A Deposition Model : Riemann Problem and Flux-Function Limits of Solutions

and Applied Analysis 3 solution containing a weighted δ-measure (i.e., delta-shock) supported on a line should be introduced in order to establish the existence in a space of measures from the mathematical point of view. Denote by BM(R) the space of bounded Borel measures on R, then the definition of a measure solution of (3) in BM(R) can be given as follows. Definition 1. A pair (u, V)(x, t) is called a measure solution of (3) if it satisfies that (a) u ∈ L∞([0,∞), L∞(R)) ∩ C([0,∞),H−s(R)) (b) V ∈ L∞([0,∞),BM(R)) ∩ C([0,∞),H−s(R)), s > 0 (c) u is measurable with respect to V almost for all t ≥ 0. And (3) is satisfied in the measure and distributional senses; that is, ∫ 0 ∫ R uψtdx dt = 0, ∫ 0 ∫ R (ψt + uψx) dV dt = 0 (11) for all ψ ∈ C∞ 0 ([0,∞) ×R). Remark 2. The continuity conditions in C([0,∞),H−s(R)) may be used to give an interpretation for (u, V)(x, t) to take on initial values [25]. Definition 3. A two-dimensional weighted delta function w(s)δL supported on a smooth curve L parameterized as x = x(s), t = t(s) (c ≤ s ≤ d) is defined by ⟨w (s) δL, ψ (x, t)⟩ = ∫ c w (s) ψ (x (s) , t (s)) ds (12) for all ψ ∈ C∞ 0 ([0,∞) ×R). We propose to find a solution of (3) of the form (u, V) (x, t) = {{{{{{ (u1, V1) (x, t) , x < x (t) , (uδ (t) , w (t) δ (x − x (t))) , x = x (t) , (u2, V2) (x, t) , x > x (t) , (13) where uδ(t), w(t), x(t) ∈ C1[0,∞), and (u1, V1)(x, t) and (u2, V2)(x, t) are respective bounded smooth solutions of (3). We assert that (13) is a measure solution of (3) if the relation dx (t) dt = uδ (t) , −uδ (t) [u] = 0, dw (t) dt = −uδ (t) [V] + [uV] (14) is satisfied, where [g] = g1 − g2 is the jump of g across the discontinuity with g being the limit value of g on the discontinuity. The proof is similar to that in [11, 17], and we omit it. System (14) is called the generalized RankineHugoniot relation. In addition, to guarantee uniqueness of solution, we propose the following admissible condition: λ2 (u2, V2) ⩽ λ1 (u2, V2) ⩽ dx (t) dt ⩽ λ1 (u1, V1) ⩽ λ2 (u1, V1) , (15) which means that all characteristics on both sides of discontinuity line are not outgoing. A discontinuity in the form (13) satisfying (14) and (15) will be called a delta-shock, symbolized by δ. Remark 4. Here the delta-shock is defined as a measure solution just as in [10, 17, 18, 25]. In fact, like [14, 15], it can also be defined as a solution in the sense of distributions, which gives a natural generalization of the classical definition of the weak L∞-solution and specifies the definition of measure solution. Remark 5. Here in a delta-shock, we assign u to be uδ(t) on the discontinuity line. Physically, for instance, in adhesion particle dynamics, the formation of delta-shock can describe the process of the concentration of particles, and such an assignment can be interpreted as the velocity of colliding particles [17]. Nowwe consider the Riemann problem (3) and (2) for the case u+ ⩽ 0 ⩽ u−. At this time, the solution is a delta-shock in the form (u, V) (x, t) = {{{{{{ (u−, V−) , x < x (t) , (uδ (t) , w (t) δ (x − x (t))) , x = x (t) , (u+, V+) , x > x (t) . (16) Solving the generalized Rankine-Hugoniot relation (14) under the admissible condition (15) gives x (t) = 0, uδ (t) = 0, w (t) = (u−V− − u+V+) t. (17) 3. Solutions of the Riemann Problem for (1) In this section, we solve the Riemann problem for system (1) with initial data (2) and examine the dependence of the Riemann solutions on the parameter ε > 0. For (1), the 4 Abstract and Applied Analysis characteristic roots and corresponding right characteristic vectors are λε1 = u − √u2 + 4εV 2 , λε2 = u + √u2 + 4εV 2 , 󳨀 →r ε1 = (1, u − √u2 + 4εV 2ε ) T , 󳨀 →r ε2 = (1, u + √u2 + 4εV 2ε ) T . (18) It is easy to calculate that ∇λεi ⋅ 󳨀 →r εi = 1 + (−1)i u √u2 + 4εV ̸ = 0 (i = 1, 2) . (19) So (1) is nonstrictly