u_{ε}(x, t), x ∈ R, t ∈ R^{+}, ε < 1, which satisfies

Lu_{ε} = O_{D′}(ε^{1−μ})  (3)
u_{ε}|_{t=0} − u|_{t=0} = O_{L_{1}^{loc}}(ε^{1−μ}),  (4)

or, more precisely, we search for u_{ε}(x, t), x ∈ R, t ∈ R^{+}, 0 < ε < 1, smooth for ε > 0 and such that for every ϕ ∈ C_{0}^{∞}(R_{x}^{1}) we have in the distributional sense,

∫ Lu_{ε}ϕdx = O(ε^{1−μ})
∥u_{ε}|_{t=0} − u|_{t=0}∥_{L_{1}^{loc}(R)} = O(ε^{1−μ}), ε → 0.

First, we approximate given initial condition by a polygon, and then, we solve equation (3) with this polygon as initial condition. It is well known that, as time goes, ends of intervals will reach one another (we will call that phenomenon interaction of points). So, sooner or later one or few shock waves will appear and ends of intervals which do not take part in shocks formation will continue to interact with the formed shocks.

In the analysis of this situation we start from the case when we have two weak discontinuities, and then when we have one weak discontinuity and one shock wave. We use such an approach to analyze the situation when we have arbitrary many weak discontinuities. Here, we additionally assume that no three consequent point will interact simultaneously. We can assume that interactions happens in the moments t_{1}^{∗} < t_{2}^{∗} < ⋯ < t_{n}^{∗} < · · ·. First, we will look for the partial solutions of equation (3) in the intervals [0, t_{1}^{∗} +δ_{1}), [t_{1}^{∗}, t_{2}^{∗} +δ_{2}), …, and prove that in common parts of every two intervals partial solutions match. Then, we will connect them by the partition of unity of the time interval.

2. Some weak asymptotic formulas

We will introduce necessary definitions and formulas. The proofs can be found in [3] or in [6].

Proposition 2.1. Let ω(z) ∈ S(R^{1}), where S is the Schwartz space. For any function η(x) ∈ C_{0}^{∞} we have

⟨\frac{1}{ε}ω(\frac{x − a}{ε}), η(x)⟩ = ∑_{k=0}^{n} Ω_{k} \frac{ε^{k}}{k!} (−1)^{k}⟨δ^{(k)}(x − a), η⟩ + O(ε^{n}), ε > 0, (5)

where

Ω_{k} = ∫ ω(z)z^{k} dz, k = 1, …, n.  (6)

Expression (5) we call the weak asymptotic of the function (distribution) \frac{1}{ε}ω(\frac{x−a}{ε}).

Definition 2.2. We denote by O_{D′}(ε^{α}) an element of D′ such that for every function η(x) ∈ C_{0}^{∞} we have

f(x, ε) = O_{D′}(ε^{α}) ⇔ ⟨f(x, ε), η(x)⟩ = O(ε^{α}).
Proposition 2.3. Let ω_{1}(z), ω_{2}(z) ∈ S(R). We have
ω_{1}(\frac{x − a_{1}}{ε} )ω_{2}(\frac{x − a_{2}}{ε} ) = \frac{1}{2}[εδ(x − a_{1}) + εδ(x − a_{2})]B(\frac{Δa}{ε}) + O_{D′}(ε^{2}), (7)

where

B(\frac{Δa}{ε}) = ∫ ω_{1}(z)ω_{2}(z − \frac{Δa}{ε}) dz = ∫ ω_{1}(z + \frac{Δa}{ε})ω_{2}(z) dz.  (8)

Remark 2.4. Under the assumptions of Proposition 2.1 and the condition that ∫ ω_{i}(z) dz = 1, the functions ω_{i}((x − a_{i})/ε) are weak approximations (weak asymptotic) of the functions εδ(x − a_{i}),

ω_{i}((x − a_{i})/ε) = εδ_{ε,i}(x − a_{i})

Hence we can rewrite (8) as

εδ_{ε,1}(x − a_{1})εδ_{ε,2}(x − a_{2}) = \frac{1}{2}[εδ(x − a_{1}) + εδ(x − a_{2})]B(Δa/ε) + O_{D′}(ε^{2}).

In a similar way, under the assumptions of Proposition 2.3 ω_{i}((x−a_{1})/ε) = θ_{ε,i}(x− a_{i}) are approximations of the Heaviside θ-function. Hence we can rewrite (8) as

θ_{ε,1}(x − a_{1})θ_{ε,2}(x − a_{2}) = θ(x − a_{1})B_{1}(Δa/ε) + θ(x − a_{2})B_{2}(Δa/ε) + O_{D′}(ε). (9)

3. Uniform in t ∈ R solution of the Hopf equation with the “simple” functions as an initial condition

We remind that our task is to connect two states of the solution of Cauchy problem (1), (2). The first state is the continuous one (i.e. before blow up of the classical solution), and the other one is discontinuous state. As we have already said, for accomplishing our task, we will use well known idea of replacing (approximating) old initial data with a more simple function. In the case of random choice method or wave front tracking method, the old initial data are replaced by piecewise constant functions, or, in other words, the old initial data are locally replaced by the Heaviside function. In our case, we have to replace the old initial data with some continuous function (since we want to inspect the passage from continuous to discontinuous state we have to start from something which is continuous). Naturally, in the beginning we have to consider Hopf equation with the initial function equal to the function we will locally replace our old initial data with.

3.1. Interaction of weak discontinuities. Generation of shock waves We will consider the Hopf equation

Lu = u_{t} + (u^{2})_{x} = 0  (10)

and pose the following initial condition

u|_{t=0} = u_{0}^{0} + u_{1}^{0}(a_{1} − x)_{+} − u_{1}^{0}(a_{2} − x)_{+},

where a_{1} > a_{2}, z_{+} = zθ(z), u_{i}^{0} = const > 0.

We shall seek the weak asymptotic solution in the form of the (regularized) broken line

u_{ε}(x, t) = u_{0}^{0} + u_{1}(t, ε)(φ_{1}(t, ε) − x)θ_{ε,1}(−x + φ_{1}(t, ε))

− u_{2}(t, ε)(φ_{2}(t, ε) − x)θ_{ε,2}(−x + φ_{2}(t, ε)).

The unknown functions φ_{i} and u_{i}, i = 1, 2, appearing in the last expression belong to C^{1}(R^{+}) for every fixed ε and they satisfy the following initial conditions:

φ_{i}(0, ε) = a_{i}, u_{i}(0, ε) = u_{1}^{0}, i = 1, 2.

Substituting the approximation of u_{ε}(x, t) into equation (10) and taking into account the definitions and calculations given in the previous section we obtain

(u_{1}(φ_{1} − x)_{+})_{t} − (u_{2}(φ_{2} − x)_{+})_{t} + (u_{1}^{2}(φ_{1} − x)_{+}^{2})_{x} + (u_{2}^{2}(φ_{2} − x)_{+}^{2})_{x}

+ 2[u_{0}u_{1}(φ_{1} − x)_{+}]_{x} − 2[u_{0}u_{2}(φ_{2} − x)_{+}]_{x}
− 2[u_{1}u_{2}(φ_{1} − x)(φ_{2} − x)θ(φ_{1} − x)]_{x}B_{1}(Δφ/ε)

− 2[u_{1}u_{2}(φ_{1} − x)(φ_{2} − x)θ(φ_{2} − x)]_{x}B_{2}(Δφ/ε) = O_{D′}(ε), Δφ= φ_{2} − φ_{1}.

Let us consider the domain φ_{2} < x ≤ φ_{1}. Notice that in this case θ_{1} = θ(φ_{1}−x) = 1 and θ_{2} = θ(φ_{2} − x) = 0. Accordingly, we have

u_{1t}(φ_{1} − x) + u_{1}φ_{1t} + 2[u_{0}u_{1}(φ_{1} − x)]_{x} + [u_{1}^{2}(φ_{1} − x)^{2}]_{x} + 2u_{1}u_{2}(φ_{1} − x)B_{1} + 2u_{1}u_{2}(φ_{2} − x)B_{1} =

u_{1t}(φ_{1}−x)+u_{1}φ_{1t}−2u_{0}u_{1}+2u_{1}^{2}(φ_{1}−x)+2u_{1}u_{2}(φ_{1}−x)B_{1}+2u_{1}u_{2}(φ_{2}−x)B_{1} = 0

Setting x = φ_{1} we obtain,

φ_{1t} − 2u_{0} + 2u_{2}ΔφB_{1} = 0.  (11)

Substituting this relation into the last equation, we arrive at the following equation for the function u_{1}:

u_{1t} − 2u_{1}^{2} + 4u_{1}u_{2}B_{1} = 0.  (12)

In a similar way, considering the domain −∞ < x ≤ φ_{2}, we obtain the other two equations

φ_{2t} − 2u_{0} + 2u_{1}ΔφB_{2}(Δφ/ε) = 0,  (13)
u_{2t} + 2u_{2}^{2} − 4u_{1}u_{2}B_{2}(Δφ/ε) = 0, Δφ = φ_{2} − φ_{1}.  (14)

Let Δφ < 0, then, up to O(ε^{N}), we have B_{1}(Δφ/ε) = 0, B_{2}(Δφ/ε) = 1 and we obtain the following system of equations describing the evolution of the broken line until it turns over (or, more formally, until φ_{1} > φ_{2}):

(φ_{10})′_{t} − 2u_{0} = 0,  (φ_{20})′_{t} − 2u_{0} + 2u_{10}(φ_{20} − φ_{10}) = 0,  (15)
(u_{10})′_{t} − 2(u_{10})^{2} = 0, (u_{20})′_{t} + 2u_{20}^{2} − 4u_{10}u_{20} = 0,

Solutions of this system have the form

u_{10}(t) = u_{20}(t) = u_{1}^{0}/(1 − 2tu_{1}^{0}),
φ_{10} = a_{1} + 2u_{0}t, φ_{20} = a_{2} + 2[u_{1}^{0}(a_{1} − a_{2}) + u_{0}]t.

We write ψ_{0} = φ_{20}(t) − φ_{10}(t).

At time t = t^{∗} such that ψ_{0}(t^{∗}) = 0 the weak discontinuities merge and a shock wave is generated. To construct formulas that are uniform in t and describe the interaction of weak discontinuities and the generation of a shock wave, we seek the functions φ_{i}, i = 1, 2, in the form

φ_{k}(t, ε) = φ_{k0}(t) + ψ_{0}ϕ_{k}(τ), τ = ψ_{0}/ε, k = 1, 2.

We also introduce the function ρ = ρ(τ):

ρ(τ) = \frac{φ_{2}(t, ε) − φ_{1}(t, ε)}{ε}.

Substituting this into (11) and (13) and letting ρ → −∞ we see that ϕ_{k} must satisfy

ϕ_{k}(τ)|_{τ_{1}→−∞} = 0, \frac{dϕ_{k}}{dτ} |_{|τ|→∞} = o(τ^{−1}).

We shall seek the functions u_{k}(t, ε), k = 1, 2, in the form

u_{k}(t, ε) = ψ_{0}(0)u_{1}^{0}/(ψ_{0} + εg_{k}(τ)), k = 1, 2.

Here we assume that the functions g_{k}(τ) behave in the same way as the functions ϕ_{k}(τ). From equations (12), (13), (14), (15) we conclude τ +g = τ +g_{1} = τ +g_{2} = ρ as well as

ρ̇ = 1 − 2B_{1}(ρ), ρ/τ|_{τ→−∞} → 1.

The stationary solution of this equation is ρ = ρ_{0}, where ρ_{0} is such that B_{1}(ρ_{0}) = 1/2. Since 0 < B_{1} < 1 we see that ρ̇ > 0. Accordingly, ρ is an increasing function which tends to ρ_{0}.

This allows us to calculate the solution for Δψ_{0} > 0 (i.e., after the interaction). Notice that in that case τ → ∞.

We introduce the function G(τ) = τ + g(τ). By the previous, Ġ = ρ̇, G/τ|_{τ→−∞} → +1, and we choose

G = − ∫_{−∞}^{∞}(1 − 2B_{1}(ρ)) dτ′ + ρ_{0}.

On the other hand, we can express the functions u_{i} via the function G:

u_{i} = \frac{ψ_{0}(0)u_{1}^{0}}{εG} →^{τ→∞} \frac{ψ_{0}(0)u_{1}^{0}}{ερ_{0}} .

We calculate the limit (φ_{k})_{t}^{+} as τ → ∞ of the velocities of the weak discontinuities

(φ_{2})_{t}^{+} = 2u_{0} − \frac{2ψ_{0}(0)u_{1}^{0}}{ερ_{0}} \frac{1}{2}ερ_{0} = 2u_{0} + (a_{1} − a_{2})u_{1}^{0},
(φ_{1})_{t}^{+} = 2u_{0} − \frac{2ψ_{0}(0)u_{1}^{0}}{ερ_{0}} \frac{1}{2}ερ_{0} = 2u_{0} + (a_{1} − a_{2})u_{1}^{0},

which coincides with the velocity of the shock wave

U(x, t) := u_{0}^{0} + (a_{1} − a_{2})u_{1}^{0}θ(−x + φ^{+}(t)),

where φ^{+} = φ_{2}^{+} = φ_{1}^{−}.

By using the explicit formula for the solution u_{ε}(x, t), we can easily show that

w − lim_{ε→0} u_{ε}(x, t) = U(x, t), t > t^{∗}.

To this end, we rewrite the above-constructed solution u_{ε}(x, t) in the form

u_{ε}(x, t) = u_{0} + u_{1}(φ_{1} − φ_{2})θ_{ε,1}(φ_{1} − x) + u_{1}(x − φ_{2})[θ_{ε,2}(φ_{2} − x) − θ_{ε,1}(φ_{1} − x)].

Consider the second term. We have

u_{1}(φ_{1} − φ_{2}) = \frac{ψ_{0}(0)u_{1}^{0}ρ}{G} = ψ_{0}(0)u_{1}^{0} = (a_{1} − a_{2})u_{1}^{0} =^{def} U_{0}.

Since φ_{1}|_{t>t^{∗}} ≃ φ^{+}, the first two terms pass into the shock wave U(x, t) for t > t^{∗}. Consider the last term

u_{1}(x − φ_{2})[θ_{ε,2}(φ_{2} − x) − θ_{ε,1}(φ_{1} − x)] = u_{1}(x − φ_{2})[\frac{θ_{ε,2}(φ_{2} − x) − θ_{ε,1}(φ_{1} − x)}{φ_{1} − φ_{2}}]

As was already shown, the coefficient of the expression in braces is a constant. The expression in square brackets is an approximation of the δ-function at the point φ_{2}. Hence the entire expression in braces is small (in a weak sense) as ε → 0.

3.2. Interaction of a weak discontinuity and a shock wave In this case mechanism of evolution is as follows: segments of the broken line

interact with the step that has already been formed.

In order to study this we consider the Hopf equation again. The initial condition corresponding to this type of interaction has the form

u|_{t=0} = u_{0}^{0}θ(a_{1}^{0} − x) + u_{1}^{0}(a_{1} − x)θ(a_{1} − x) − u_{1}^{0}(a_{2} − x)θ(a_{2} − x), (16)

where u_{0}^{0}, u_{1}^{0} are positive constants and a_{1} > a_{2}. Just as before, we construct the weak asymptotic solution in the form

u_{ε}(x, t) = u_{0}(t, ε)θ_{ε,1}(φ_{1}(t, ε) − x) + u_{1}(t, ε)(φ_{1}(t, ε) − x)θ_{ε,1}(φ_{1}(t, ε) − x) − u_{1}(t, ε)(φ_{2}(t, ε) − x)θ_{ε,2}(φ_{2}(t, ε) − x), (17) where u_{i} = u_{i}(t, ε), φ_{i} = φ_{i}(t, ε) are unknown C^{1}(R^{+}) functions for every fixed ε which satisfy the following initial conditions:

u_{i}(0, ε) = u_{i}^{0}, φ_{i}(0, ε) = a_{i}, i = 1, 2.

Denote Δφ = φ_{2} − φ_{1}. Substituting expression for u_{ε}(x, t) into the Hopf equation and repeating the procedure from the previous subsection we obtain the following system of equations determining the unknown functions from (17)

φ_{1t} − u_{0} + 2u_{1}(φ_{2} − φ_{1})B_{1}(Δφ/ε) = 0,  (18)
u_{0t} − u_{0}u_{1}(1 − 2B_{1}(Δφ/ε)) = 0,  (19)
u_{1t} − 2u_{1}^{2} + 4u_{1}^{2}B_{1}(Δφ/ε) = 0,  (20)
φ_{2t} − 2u_{0}B_{2} + 2u_{1}ψB_{2}(Δφ/ε) = 0.  (21)

Before the interaction, we have φ_{2} < φ_{1}, Δφ/ε ∼ −∞, and B_{1} = 0, B_{2} = 1 with arbitrary accuracy in ε. Denoting by φ_{10}, u_{10}, φ_{20}, u_{00} the solution of system (18)–(21) with B_{1} = 0, B_{2} = 1 (i.e. before the interaction), we obtain the following system of equations for these functions:

φ_{10t} = u_{00},
u_{10t} = 2u_{10}^{2},
u_{00t} = u_{00}u_{10},
φ_{20t} = 2(u_{00} − u_{10}ψ_{0}),  ψ_{0} = φ_{20} − φ_{10}.

The solutions of this system are:

u_{10} = \frac{u_{1}^{0}}{1 − 2u_{1}^{0}t}, u_{00} = \frac{u_{0}^{0}}{(1 − 2u_{1}^{0}t)^{1/2}} ,

φ_{20} = a_{2} + 2Ut,  φ_{10} = a_{1} + ∫_{0}^{t} u_{00} dt,  (22)
ψ_{0} = \frac{1}{u_{1}^{0}} [(ψ_{0}^{0}u_{1}^{0} − u_{0}^{0})(1 − 2u_{0}^{1}t) − u_{0}^{0}\sqrt{1 − 2u_{0}^{1}t}],  (23)
U = u_{0}^{0} + u_{1}^{0}(a_{1} − a_{2}).

One can easily see that the function ψ_{0}(t) vanishes at the two points t_{1} = 1/2u_{0}^{1} and t^{∗} such that

\sqrt{1 − 2u_{0}^{1}t^{∗}} = \frac{u_{0}^{0}}{U} .

Obviously, t^{∗} < t_{1} and x = φ_{10} and x = φ_{20} merge at t = t^{∗}. In this case we have

u_{00}(t^{∗}) = u_{00}^{∗} ≡ U, u_{10}(t^{∗}) = \frac{U^{2}}{(u_{0}^{0})^{2}} u_{1}^{0} < ∞.

Thus, in this example the mechanism of formation of a new shock wave consists not in turning over the inclined segment of the broken line, as in the preceding example, but in the disappearance of this inclined segment due to increasing vertical segment.

Subtracting equation (20) from equation (18), we obtain the following equation for the function ψ:

ψ_{t} = (u_{0} − 2ψu_{1})(1 − 2B_{1}(Δφ/ε))

or, denoting ρ = Δφ/ε = (ψ_{0} + ψ_{0}ψ_{1}(τ))/ε, τ = ψ_{0}/ε,

ψ′_{0}ρ̇ = [u_{0} − 2ψu_{1}](1 − 2B_{1}(ρ)), \frac{ρ}{τ} |_{τ→−∞} → 1.

Note that we can use the formula for ψ_{0} (and for the functions u_{00} and u_{10}) only for t ∈ [0, t^{∗} + δ], where δ > 0 is any number such that δ < t_{1} − t^{∗}.

To obtain formulas that are global in t, we need to choose a number δ and continue the functions u_{00}, u_{10}, and ψ_{0} smoothly to the time t ≥ t^{∗} + δ so that the function u_{ε} remains the (weak asymptotic) solution of Cauchy problem (10), (16). Since u_{0} − 2ψu_{1} = \frac{2u_{1}^{0}}{\sqrt{1−2u_{1}^{0}t}} + u_{0}^{0} + (a_{1} − a_{2})u_{1}^{0} > 0, for t < t^{∗} we see that ρ̇ > 0 in

that interval. Like in Section 3.1 we conclude that there exists a solution ρ → ρ_{0} while τ → +∞, where ρ_{0} is a root of the equation B_{1}(ρ) = 1/2. This implies that ρ̇ → 0, τ → +∞ which means that when t > t^{∗} we have φ_{1}(t) = φ_{2}(t). From (20) it is easy to compute φ_{i}(t), i = 1, 2 for t > t^{∗}. We have, φ_{i}(t) = U, i = 1, 2, and from initial conditions for φ_{i}, i = 1, 2, one sees that for t > t^{∗}

φ_{1}(t, ε) = φ_{10}(t^{∗}) + Ut + O(ε), φ_{2} = φ_{20}(t^{∗}) + Ut + O(ε).  (24)

Let us consider the system of equations for the functions u_{0} and u_{1}. From (19) and (20) we see that

u_{1}(t, ε) = \frac{u_{1}^{0}}{1 − 2u_{1}^{0} ∫_{0}^{t}[1 − 2B_{1}(ρ(τ))] dt′} .