hyperbolic because of λε1 = λε2 as u = V = 0, and both characteristic fields are genuinely nonlinear in virtue of (19). As usual, we seek the self-similar solutions (u, V)(x, t) = (u, V)(ξ), where ξ = x/t. Then the Riemann problem turns into the two-point boundary value problem −ξuξ + (εV)ξ = 0, −ξVξ + (uV)ξ = 0, (20) (u, V) (±∞) = (u±, V±) . (21) For any smooth solution, (20) becomes (−ξ ε V u − ξ)(uV) ξ = 0. (22) Besides the constant states, the smooth solutions are composed of the backward rarefaction waves ξ = λε1 = u − √u2 + 4εV 2 , dV du = u − √u2 + 4εV 2ε , (23) and the forward rarefaction waves ξ = λε2 = u + √u2 + 4εV 2 , dV du = u + √u2 + 4εV 2ε . (24) For them, we have dλεi du = ∂λεi ∂u + ∂λεi ∂V dV du = 1 + (−1)i u √u2 + 4εV > 0, i = 1, 2. (25) Let (ul, Vl) and (ur, Vr) denote the states connected by a rarefaction wave on the left and right sides, respectively. Then the conditions λε1(ur, Vr) > λε1(ul, Vl) and λε2(ur, Vr) > λε2(ul, Vl) are required for the forward and backward rarefactionwaves, respectively. From (25), we have that both forward and backward rarefaction wave should satisfy ur > ul. (26) By solving the differential equations in (23) and (24), we further have the backward and forward rarefaction waves as follows: ←󳨀 R : {{{{ ξ = λε1 (u, V) = u − √u2 + 4εV 2 , √√u2 r + 4εVr − ur (√u2 r + 4εVr + 2ur) = √√u2 l + 4εVl − ul (√u2 l + 4εVl + 2ul) , ur > ul, 󳨀 →R : {{{{ ξ = λε2 (u, V) = u + √u2 + 4εV 2 , √√u2 r + 4εVr + ur (√u2 r + 4εVr − 2ur) = √√u2 l + 4εVl + ul (√u2 l + 4εVl − 2ul) , ur > ul. (27) For a given state (ul, Vl), all possible states which can connect to (ul, Vl) on the right by a backward rarefactionwave must be located on the curve ←󳨀 R (ul, Vl) : √√u2 + 4εV − u (√u2 + 4εV + 2u) = √√u2 l + 4εVl − ul (√u2 l + 4εVl + 2ul) , u > ul. (28) For a given state (ur, Vr), all possible states which can connect to (ur, Vr) on the left by a forward rarefaction wave must be located on the curve 󳨀 →R (ur, Vr) : √√u2 + 4εV + u (√u2 + 4εV − 2u) Abstract and Applied Analysis 5 = √√u2 r + 4εVr + ur (√u2 r + 4εVr − 2ur) , u < ur. (29) It is easy to check that the backward shock curve←󳨀 R(ul, Vl) is monotonously decreasing and convex, and the forward shock curve 󳨀 →R(ur, Vr) is monotonously increasing and convex. Furthermore, ←󳨀 R(ul, Vl) has the u-axis as the asymptote as Kl > 0 and interacts with the u-axis at u = − 3 √K2 l /2 asKl ⩽ 0, where Kl = √√u2 l + 4εVl − ul (√u2 l + 4εVl + 2ul) ; (30) 󳨀 →R(ur, Vr) has the u-axis as the asymptote as Mr > 0 and interacts with the u-axis at u = 3 √M2 r /2 asMr ⩽ 0, where Mr = √√u2 r + 4εVr + ur (√u2 r + 4εVr − 2ur) . (31) Let us turn to the discontinuous solutions. For a bounded discontinuity at x = x(t), the Rankine-Hugoniot relation reads −σ [u] + [εV] = 0, −σ [V] + [uV] = 0, (32) where σ = dx/dt, [u] = ul − ur with ul = u(x(t) − 0, t) and ur = u(x(t) + 0, t), and so forth. From (32), one easily obtains (ε [V] [u])2 − ε[uV] [u] = 0. (33)and Applied Analysis 5 = √√u2 r + 4εVr + ur (√u2 r + 4εVr − 2ur) , u < ur. (29) It is easy to check that the backward shock curve←󳨀 R(ul, Vl) is monotonously decreasing and convex, and the forward shock curve 󳨀 →R(ur, Vr) is monotonously increasing and convex. Furthermore, ←󳨀 R(ul, Vl) has the u-axis as the asymptote as Kl > 0 and interacts with the u-axis at u = − 3 √K2 l /2 asKl ⩽ 0, where Kl = √√u2 l + 4εVl − ul (√u2 l + 4εVl + 2ul) ; (30) 󳨀 →R(ur, Vr) has the u-axis as the asymptote as Mr > 0 and interacts with the u-axis at u = 3 √M2 r /2 asMr ⩽ 0, where Mr = √√u2 r + 4εVr + ur (√u2 r + 4εVr − 2ur) . (31) Let us turn to the discontinuous solutions. For a bounded discontinuity at x = x(t), the Rankine-Hugoniot relation reads −σ [u] + [εV] = 0, −σ [V] + [uV] = 0, (32) where σ = dx/dt, [u] = ul − ur with ul = u(x(t) − 0, t) and ur = u(x(t) + 0, t), and so forth. From (32), one easily obtains (ε [V] [u])2 − ε[uV] [u] = 0. (33) By