Since ∫_{0}^{t^{∗}}(1 − 2B_{1}) dt′ ≤ t^{∗}, we have u_{1}(t^{∗}, ε) ≤ u_{10}(t^{∗}). On the other hand, we have t > t^{∗} for ρ → ρ_{0}. Therefore, ψ_{1}(τ) → −1 as τ → ∞ and hence (Δφ) → 0 as τ → ∞ (i.e., for t > t^{∗} and ε → 0). This implies that for t > t^{∗} we have

u_{1}(t, ε) = u_{10}(t^{∗}) + o(1), ε → 0.

We represent the above-constructed solution in the form

u_{ε}(x, t) = u_{0}θ_{ε,1}(φ_{1} − x) + u_{1}(φ_{1} − x)θ_{ε,1}(φ_{1} − x) − u_{1}(φ_{2} − x)θ_{ε,2}(φ_{2} − x)
=Uθ_{ε,1}(φ_{1} − x) + (u_{0} − U)θ_{ε,1}(φ_{1} − x) + u_{1}(φ_{1} − φ_{2})θ_{ε,1}(φ_{1} − x)
+ [θ_{ε,1}(φ_{1} − x) − θ_{ε,2}(φ_{2} − x)](φ_{2} − x)u_{1}.

Since when t > t^{∗} we have U − u_{0} → 0 and φ_{2} − φ_{1} → 0 the second and the third term in the previous expression tends to zero while the fourth term tends to zero because

[θ_{ε,1}(φ_{1} − x) − θ_{ε,2}(φ_{2} − x)](φ_{2} − x)u_{1}

= \frac{θ_{ε,1}(φ_{1} − x) − θ_{ε,2}(φ_{2} − x)}{φ_{2} − φ_{1}}(φ_{2}−x)(φ_{2}−φ_{1})u_{1} → δ(φ−x)(φ_{2}−x)(U −u_{0}) = 0

Since for t > t^{∗}, ε → 0, from (24) we have φ_{1} = a_{1} + Ut, the first term, and therefore constructed function u_{ε}(x, t) when t > t^{∗}, approximates the shock wave

u = Uθ(a_{1} + Ut − x).

4. Standard examples of Cauchy problems for Hopf equation

In the next four sections we will inspect only the development of the initial condition before the second interaction. After the first interaction we will have the analogous problem as in the start of our considerations or the problem will be deduced to some of situations we have already analyzed (more precisely, equation is the same while initial condition is analogous to one of those we have already considered). Using same methods as before one can prove that the functions which describe the solution of appropriate problem from the k-th till the k + 2-nd interaction and from the k + 1-st till the k + 3-rd interaction match in the common part of the intervals (t_{k}^{∗}, t_{k+1}^{∗} + δ_{k}), δ_{k} < t_{k+2}^{∗} − t_{k+1}^{∗}, k ∈ N, where, as usual, t_{k}^{∗} is the time of k-th interaction. Therefore, we will be able to write the uniform solution using partition of unity of the time axis.

In what follows, functions ϕ which we were using for “smoothing” functions describing moving of points a_{i} will depend not only on the “fast” variable τ but on the variable t, too. Therefore, we will need the following lemma:

Lemma 4.1. Suppose that for the functions φ(t, τ) and F (t, τ) where τ = \frac{ψ_{0}(t)}{ε}, ψ_{0} a smooth increasing function which has unique zero in t = t^{∗}, we have

(φ(t, τ))_{t} = F (t, τ),  (25)

as well as F (·, τ) − F (·, −∞) ∈ S(R) and F (·, τ) − F (·, ∞) ∈ S(R). Then there exist functions ϕ(t, τ), t ∈ R^{+}, τ ∈ R such that

φ(t, τ) = φ^{−}(t) + ψ_{0}(t)ϕ(t, τ) + O(ε),  (26)

where φ^{−}(t) = φ(t, −∞), t ∈ R^{+}. Furthermore, we have

ϕ = \frac{1}{ψ′_{0}τ} ∫_{0}^{τ} (F (t, τ) − φ^{−}(t) − ψ_{0}(t)ϕ_{t}^{+}(t)ω(τ)) dτ′,  (27)

where ϕ^{+} is given by

ψ_{0}(t)ϕ^{+}(t) = ∫_{t^{∗}}^{t} (F (t, +∞) − φ_{t}^{−}(t)) dt′

while ω is such that lim_{z→∞} w(z) = 1, lim_{z→−∞} w(z) = 0, and \frac{dω}{dz} ∈ S(R).

Proof. Substituting (26) in (25) we obtain

φ_{t}^{−} + ψ_{0t}(τϕ)_{τ} + ψ_{0} \frac{∂ϕ}{∂t} = F.

So, for proving that (27) is the function satisfying the lemma, we have to prove that

φ_{t}^{−} + (F − φ_{t}^{−} − ψ_{0}ωϕ_{t}^{+}) + ψ_{0} \frac{∂ϕ}{∂t} = F + O(ε),

i.e. that,

ε\frac{∂}{∂t} (τϕ − τωϕ^{+}) = O(ε).

Substituting (27) in the above equation we have

ε\frac{∂}{∂t} (\frac{1}{ψ′_{0}} ∫_{0}^{τ} (F (t, τ̃) − φ_{t}^{−}(t) − ψ_{0}(t)ϕ_{t}^{+}(t)ω(τ̃) − ψ′_{0}(t)ϕ^{+}(t)ω(τ̃)) dτ̃) = O(ε).

Using the fact that φ^{−} solves the equation φ_{t} = F^{−} = F (: ·: , −∞) and that φ_{t}^{−} + ψ_{0}ϕ_{t}^{+} + ψ′_{0}ϕ^{+} = (φ^{−}ψ_{0}ϕ^{+})_{t} = F^{+} = F (: ·: , ∞) we have,

ε\frac{∂}{∂t} (\frac{1}{ψ′_{0}} ∫_{−∞}^{τ} (F (t, τ̃) − F^{+}(t)ω(τ̃) − F^{−}(t)(1 − ω(τ̃))) dτ̃) = O(ε).  (28)

Since we have F (·, τ) − F (·, −∞) ∈ S(R) and F (·, τ) − F (·, ∞) ∈ S(R), for every t ∈ R^{+}, we also have \frac{∂}{∂t} (F (·, τ) − F (·, −∞)) ∈ S(R) and \frac{∂}{∂t} (F (·, τ) − F (·, ∞)) ∈ S(R). Furthermore, since in (28) we have partial derivative in t (derivative does not affect τ), we conclude that the last integral is bounded and, consequently, that equality given by (28) is correct.

In what follows we will need the following functions:

τ^{ij} = \frac{φ_{j0}(t) − φ_{i0}(t)}{ε},
ρ^{ij} = \frac{Δ^{ij}φ(t)}{ε} = \frac{φ_{j}(t) − φ_{i}(t)}{ε},
B_{k}^{ij} = B_{k}(ρ^{ij}), t ∈ R^{+}.

4.1. Confluence of weak discontinuities and the shock wave, four point case; the first possibility

In this section we will search asymptotic solution of equation (10) with the initial condition which has three weak discontinuities and one shock wave. We will analyze the transformation of the initial condition till the second interaction. The initial condition is:

u(x, 0) = h + u_{1}^{0}(a_{1} − x) − u_{1}^{0}(a_{2} − x)_{+} + u_{0}^{0}θ(a_{2} − x)

+ u_{2}^{0}(a_{2} − x)_{+} − u_{2}^{0}(a_{3} − x)_{+} + u_{3}^{0}(a_{3} − x)_{+} − u_{3}^{0}(a_{4} − x)_{+}, (29)

where a_{1} > a_{2} > a_{3} > a_{4} as well as v_{0}^{0}, v_{1}^{0}, v_{2}^{0}, h > 0, u_{0}^{0} > 0, u_{1}^{0} > 0, u_{2}^{0} > 0 and u_{3}^{0} are constants in R such that (a_{1} − a_{2})u_{1}^{0} = v_{0}^{0}, (a_{2} − a_{3})u_{2}^{0} = v_{1}^{0} and (a_{3} − a_{4})u_{3}^{0} = v_{2}^{0}. The weak asymptotic solution of problem (10), (29) we have in the form:

u_{ε}(x, t) = h + u_{1}(φ_{1} − x)θ_{ε}(φ_{1} − x) − u_{1}(φ_{2} − x)θ_{ε}(φ_{2} − x)

+ u_{0}θ_{ε}(φ_{2} − x) + u_{2}(φ_{2} − x)θ_{ε}(φ_{2} − x) − u_{2}(φ_{3} − x)θ_{ε}(φ_{3} − x) + u_{3}(φ_{3} − x)θ_{ε}(φ_{3} − x) − u_{3}(φ_{4} − x)θ_{ε}(φ_{4} − x), (30)

where φ_{i}(0) = a_{i}, i = 1, 2, 3, 4, and u_{i}(0) = u_{i}^{0}, i = 0, 1, 2, 3. Like in the previous section we will just write down the equations for the unknown functions φ_{i} and u_{i−1}, i = 1, 2, 3, 4, in the interval [0, t_{1}^{∗} + δ). Here, we will assume that the first interaction will happen between the points a_{2} and a_{3}.

34 V. G. Danilov, D. Mitrović

Accordingly, system of equations determining unknown functions in (30) in the interval [0, t_{1}^{∗}), δ < t_{2}^{∗} − t_{1}^{∗}, where t_{2}^{∗} is time of the second interaction, is:

φ_{1t} = 2h
φ_{2t} = u_{0} + 2h + 2v_{0}^{0} − 2u_{2}Δ^{23}φB_{1}^{23} + 2u_{3}Δ^{23}φB_{1}^{23} = 0
φ_{3t} = 2h + 2v_{0}^{0} + 2u_{1}Δ^{23}φ + 2u_{0}B_{2}^{23} − 2u_{1}Δ^{23}φB_{2}^{23} − 2u_{2}Δ^{23}φB_{2}^{23} = 0
φ_{4t} = 2h + 2v_{0}^{0} + 2v_{1}^{0} + v_{2}^{0} + 2u_{0}  (31)
u_{0t} − u_{0}(u_{2} + u_{1}) + 2u_{0}u_{2}B_{1}^{23} − 2u_{0}u_{3}B_{1}^{23} = 0
u_{1t} − u_{1}^{2} = 0
u_{2t} − 2u_{2}^{2} − 4u_{1}u_{2}B_{1}^{23} + 4u_{1}u_{3}B_{1}^{23} − 4u_{2}u_{3}B_{1}^{23} − 4u_{2}^{2}B_{1}^{23} = 0
u_{3t} − 2u_{3}^{2} = 0.

The solutions of the system are:

φ_{1}(t) = 2ht + a_{1}, φ_{4}(t) = 2(h + v_{0}^{0} + v_{1}^{0} + v_{2}^{0})t + 2 ∫_{0}^{t} u_{0}(t′)dt′ + a_{4}

u_{1}(t) = \frac{u_{1}^{0}}{1 − 2u_{1}^{0}t}, u_{3}(t) = \frac{u_{3}^{0}}{1 − 2u_{3}^{0}t}

u_{0}(t) = exp(∫_{0}^{t} (u_{2} + u_{1} − 2u_{2}B_{1}^{23} + 2u_{3}B_{1}^{23}) dt′ + u_{0}^{0}),

φ_{2}(t) = φ_{20}(t) + ψ_{0}^{23}(t)ϕ_{2}(t, τ^{23})  (32)

φ_{3}(t) = φ_{30}(t) + ψ_{0}^{23}(t)ϕ_{3}(t, τ^{23}) u_{2}(t) = u_{20}(t) + ψ_{0}^{23}(t)κ(t, τ),

where ψ_{0}^{23} = φ_{30} − φ_{20}, τ^{23} = \frac{ψ_{0}^{23}}{ε} and

φ_{20}(t) = (2h + 2v_{0}^{0})t + ∫_{0}^{t} 2u_{0}(t)dt + a_{2},

φ_{30}(t) = 2(h + v_{0}^{0} + v_{1}^{0})t + ∫_{0}^{t} 2u_{0}(t)dt + a_{3},

u_{20}(t) = \frac{u_{2}^{0}}{1 − 2u_{2}^{0}t},

ρ̇^{23}ψ_{0t}^{23} = (u_{0} − 2u_{2}Δ^{23})(1 − 2B_{1}^{23}) + 2u_{1}Δ^{23}φB_{1}^{23} − 2u_{3}Δ^{23}φB_{1}^{23}. Functions ρ^{23}(τ^{23}), ϕ_{2}(t, τ^{23}), ϕ_{3}(t, τ^{23}) and κ(t, τ) are such that

\frac{ρ^{23}(τ^{23})}{τ^{23}} |_{τ^{23}→−∞} = 1,

B_{i}(ρ^{23}(t, τ^{23})|_{τ^{23}→+∞}) = 1/2, i = 1, 2, ϕ_{i}(t, τ^{23})|_{τ^{23}→−∞} = 0,

\frac{dϕ_{i}(t, τ^{23}}{dτ^{23}} |_{τ^{23}→−∞} = o((τ^{23})^{−1}), i = 2, 3,

κ(t, τ^{23})|_{τ^{23}→−∞} = 0,

\frac{dκ(t, τ^{23})}{τ^{23}} = o((τ^{23})^{−1}).

Expressions for functions ϕ_{2}(t, τ^{23}), ϕ_{3}(t, τ^{23}) and κ(t, τ) we obtain from system (31) and Lemma 4.1.

Remark 4.2. While we obtain expressions for ϕ_{2}(t, τ^{23}) and ϕ_{3}(t, τ^{23}) explicitly from Lemma 4.1, for κ(t, τ) we have u_{2t} = F (t, τ, u_{2}) wherefrom we obtain the relation

a(t, τ)κ^{2} + b(t, τ)κ + c(t, τ) = 0,

for appropriate C^{1}(R^{+} × R) functions a, b and c. From here, using boundary condition for the function κ we find the unknown function κ.

4.2. Confluence of weak discontinuities and the shock wave, four point case; the second possibility

The disposure of discontinuities in the initial data is here different than in the previous section. Here, we want to explore interaction of weak discontinuity and the shock wave in the case when the shock wave is above the weak discontinuity. We will analyze the evolution of the initial condition till the second interaction. The initial condition is:

u(x, 0) = h + u_{1}^{0}(a_{1} − x) − u_{1}^{0}(a_{2} − x)_{+} + u_{2}^{0}(a_{2} − x)_{+}

+ u_{0}^{0}(a_{3} − x)_{+} − u_{2}^{0}(a_{3} − x)_{+} + u_{3}^{0}(a_{3} − x)_{+} − u_{3}^{0}(a_{4} − x)_{+}, (33)

where a_{1} > a_{2} > a_{3} > a_{4} are constants in R, u_{0}^{0}, u_{1}^{0}, u_{2}^{0} and u_{3}^{0} are positive constants such that (a_{1} − a_{2})u_{1}^{0} = v_{0}^{0}, (a_{2} − a_{3})u_{2}^{0} = v_{1}^{0} and (a_{3} − a_{4})u_{3}^{0} = v_{2}^{0}. The weak asymptotic solution of problem (10), (33) we will search in the form:

u_{ε}(x, t) = h + u_{1}(φ_{1} − x)θ_{ε}(φ_{1} − x) − u_{1}(φ_{2} − x)θ_{ε}(φ_{2} − x) + u_{2}(φ_{2} − x)θ_{ε}(φ_{2} − x)

− u_{2}(φ_{3} − x)θ_{ε}(φ_{3} − x) + u_{0}θ_{ε}(φ_{3} − x) + u_{3}(φ_{3} − x)θ_{ε}(φ_{3} − x) − u_{3}(φ_{4} − x)θ_{ε}(φ_{4} − x), (34)

where φ_{i}(0) = a_{i}, i = 1, 2, 3, 4, and u_{i}(0) = u_{i}^{0}, i = 0, 1, 2, 3. Here, we will assume that the first interaction will happen between the points a_{2} and a_{3}.

Accordingly, system of equations determining unknown functions in (34) in the interval [0, t_{1}^{∗} + δ), δ < t_{2}^{∗} − t_{1}^{∗}, where t_{2}^{∗} is time of the second interaction, is:

φ_{1t} = 2h
φ_{2t} = 2h + 2v_{0}^{0} + 2u_{0}B_{1}^{23} − 2u_{2}Δ^{23}φB_{1}^{23} + 2u_{3}Δ^{23}φB_{1}^{23}
φ_{3t} = 2h + u_{0} + 2v_{0}^{0} − 2u_{1}Δ^{23}φ + 2u_{1}Δ^{23}φB_{2}^{23} − 2u_{2}Δ^{23}φB_{2}^{23} = 0
φ_{4t} = 2h + 2v_{0}^{0} + 2v_{1}^{0} + v_{2}^{0} + 2u_{0}

u_{0t} − u_{0}(u_{2} + u_{1}) + 2u_{0}u_{2}B_{1}^{23} − 2u_{0}u_{3}B_{1}^{23} = 0
u_{1t} − u_{1}^{2} = 0
u_{2t} − 2u_{2}^{2} − 4u_{1}u_{2}B_{1}^{23} + 4u_{1}u_{3}B_{1}^{23} − 4u_{2}u_{3}B_{1}^{23} = 4u_{2}^{2}B_{1}^{23} = 0
u_{3t} − 2u_{3}^{2} = 0.

The solutions of the system are:

φ_{1}(t) = 2ht + a_{1}, φ_{4}(t) = 2(h + v_{1}^{0} + v_{2}^{0} + v_{3}^{0})t + 2 ∫_{0}^{t} u_{0}(t′)dt′ + a_{4}

u_{1}(t) = \frac{u_{1}^{0}}{1 − 2u_{1}^{0}t}, u_{3}(t) = \frac{u_{3}^{0}}{1 − 2u_{3}^{0}t}

φ_{2}(t) = φ_{20}(t) + ψ_{0}^{23}(t)ϕ_{2}(t, τ^{23})  (35)
u_{0}(t) = exp(∫_{0}^{t} (u_{2} + u_{1} − 2u_{2}B_{1}^{23} + 2u_{3}B_{1}^{23}) dt′ + u_{0}^{0}),

φ_{3}(t) = φ_{30}(t) + ψ_{0}^{23}(t)ϕ_{3}(t, τ^{23})

u_{2}(t) = u_{20}(t) + ψ_{0}^{23}(t)κ(t, τ),

where ψ_{0}^{23} = φ_{30} − φ_{20}, τ^{23} = \frac{ψ_{0}^{23}}{ε} and

φ_{20}(t) = 2(h + v_{0}^{0})t + ∫_{0}^{t} 2u_{0}(t)dt + a_{2}
φ_{20}(t) = 2(h + v_{1}^{0} + v_{0}^{0})t + ∫_{0}^{t} 2u_{0}(t)dt + a_{3}

u_{20}(t) = \frac{u_{2}^{0}}{1 − 2u_{2}^{0}t},

ρ̇^{23}ψ_{0t}^{23} = (u_{0} − 2u_{2}Δ^{23})(1 − 2B_{1}^{23}) − 2u_{1}Δ^{23}φB_{1}^{23} − 2u_{3}Δ^{23}φB_{1}^{23}.

Functions ρ^{23}(τ^{23}), ϕ_{2}(t, τ^{23}), ϕ_{3}(t, τ^{23}) and κ(t, τ^{23}) have the same properties as in the previous section and we obtain them in the completely same manner.

4.3. Confluence of weak discontinuities, four point case

In this section we will search asymptotic solution of equation (10) with the initial condition which has four weak discontinuities. We will analyze the transformation of the initial condition till the second interaction. The initial condition is:

u(x, 0) = h + u_{1}^{0}(a_{1} − x) − u_{1}^{0}(a_{2} − x)_{+} + u_{2}^{0}(a_{2} − x)_{+}

− u_{2}^{0}(a_{3} − x)_{+} + u_{3}^{0}(a_{3} − x)_{+} − u_{3}^{0}(a_{4} − x)_{+}, (36)

where a_{1} > a_{2} > a_{3} > a_{4} are constants in R, u_{1}^{0}, u_{2}^{0} and u_{3}^{0} are positive constants such that (a_{1} − a_{2})u_{1}^{0} = v_{0}^{0}, (a_{2} − a_{3})u_{2}^{0} = v_{1}^{0} and (a_{3} − a_{4})u_{3}^{0} = v_{2}^{0}. The weak asymptotic solution of problem (10), (36) we will search in the form:

u_{ε}(x, t) = h + u_{1}(φ_{1} − x)θ_{ε}(φ_{1} − x) − u_{1}(φ_{2} − x)θ_{ε}(φ_{2} − x) + u_{2}(φ_{2} − x)θ_{ε}(φ_{2} − x) − u_{2}(φ_{3} − x)θ_{ε}(φ_{3} − x) + u_{3}(φ_{3} − x)θ_{ε}(φ_{3} − x) − u_{3}(φ_{4} − x)θ_{ε}(φ_{4} − x), (37) where φ_{i}(0) = a_{i}, i = 1, 2, 3, 4, and u_{i}(0) = u_{i}^{0}, i = 1, 2, 3. Like in the previous section we will just write down the equations for the unknown functions φ_{i} and u_{i−1}, i = 1, 2, 3, 4, in the interval [0, t_{1}^{∗} + δ). We will assume that the first interaction will happen between the points a_{2} and a_{3}.

Accordingly, system of equations determining unknown functions in (37) in the interval [0, t_{1}^{∗} + δ), δ < t_{2}^{∗} − t_{1}^{∗}, where t_{2}^{∗} is time of the second interaction, is:

φ_{1t} = 2h
φ_{2t} = 2(h + v_{0}^{0}) − 2u_{2}Δ^{23}φB_{1}^{23} + 2u_{3}Δ^{23}φB_{1}^{23}
φ_{3t} = 2(h + v_{0}^{0}) − 2u_{1}Δ^{23}φ + 2u_{1}Δ^{23}φB_{2}^{23} − 2u_{2}Δ^{23}φB_{2}^{23} = 0
φ_{4t} = 2h + 2v_{0}^{0} + 2v_{1}^{0} + v_{2}^{0}
u_{1t} − u_{1}^{2} = 0
u_{2t} − 2u_{2}^{2} − 4u_{1}u_{2}B_{1}^{23} + 4u_{1}u_{3}B_{1}^{23} − 4u_{2}u_{3}B_{1}^{23} = 4u_{2}^{2}B_{1}^{23} = 0
u_{3t} − 2u_{3}^{2} = 0.

The solutions of the system are:

φ_{1}(t) = 2ht + a_{1}, φ_{4}(t) = 2(h + v_{1}^{0} + v_{2}^{0} + v_{3}^{0})t + 2 ∫_{0}^{t} u_{0}(t′) dt′ + a_{4}

u_{1}(t) = \frac{u_{1}^{0}}{1 − 2u_{1}^{0}t}, u_{3}(t) = \frac{u_{3}^{0}}{1 − 2u_{3}^{0}t}

φ_{2}(t) = φ_{20}(t) + ψ_{0}^{23}(t)ϕ_{2}(t, τ^{23})  (38)

φ_{3}(t) = φ_{30}(t) + ψ_{0}^{23}(t)ϕ_{3}(t, τ^{23})

u_{2}(t) = u_{20}(t) + ψ_{0}^{23}(t)κ(t, τ),

where ψ_{0}^{23} = φ_{30} − φ_{20}, τ^{23} = \frac{ψ_{0}^{23}}{ε} and

φ_{20}(t) = 2(h + v_{0}^{0})t + a_{2}

φ_{20}(t) = 2(h + v_{1}^{0} + v_{0}^{0})t + a_{3}

u_{20}(t) = \frac{u_{2}^{0}}{1 − 2u_{2}^{0}t},

ρ̇^{23} = 1 − 2B_{1}^{23}

Functions ρ^{23}(τ^{23}), ϕ_{2}(t, τ^{23}), ϕ_{3}(t, τ^{23}) and κ(t, τ^{23}) have the same properties as in the previous section and we obtain them in the completely same manner.

4.4. Confluence of two shock waves connected with the straight line We will analyze the transformation of the initial condition till the second in-

teraction. The initial condition is:

u(x, 0) = h + u_{1}^{0}(a_{1} − x)_{+} − u_{1}^{0}(a_{2} − x)_{+}u_{1}^{0}(a_{2} − x)_{+} + u_{2}^{0}(a_{2} − x)_{+}

+ u_{2}^{0}(a_{3} − x)_{+} − u_{2}^{0}(a_{3} − x)_{+} + u_{3}^{0}(a_{3} − x)_{+} − u_{3}^{0}(a_{4} − x)_{+}, (39) where a_{1} > a_{2} > a_{3} > a_{4} are constants in R, u_{0}^{0}, u_{1}^{0}, u_{2}^{0} and u_{3}^{0} are positive constants such that (a_{1} − a_{2})u_{1}^{0} = v_{0}^{0}, (a_{2} − a_{3})u_{2}^{0} = v_{1}^{0} and (a_{3} − a_{4})u_{3}^{0} = v_{2}^{0}. The weak asymptotic solution of problem (10), (39) we have in the form:

u_{ε}(x, t) = h + u_{1}(φ_{1} − x)θ_{ε}(φ_{1} − x) − u_{1}(φ_{2} − x)θ_{ε}(φ_{2} − x) + u_{01}θ_{ε}(φ_{2} − x)

+ u_{2}(φ_{2} − x)θ_{ε}(φ_{2} − x) + −u_{2}(φ_{3} − x)θ_{ε}(φ_{3} − x) + u_{02}θ_{ε}(φ_{3} − x) + u_{3}(φ_{3} − x)θ_{ε}(φ_{3} − x) − u_{3}(φ_{4} − x)θ_{ε}(φ_{4} − x), (40)

where φ_{i}(0) = a_{i}, i = 1, 2, 3, 4, and u_{i}(0) = u_{i}^{0}, i = 0, 1, 2, 3. Like in the previous section we will just write down the equations for the unknown functions φ_{i} and u_{i−1}, i = 1, 2, 3, 4, in (40) in the interval [0, t_{1}^{∗} + δ). Here, we will assume that the first interaction will happen between the points a_{2} and a_{3}.

System of equations determining unknown functions in (40) in the interval [0, t_{1}^{∗} + δ), δ < t_{2}^{∗} − t_{1}^{∗}, where t_{2}^{∗} is time of the second interaction, is:

φ_{1t} = 2h
φ_{2t} = 2h + 2v_{0}^{0} + u_{01} + 2u_{02}B_{1}^{23} − 2u_{2}Δ^{23}φB_{1}^{23} + 2u_{3}Δ^{23}φB_{1}^{23}
φ_{3t} = 2h + u_{02} − 2u_{01}B_{2}^{23} + 2v_{0}^{0} − 2u_{1}Δ^{23}φB_{1}^{23} − 2u_{2}Δ^{23}φB_{2}^{23}
φ_{4t} = 2h + 2v_{0}^{0} + 2v_{1}^{0} + v_{2}^{0} + 2u_{01} + 2u_{02}
u_{01t} − u_{01}(u_{2} + u_{1}) + 2u_{01}u_{2}B_{1}^{23} − 2u_{01}u_{3}B_{1}^{23} = 0
u_{02t} − u_{02}(u_{3} + u_{2}) + 2u_{02}u_{1}B_{1}^{23} − 2u_{02}u_{2}B_{1}^{23} = 0
u_{1t} − u_{1}^{2} = 0
u_{2t} − 2u_{2}^{2} − 4u_{1}u_{2}B_{1}^{23} + 4u_{1}u_{3}B_{1}^{23} − 4u_{2}u_{3}B_{1}^{23} = 4u_{2}^{2}B_{1}^{23} = 0
u_{3t} − 2u_{3}^{2} = 0.

The solutions of the system are:

φ_{1}(t) = 2ht + a_{1}, φ_{4}(t)=2(h + v_{1}^{0} + v_{2}^{0} + v_{3}^{0})t+2 ∫_{0}^{t}(u_{01}(t′) + u_{02}(t′))dt′ + a_{4}

u_{1}(t) = \frac{u_{1}^{0}}{1 − 2u_{1}^{0}t}, u_{3}(t) = \frac{u_{3}^{0}}{1 − 2u_{3}^{0}t}

u_{0}(t) = exp(∫_{0}^{t} (u_{2} + u_{1} − 2u_{2}B_{1}^{23} + 2u_{3}B_{1}^{23}) dt′ + u_{0}^{0}),

φ_{2}(t) = φ_{20}(t) + ψ_{0}^{23}(t)ϕ_{2}(t, τ^{23})

φ_{3}(t) = φ_{30}(t) + ψ_{0}^{23}(t)ϕ_{3}(t, τ^{23})

u_{2}(t) = u_{20}(t) + ψ_{0}^{23}(t)κ(t, τ),

where ψ_{0}^{23} = φ_{30} − φ_{20}, τ^{23} = \frac{ψ_{0}^{23}}{ε} and

φ_{20}(t) = 2(h + v_{0}^{0})t + ∫_{0}^{t} 2u_{0}(t)dt + a_{2}

φ_{20}(t) = 2(h + v_{1}^{0} + v_{0}^{0})t + ∫_{0}^{t} 2u_{0}(t)dt + a_{3}

u_{20}(t) = \frac{u_{2}^{0}}{1 − 2u_{2}^{0}t},

ρ̇^{23}ψ_{0t}^{23} = (u_{0} − 2u_{2}Δ^{23})(1 − 2B_{1}^{23}) − 2u_{1}Δ^{23}φB_{1}^{23} − 2u_{3}Δ^{23}φB_{1}^{23}.

Functions ρ^{23}(τ^{23}), ϕ_{2}(t, τ^{23}), ϕ_{3}(t, τ^{23}) and κ(t, τ) have the same properties and we obtain them in the same way as before.

5. Interaction of weak discontinuities and generation of shock waves; n-point case.

In this section we will solve equation (10) with the polygon with n knots as an initial condition. First, we will write down the solution of the problem in the interval [0, t_{1}^{∗} + δ_{1}), where, as usual, t_{i}^{∗} is the time of the i-th interaction and δ_{i} < t_{i+1}^{∗} − t_{i}^{∗}. Using results from Section 4 we will then recursively write down the solutions in the intervals [t_{i}^{∗}, t_{i+1}^{∗} + δ_{i+1}), i = 1, …, n − 2, and [t_{n−1}^{∗}, ∞).

In the beginning we will introduce some auxiliary notations.

u_{l,0}(t) = \frac{u_{l}^{0}}{1 − 2u_{l}^{0}t}, l = 1, 2, …, n,

v_{0}^{0} = 0, v_{k}^{0} = (a_{k−1} − a_{k})u_{k−1}^{0}, k = 2, …, n + 1,

φ_{1,0}(t) = a_{1}, φ_{l−1,0}(t) = 2 ∑_{k=1}^{l−1} v_{k}^{0}t + a_{l}, l = 2, …, n + 1,

ψ_{l−1,0} = φ_{l,0} − φ_{l−1,0}, τ^{l−1,l} = \frac{ψ_{l,0}}{ε} , l = 2, …, n + 1,

ρ_{l}^{[i]}(τ^{l−1,l}) = \frac{φ_{l}^{[i]} − φ_{l−1}^{[i]}}{ε} , l = 2, …, n + 1

B_{[i]k}^{l−1,l} = B_{k}(ρ_{l}^{[i]}), k = 1, 2, l = 2, …, n + 1, i = 1, …, n,

u_{s}^{[i]} ≡ 0, s ∈ Z_{0}^{−}, i = 1, …, n + 1.

Let the initial condition for Hopf equation be:

u|_{t=0} = u_{1}^{0}(a_{1} − x)_{+} + (u_{2}^{0} − u_{1}^{0})(a_{2} − x)_{+} + (u_{3}^{0} − u_{2}^{0})(a_{3} − x)_{+} + ⋯

+ (u_{n−1}^{0} − u_{n−2}^{0})(a_{n−1} − x)_{+} + (u_{n}^{0} − u_{n−1}^{0})(a_{n} − x)_{+} − u_{n}^{0}(a_{n+1} − x)_{+} (41)

We know that after certain time point a_{i} will reach point a_{j} or oppositely. Let us suppose that the moments of interactions are t_{1}^{∗} < t_{2}^{∗} < ⋯ < t_{n}^{∗} and that in the moment t = t_{i}^{∗} points a_{l(i−1)} and a_{l(i)} will interact. By an induction argument, we can write the uniform formula which describes propagation of initial wave. Like before, we will analyze behavior of solution in the intervals [0, t_{1}^{∗} + δ_{1}), [t_{1}^{∗}, t_{2}^{∗} + δ_{2}), …, [t_{n−2}^{∗}, ∞) where t_{0}^{∗} = 0 and δ_{i} < t_{i+1}^{∗} − t_{i}^{∗}, i = 1, …, n − 2. The weak asymptotic solution in the interval [t_{i}^{∗}, t_{i+1}^{∗} + δ_{i}) we will denote with u_{ε}^{[i]}.

As earlier, we assume that the approximate solution in the interval [0, t_{1}^{∗} + δ_{1}) is in the form

u_{ε}^{[1]}(x, t) = u_{1}^{[1]}(φ_{1}^{[1]} − x)θ_{1,ε}(φ_{1}^{[1]} − x) − u_{1}^{[1]}(φ_{2}^{[1]} − x)θ_{2ε}(φ_{2}^{[1]} − x)

+ ⋯ + u_{l(1)−1}^{[1]}(φ_{l(1)−1}^{[1]} − x)θ_{l(1)−1,ε}(φ_{l(1)−1}^{[1]} − x) − u_{l(1)−1}^{[1]}(φ_{l(1)}^{[1]} − x)θ_{l(1),ε}(φ_{l(1)}^{[1]} − x)

+ u_{l(1)}^{[1]}(φ_{l(1)−1}^{[1]} − x)θ_{l(1),ε}(φ_{l(1)}^{[1]} − x) − ⋯ + u_{n}^{[1]}(φ_{n}^{[1]} − x)θ_{n,ε}(φ_{n}^{[1]} − x)

− u_{n}^{[1]}(φ_{n+1} − x)θ_{n+1,ε}(φ_{n+1} − x). (42)

Since in the first step only the interaction of weak discontinuities is possible, by the analogy with Section 4.3 we have:

φ_{s}^{[1]}(t) = φ_{s,0}(t) s ≠ l(1) and s ≠ l(0), s = 1, …, n + 1,
φ_{l(0)}(t, ε) = φ_{l(0),0}(t) + ψ_{l(0),0}(t)ϕ_{1}^{[1]}(τ^{l(0),l(1)}, t)
φ_{l(1)}(t, ε) = φ_{l(1),0}(t) + ψ_{l(0),0}(t)ϕ_{2}^{[1]}(τ^{l(0),l(1)}, t)
u_{s}^{[1]}(t) = u_{s,0}(t), s ≠ l(1) − 1, s = 1, …, n,
u_{l(0)}(t, ε) = u_{l(0),0}(t) + κ^{[1]}(τ^{l(0),l(1)}, t),

where functions ϕ_{k}^{[1]}(τ^{l(0),l(1)}, t), k = 1, 2, and κ^{[1]}(τ^{l(0),l(1)}, t) for every fixed t satisfy (τ^{23} is independent variable here),

(φ_{l(0)})_{t} = ∑_{k=0}^{l(0)} v_{k}^{0} + 2u_{0}^{[1]}B_{[1]1}^{l(0),l(1)}

(φ_{l(1)})_{t} = ∑_{k=0}^{l(0)} v_{k}^{0} + 2u_{0}^{[1]}B_{[1]1}^{l(0),l(1)} + 2u_{0}^{[1]}B_{[1]2}^{l(0),l(1)},
u_{0}^{[1]} = u_{l(0)}^{[1]} (φ_{l(0)}^{[1]} − φ_{l(1)}^{[1]})
u_{0t}^{[1]} = −2u_{l(0)−1}^{[1]}u_{0}^{[1]}B_{[1]1}^{23} + 2u_{l(1)}^{[1]}u_{0}^{[1]}B_{[1]1}^{23}.

with the initial conditions κ^{[1]}(τ, t) = ϕ_{j}^{[1]}(τ) = O(τ^{−N}), j = l(0) and j = l(1), τ → −∞, N ∈ N arbitrary.

Now, we will analyze the Hopf equation in the intervals [t_{i}^{∗}, t_{i+1}^{∗} + δ_{i+1}), i = 2, …, n − 2, and [t_{n−1}^{∗}, +∞). By an induction argument, from previous sections we conclude that after i-th interaction we have:

φ_{1} = ⋯ = φ_{j(1)},
φ_{j(1)+1} = ⋯ = φ_{j(2)}
…
φ_{j(s−1)+1} = ⋯ = φ_{j(s)},

where j(s) = n + 1. Let us assume that in the moment t = t_{i+1}^{∗} the points a_{j(k)} and a_{j(k+1)} interact. Then we have the following possibilities:

a) φ_{j(k)}(t) ≠ φ_{m}(t) for every m ∈ {1, 2, …, n + 1}\{j(k)} and φ_{j(k+1)}(t) ≠ φ_{m}(t) for every m ∈ {1, 2, …, n + 1}\{j(k + 1)} in the interval [t_{i}^{∗}, t_{i}^{∗} + δ_{i}). In this case we have the interaction of weak discontinuities (notice that therefore j(k + 1) = j(k) + 1). The solution will have the form:

1. x ≤ φ_{j(k+2)} and x ≥ φ_{j(k−1)}. In this case we take u_{ε}^{[i+1]}(x, t) = u_{ε}^{[i]}(x, t).

2. φ_{j(k+2)} ≥ x ≥ φ_{j(k−1)}. In this interval the solution looks like:

u_{ε}^{[i+1]}(x, t) =

∑_{p=1}^{j(k−1)} v_{p}^{0} + u_{j(k−1)}^{[i+1]}(φ_{j(k−1)}^{[i+1]} − x)θ_{ε}(φ_{j(k−1)}^{[i+1]} − x)

− u_{j(k−1)}^{[i+1]}(φ_{j(k)}^{[i+1]} − x)θ_{ε}(φ_{j(k)}^{[i+1]} − x) + u_{j(k)}^{[i+1]}(φ_{j(k)}^{[i+1]} − x)θ_{ε}(φ_{j(k)}^{[i+1]} − x)

− u_{j(k)}^{[i+1]}(φ_{j(k+1)}^{[i+1]} − x)θ_{ε}(φ_{j(k+1)}^{[i+1]} − x) + u_{j(k+1)}^{[i+1]}(φ_{j(k+1)}^{[i+1]} − x)θ_{ε}(φ_{j(k+1)}^{[i+1]} − x)

− u_{j(k+1)}^{[i+1]}(φ_{j(k+2)}^{[i+1]} − x)θ_{ε}(φ_{j(k+2)}^{[i+1]} − x).

This is the situation as in Section 4.3. The difference is in the fact that here we do not begin from the moment t = 0 but from t = t_{i}^{∗}. Therefore, for initial condition here we will have:

φ_{j(k−1)}^{[i+1]}(t_{i}^{∗}) = φ_{j(k−1)}^{[i]}(t_{i}^{∗}) φ_{j(k)}^{[i+1]}(t_{i}^{∗}) = φ_{j(k)}^{[i]}(t_{i}^{∗})
φ_{j(k+1)}^{[i+1]}(t_{i}^{∗}) = φ_{j(k+1)}^{[i]}(t_{i}^{∗}) φ_{j(k+2)}^{[i+1]}(t_{i}^{∗}) = φ_{j(k+2)}^{[i]}(t_{i}^{∗})
u_{j(k−1)}^{[i+1]}(t_{i}^{∗}) = u_{j(k−1)}^{[i]}(t_{i}^{∗})  u_{j(k)}^{[i+1]}(t_{i}^{∗}) = u_{j(k)}^{[i]}(t_{i}^{∗})
u_{j(k+1)}^{[i+1]}(t_{i}^{∗}) = u_{j(k+1)}^{[i]}(t_{i}^{∗}).

The unknown functions are given by set of formulas (38) with an obvious difference in indexing.

b) φ_{j(k)}(t) ≠ φ_{m}(t) for every m ∈ {1, 2, …, n + 1}\{j(k)} and φ_{j(k+1)}(t) = φ_{m}(t) for some m ∈ {1, 2, …, n + 1}\{j(k + 1)} in the interval (t_{i}^{∗}, t_{i}^{∗} + δ)_{i}).

In this case we have interaction of weak discontinuity and the shock wave whose disposure is analogous to the one from Section 4.1. The solution we have in the form:

1. x ≤ φ_{j(k+2)} and x ≥ φ_{j(k−1)}. In this case we take u_{ε}^{[i+1]}(x, t) = u_{ε}^{[i]}(x, t).

2. φ_{j(k+2)} ≥ x ≥ φ_{j(k−1)}. In this interval the solution looks like:

u_{ε}^{[i+1]}(x, t) =

∑_{p=1}^{j(k−1)−1} v_{p}^{0} + u_{j(k−1)}^{[i+1]}(φ_{j(k−1)}^{[i+1]} − x)θ_{ε}(φ_{j(k−1)}^{[i+1]} − x)

− u_{j(k−1)}^{[i+1]}(φ_{j(k)}^{[i+1]} − x)θ_{ε}(φ_{j(k)}^{[i+1]} − x) + u_{0,j(k)}^{[i+1]}θ_{ε}(φ_{j(k)}^{[i+1]} − x)

+ u_{j(k)}^{[i+1]}(φ_{j(k)}^{[i+1]} − x)θ_{ε}(φ_{j(k)}^{[i+1]} − x) − u_{j(k)}^{[i+1]}(φ_{j(k+1)}^{[i+1]} − x)θ_{ε}(φ_{j(k+1)}^{[i+1]} − x)

+ u_{j(k+1)}^{[i+1]}(φ_{j(k+1)}^{[i+1]} − x)θ_{ε}(φ_{j(k+1)}^{[i+1]} − x) − u_{j(k+1)}^{[i+1]}(φ_{j(k+2)}^{[i+1]} − x)θ_{ε}(φ_{j(k+2)}^{[i+1]} − x).

The difference from the situation from Section 4.1 is in the fact that for initial condition here we will have:

φ_{j(k−1)}^{[i+1]}(t_{i}^{∗}) = φ_{j(k−1)}^{[i]}(t_{i}^{∗}) φ_{j(k)}^{[i+1]}(t_{i}^{∗}) = φ_{j(k)}^{[i]}(t_{i}^{∗})
φ_{j(k+1)}^{[i+1]}(t_{i}^{∗}) = φ_{j(k+1)}^{[i]}(t_{i}^{∗}) φ_{j(k+2)}^{[i+1]}(t_{i}^{∗}) = φ_{j(k+2)}^{[i]}(t_{i}^{∗})
u_{j(k−1)}^{[i+1]}(t_{i}^{∗}) = u_{j(k−1)}^{[i]}(t_{i}^{∗}) u_{j(k)}^{[i+1]}(t_{i}^{∗}) = u_{j(k)}^{[i]}(t_{i}^{∗})
u_{j(k+1)}^{[i+1]}(t_{i}^{∗}) = u_{j(k+1)}^{[i]}(t_{i}^{∗}) u_{0,j(k)}^{[i+1]}(t_{i}^{∗}) = ∑_{p=j(k−1)+1}^{j(k)−1} v_{p}^{0}.

The unknown functions are given by (35) with an obvious difference in indexing.

c) φ_{j(k)}(t) = φ_{m}(t) for some m ∈ {1, 2, …, n + 1}\{j(k)} and φ_{j(k+1)}(t) ≠ φ_{m}(t) for every m ∈ {1, 2, …, n + 1}\{j(k + 1)} in the interval (t_{i}^{∗}, t_{i}^{∗} + δ_{i}). In this case we have interaction of weak discontinuity and the shock wave whose disposure is analogous to the one from Section 4.2. The solution we have in the form:

1. x ≤ φ_{j(k+2)} and x ≥ φ_{j(k−1)}. In this case we take u_{ε}^{[i+1]}(x, t) = u_{ε}^{[i]}(x, t).

2. φ_{j(k+2)} ≥ x ≥ φ_{j(k−1)}. In this interval the solution will look like:

u_{ε}^{[i+1]}(x, t) = ∑_{p=1}^{j(k−1)−1} v_{p}^{0}

+ u_{j(k−1)}^{[i+1]}(φ_{j(k−1)}^{[i+1]} − x)θ_{ε}(φ_{j(k−1)}^{[i+1]} − x) − u_{j(k−1)}^{[i+1]}(φ_{j(k)}^{[i+1]} − x)θ_{ε}(φ_{j(k)}^{[i+1]} − x)

+ u_{j(k)}^{[i+1]}(φ_{j(k)}^{[i+1]} − x)θ_{ε}(φ_{j(k)}^{[i+1]} − x) − u_{j(k)}^{[i+1]}(φ_{j(k+1)}^{[i+1]} − x)θ_{ε}(φ_{j(k+1)}^{[i+1]} − x)
+ u_{0,j(k+1)}^{[i+1]}θ_{ε}(φ_{j(k+1)}^{[i+1]} − x) + u_{j(k+1)}^{[i+1]}(φ_{j(k+1)}^{[i+1]} − x)θ_{ε}(φ_{j(k+1)}^{[i+1]} − x)
− u_{j(k+1)}^{[i+1]}(φ_{j(k+2)}^{[i+1]} − x)θ_{ε}(φ_{j(k+2)}^{[i+1]} − x).

The difference from the situation from Section 4.2 is in the fact that for initial condition here we will have:

φ_{j(k−1)}^{[i+1]}(t_{i}^{∗}) = φ_{j(k−1)}^{[i]}(t_{i}^{∗}) φ_{j(k)}^{[i+1]}(t_{i}^{∗}) = φ_{j(k)}^{[i]}(t_{i}^{∗})
φ_{j(k+1)}^{[i+1]}(t_{i}^{∗}) = φ_{j(k+1)}^{[i]}(t_{i}^{∗}) φ_{j(k+2)}^{[i+1]}(t_{i}^{∗}) = φ_{j(k+2)}^{[i]}(t_{i}^{∗})
u_{j(k−1)}^{[i+1]}(t_{i}^{∗}) = u_{j(k−1)}^{[i]}(t_{i}^{∗}) u_{j(k)}^{[i+1]}(t_{i}^{∗}) = u_{j(k)}^{[i]}(t_{i}^{∗})
u_{j(k+1)}^{[i+1]}(t_{i}^{∗}) = u_{j(k+1)}^{[i]}(t_{i}^{∗}) u_{0,j(k)}^{[i+1]}(t_{i}^{∗}) = ∑_{p=j(k)+1}^{j(k+1)−1} v_{p}^{0}.

The unknown functions are given by (35) with an obvious difference in indexing.

d) φ_{j(k)}(t) = φ_{m}(t) for some m ∈ {1, 2, …, n + 1}\{j(k)} and φ_{j(k+1)}(t) = φ_{m}(t) for some m ∈ {1, 2, …, n + 1}\{j(k + 1)} in the interval [t_{i}^{∗}, t_{i}^{∗} + δ)i − 1). In this case we have the situation analogous to one from Section 4.4. The solution we have in the form:

1. x ≤ φ_{j(k+2)} and x ≥ φ_{j(k−1)}. In this case we take u_{ε}^{[i+1]}(x, t) = u_{ε}^{[i]}(x, t).

2. φ_{j(k+2)} ≥ x ≥ φ_{j(k−1)}. In this interval the solution will look like:

u_{ε}^{[i+1]}(x, t) = ∑_{p=1}^{j(k−1)−1} v_{p}^{0}

+ u_{j(k−1)}^{[i+1]}(φ_{j(k−1)}^{[i+1]} − x)θ_{ε}(φ_{j(k−1)}^{[i+1]} − x) − u_{j(k−1)}^{[i+1]}(φ_{j(k)}^{[i+1]} − x)θ_{ε}(φ_{j(k)}^{[i+1]} − x)

+ u_{0,j(k)}^{[i+1]}θ_{ε}(φ_{j(k)}^{[i+1]} − x) + u_{j(k)}^{[i+1]}(φ_{j(k)}^{[i+1]} − x)θ_{ε}(φ_{j(k)}^{[i+1]} − x)
− u_{j(k)}^{[i+1]}(φ_{j(k+1)}^{[i+1]} − x)θ_{ε}(φ_{j(k+1)}^{[i+1]} − x) + u_{0,j(k+1)}^{[i+1]}θ_{ε}(φ_{j(k+1)}^{[i+1]} − x)
+ u_{j(k+1)}^{[i+1]}(φ_{j(k+1)}^{[i+1]} − x)θ_{ε}(φ_{j(k+1)}^{[i+1]} − x)
− u_{j(k+1)}^{[i+1]}(φ_{j(k+2)}^{[i+1]} − x)θ_{ε}(φ_{j(k+2)}^{[i+1]} − x).

The difference from the situation from Section 4.4 is in the fact that for initial condition here we will have:

φ_{j(k−1)}^{[i+1]}(t_{i}^{∗}) = φ_{j(k−1)}^{[i]}(t_{i}^{∗})  φ_{j(k)}^{[i+1]}(t_{i}^{∗}) = φ_{j(k)}^{[i]}(t_{i}^{∗})
φ_{j(k+1)}^{[i+1]}(t_{i}^{∗}) = φ_{j(k+1)}^{[i]}(t_{i}^{∗}) φ_{j(k+2)}^{[i+1]}(t_{i}^{∗}) = φ_{j(k+2)}^{[i]}(t_{i}^{∗})
u_{j(k−1)}^{[i+1]}(t_{i}^{∗}) = u_{j(k−1)}^{[i]}(t_{i}^{∗})  u_{j(k)}^{[i+1]}(t_{i}^{∗}) = u_{j(k)}^{[i]}(t_{i}^{∗})
u_{j(k+1)}^{[i+1]}(t_{i}^{∗}) = u_{j(k+1)}^{[i]}(t_{i}^{∗})  u_{0,j(k)}^{[i+1]}(t_{i}^{∗}) = ∑_{p=j(k)+1}^{j(k+1)−1} v_{p}^{0}
u_{0,j(k+1)}^{[i+1]}(t_{i}^{∗}) = ∑_{p=j(k+1)+1}^{j(k+2)−1} v_{p}^{0}

The unknown functions are given by (40) with an obvious difference in indexing.

Since in the intervals (t_{i}^{∗}, t_{i} + δ_{i}) we have u_{ε}^{[i]} = u_{ε}^{[i+1]}, the uniform solution of equation (10) with the initial condition (41) we have in the form:

u_{ε} = ∑_{i=1}^{n} η_{i}u_{ε}^{[i]}  (43)

where {η_{1}, …, η_{n}} is the partition of unity of the nonnegative part of real line such that

η_{1}(t) = 1, for t ∈ [0, t_{1}^{∗}),
η_{1}(t) = 0, for t ∈ [t_{1}^{∗} + δ_{1}, +∞),
η_{i}(t) = 1, for t ∈ [t_{i−1}^{∗} + δ_{i−1}, t_{i}^{∗}),
η_{i}(t) = 0, for t ∉ [t_{i−1}^{∗}, t_{i}^{∗} + δ_{i}), i = 2, …, n − 1,
η_{n}(t) = 1, for t ∈ [t_{n−1}^{∗} + δ_{n−1}, ∞),
η_{n}(t) = 0, for t ∈ [0, t_{n−1}^{∗}).

6. Arbitrary decreasing initial condition

In this section we will solve Hopf equation (10) with initial condition (2) where u_{0}(x), x ∈ R, is an arbitrary decreasing Lipschitz continuous function which takes values in some compact subset of real line. The process of solving will be as follows.

First, we approximate given initial data with a sequence of polygons and then for each of these polygons as initial data we write down the solution of the Hopf equation (we use the results from the previous sections). In choosing the snags of the polygon we have to take care that any three consecutive points must not interact simultaneously (the proof that such a choice is possible one can find in [6] in the case of more general equation u_{t} + (f(u))_{x} = 0, f′′ > 0). We will obtain a sequence of solutions and we will prove that it converges in the weak sense to the solution of Hopf equation with the original initial condition. More precisely, we will prove the following theorem:

Theorem 6.1. For problem (10), (2) there exists a function û_{ε} = û(x, t, ε) ∈ C^{∞}(R × R^{+} × (0, 1)) given by (45) such that we have

∫ Lû_{ε} · ϕdx = O(ε^{1/3}), ϕ ∈ D(R),
∥û_{ε}(x, 0) − u_{0}(x)∥_{L_{1}(X)} = O(ε^{1−μ}), x ∈ R,

for every compact X ⋐ R, some μ ∈ (0, 1) and every t > 0.

Furthermore, the weak entropy admissible solution û(x, t)^{1} of problem (10), (2) satisfies

∥û_{ε}(·, t) − û(·, t)∥_{L^{1}(|x|<R)} = O(ε^{1−μ}).

Proof. Denote by û(x, t) weak entropy admissible solution of the given problem. Divide the interval (−\frac{1}{ε^{μ_{1}}} , \frac{1}{ε^{μ_{1}}} ) by points a_{i}, i ∈ N, to subintervals of length Δa_{i} = a_{i−1} − a_{i} = Mε^{1−μ}, 0 < μ < 1, M ∈ (1, 2). Then we approximate initial condition (2) by the polygon u_{0,ε}(x) with the edges u_{0}(a_{i}), i = 1, 2, …, [\frac{2}{ε^{1−μ−μ_{1}}} ] + 1, a_{i} < a_{j} for i < j. For x > \frac{1}{ε^{μ_{1}}} we define u_{0,ε}(x) = u_{0,ε}(\frac{1}{ε^{μ_{1}}} ) and for x < −\frac{1}{ε^{μ_{1}}} we define u_{0,ε}(x) = u_{0,ε}(−\frac{1}{ε^{μ_{1}}} ). We select μ_{1} and μ from the interval (0, 1) such that 1+μ_{1}−μ < 1/3. We select points a_{i}, i ∈ N, in a such way that no three consecutive points interact simultaneously. In other words, we do not have φ_{i−1}(t) = φ_{i}(t) = φ_{i+1}(t) for any t < t_{i}^{∗}, i = 2, …, n − 1, where φ_{i} describes moving of the point a_{i} while t_{i}^{∗} denotes the time of interaction of the points a_{i} and a_{i+1} (we repeat that the proof that this can be done is given in [6]). According to the previous, approximate solution of the Hopf equation with the initial condition

u|_{t=0} = u_{0ε}(x),  (44)

will be of the form

û_{ε}(x, t) = ∑_{−∞}^{+∞} η_{i}(t)u_{iε}(x, t),  (45)

1For the proof that there exists a weak entropy admissible solution of the considered problem see [1].

where {η_{i} : i ∈ N} is partition of unity of real line such that supp η_{i} ∩ supp η_{j} = ∅, |i − j| > 1. We will prove that û_{ε}(x, t) is the weak asymptotic solution of problem (10), (2) with the accuracy ε^{1−μ}.

Accordingly, we have to check:

a) ∫_{−∞}^{+∞} [(û_{ε})_{t} + (û_{ε}^{2})_{x}] ϕ(x)dx = O(ε^{1/3}) for every ϕ ∈ C_{0}^{∞}(R,

b) ∥û_{ε}(·, 0) − u_{0}(·)∥_{L^{1}(R)} = O(ε^{1/3}),

To prove a) we substitute solution (45) in equation (2). We obtain O(\frac{1}{ε^{2(1+μ_{1}−μ)}} ) products of the form θ_{ε}(x − φ_{i})θ_{ε}(x − φ_{j}), i ≠ j. Using the results from Section 2 we see that each of these products generates remainder equal to O(ε). Accordingly, summing all remainders we get that the total remainder is O(ε^{1−2(1+μ_{1}−μ)}) = O(ε^{1/3}). So, expression given in a) is of order ε^{1/3} what we wanted to prove.

Concerning initial condition (2) on a bounded subset X of R we have,

∫_{X}(u_{0ε}(·) − u_{0}(·))dx = sup_{x∈X} |u_{0ε}(x) − u_{0}(x)| · diam X · ε^{1−μ} = O(ε^{1−μ}). (46)

This proves b) and finishes the proof that û_{ε} is a weak asymptotic solution of (10), (2).

At the end, we will prove that our weak asymptotic solution is in the sense of L^{1}(R) convergence “close” to the admissible weak solution of (10), (2). For that reason we will prove that the weak asymptotic solution constructed in the first part of the proof weakly converges to a weak solution of (10), (2). We have for every ψ ∈ C_{0}^{∞}([0, T) × R)

∫_{R}[u_{t} + (u^{2})_{x}]ψ(x, t)dx = O(ε), t is fixed.

If we apply ∫_{0}^{T} dt to the last relation we see that

∫_{0}^{T} ∫_{R}[u_{t} + (u^{2})_{x}]ψ(x, t)dxdt = O(ε).

If we apply the same procedure to the initial condition we conclude that w-lim_{ε→0} u_{ε}(x, t) = u(x, t), x ∈ R, t ∈ R^{+} is the weak solution of problem (10), (2). If we prove that the function u is piecewise continuous, from its construction we see that it satisfies Oleinik admissibility condition (see [7]) since the initial condition is decreasing. From [1] one can see that the Oleinik admissibility condition and entropy admissibility condition:

∫_{0}^{T} ∫_{R} [∂_{t}ψη(u) + ∂_{x}ψq(u)] dxdt + ∫_{R} ψ(x, 0)η(u_{0}(x)) dx ≥ 0,  (47)

where q(u) = ∫ η′(u) and η ∈ C^{1}(R) is an arbitrary convex function. Since w-lim_{ε→0} u_{ε}(x, t) = u(x, t) it is not difficult to conclude that we have

∫_{0}^{T} ∫_{R} [∂_{t}ψη(u_{ε}) + ∂_{x}ψq(u_{ε})] dx dt + ∫_{R} ψ(x, 0)η(u_{ε0}(x)) dx ≥ ε^{1−μ}O(1), (48)

where q(u) = ∫ 2uη′(u)du, are equivalent in the case of piecewise continuous functions.

If we revisit Theorem 6.2.2 in [1] using (48) instead of Definition 6.2.1 of the same book, we obtain

∥ǔ_{ε}(·, t)−û_{ε}(·, t)∥_{L^{1}(|x|<R)} ≤ ∥ǔ_{ε}(·, 0)−û_{ε}(·, 0)∥_{L^{1}(|x|<R+st)} + O(ε^{1−μ}) = O(ε^{1−μ}),

since in this case ǔ_{ε}(x, 0) = û_{ε}(x, 0). This implies

∥û_{ε}(·, t) − û(·, t)∥_{L^{1}(|x|<R)} ≤ ∥ǔ_{ε}(·, t) − û(·, t)∥_{L^{1}(|x|<R)}

+ ∥ǔ_{ε}(·, t) − û_{ε}(·, t)∥_{L^{1}(|x|<R)} = O(ε^{1−μ}),

which implies what we wanted.

Remark 6.2. For additional explanation see [1], Chapter 6.

So, it remains to prove that the function u is piecewise continuous. According to definition we know that two points interacted if they were distanced for the quantity O(ε). In every finite interval for fixed ε ∈ (0, 1) and t ∈ R^{+} we have finite number of points, in the interactions of which we are interested (the points of the partition of the x-axis). Therefore, number of formed “shocks” is finite for every fixed t ∈ R^{+} (we put here the quotation marks since in the case of the functions u_{ε}, ε ∈ (0, 1) we cannot have proper shocks due to smoothness of appropriate functions; here we have only very steep parts of graphs of the functions u_{ε}, ε ∈ (0, 1) from which, when we let ε → 0, become shocks). When we let ε → 0 the parts between the shocks will become continuous since those points originates from the initial condition. More precisely, since they do not form any shock, they reached the appropriate state by moving along the characteristics which makes them the points of continuity. So, it is obvious that u must be piecewise continuous.

REFERENCES

[1] C. M. Dafernos Hyperbolic Conservation Laws in Continuum Physics, Springer, 2000.

[2] V. G. Danilov, A new definition of weak solutions of semilinear equations with a small parameter, Uspekhi Mat. Nauk, 51:5 (1997), 184 (in Russian). English transl. in Russian Math. Surveys.

[3] V. G. Danilov, Generalized solution describing singularity interaction, Int. Journal of Mathematics and Math. Sciences, 29, 22 (2002), 481–494.

[4] G. B. Whitham, Linear and Nonlinear Waves, John Wiley, New York, 1974.

[5] A. M. Il’in, Compound expansions, Matem. Sb. (N.S.) 99 (141), 4 (1976), 514–537.

[6] D. Mitrović Uniform Description of Singularity Interaction in Scalar Conservation Laws, Faculty of Mathematics and Natural Sciences in Podgorica, PhD thesis, 2004.

[7] A. Y. Goritskii, S. N. Kruzhkov, G. A. Ceckin Partial Differential Equation of the First Order, Russian edition, Moscow State University, Faculty of Mathematics and Mechanics, Moscow, 1999.

[8] A. Bressan, Hyperbolic System of Conservation Laws, Preprints S.I.S.S.A., Trieste, Italy.

(received 19.03.2003)

Moscow Technical University of Communication and Informatics, Russia Faculty of Mathematics and Natural Sciences, University of Montenegro, Podgorica, Serbia & Montenegro

E-mail: matematika@cg.yu