noticing [uV] [u] = Vl + ur [V] [u] = Vr + ul [V] [u] , (34) we solve (33) to obtain ε [V] [u] = ur ± √u 2 r + 4Vl 2 = ul ± √u 2 l + 4εVr 2 . (35) Then we obtain two kinds of discontinuities σ1 = ur − √u2 r + 4εVl 2 , ul − √u2 l + 4εVr = ur − √u2 r + 4εVl, (36) σ2 = ur + √u2 r + 4εVl 2 , ul + √u2 l + 4εVr = ur + √u2 r + 4εVl. (37) Notice that the second equations in (36) and (37) are equivalent to Vr − Vl = (ur − √u2 r + 4εVl 2ε ) (ur − ul) = (ul − √u2 l + 4εVr 2ε ) (ur − ul) , (38) Vr − Vl = (ur + √u2 r + 4εVl 2ε ) (ur − ul) = (ul + √u2 l + 4εVr 2ε ) (ur − ul) , (39) respectively. In order to identity the admissible solutions, the discontinuity (36) associating with λ1 should satisfy σ1 < λε1 (ul, Vl) < λε2 (ul, Vl) , λε1 (ur, Vr) < σ1 < λε2 (ur, Vr) , (40) while the discontinuity (37) associating with λ2 should satisfy λε1 (ul, Vl) < σ2 < λε2 (ul, Vl) , λε1 (ur, Vr) < λε2 (ur, Vr) < σ2. (41) Then one can check that both inequalities (40) and (41) are equivalent to ur < ul. (42) The discontinuity (36) with (42) is called a backward shock and symbolized by ←󳨀 S , and (37) with (42) is called a forward shock and symbolized by 󳨀 →S . For a given state (ul, Vl), all possible states which can connect to (ul, Vl) on the right by a backward shock must be located on the curve ←󳨀 S (ul, Vl) : ul − √u2 l + 4εV = u − √u2 + 4εVl, u < ul. (43) For a given state (ur, Vr), all possible states which can connect to (ur, Vr) on the left by a forward shock must be located on the curve 󳨀 →S (ur, Vr) : u + √u2 + 4εVr = ur + √u2 r + 4εV, u > ur. (44) One can check that the backward shock curve ←󳨀 S (ul, Vl) is monotonously decreasing, convex, and limu→−∞V = +∞, 6 Abstract and Applied Analysis and the forward shock curve 󳨀 →S (ur, Vr) is monotonously increasing, convex, and limu→+∞V = +∞. Denote←󳨀 W(ul, Vl) = ←󳨀 R(ul, Vl) ∪ ←󳨀 S (ul, Vl) and 󳨀→ W(ur, Vr) = 󳨀 →R(ur, Vr) ∪ 󳨀 →S (ur, Vr). Then the curve ←󳨀 W(ul, Vl) is monotonously decreasing, convex, and limu→−∞V = +∞; besides, it has the u-axis as the asymptote asKl > 0 and interacts with the u-axis at u = − 3 √K2 l /2 as Kl ⩽ 0. The curve 󳨀→ W(ur, Vr) is monotonously increasing, convex, and limu→+∞V = +∞; besides, it has the u-axis as the asymptote as Mr > 0 and interacts with the u-axis at u = 3 √M2 r /


Introduction
Consider the following deposition model of conservation laws: where V ⩾ 0 denotes the density of the population performing the deposition,  = −  ℎ(, ) with ℎ(, ) being the deposition height, and  is a positive parameter.The first equation describes the conservation of total population.The second one is derived from the rules governing the time evolution of the deposition system: the deposition rate is proportional to the density of the population, and the population is driven by a velocity field proportional to the negative gradient of height.It was also derived as a decent hydrodynamic limit of some systems of interacting particles with two conserved quantities [1,2].This system can also describe the macroscopic behaviors of some (so-called chemotactic) bacterial populations, which are attracted by a chemical substrate [3][4][5].
The first task of this paper is to solve the Riemann problem, one of the fundamental problems associated with nonlinear hyperbolic conservation laws, for (1) with initial data (, V) (,  = 0) = ( ± , V ± ) , ± > 0, where V ± > 0. System (1) is nonstrictly hyperbolic, and both characteristic fields are genuinely nonlinear.The elementary waves include shocks and rarefaction waves.By the analysis method in phase plane, the unique global Riemann solution is constructed with five different kinds of structures containing shock(s) and/or rarefaction wave(s).
As  → 0 + , system (1) formally becomes which can be rewritten in the form of scalar conservation law with a linear flux function involving discontinuous coefficient The scalar hyperbolic conservation laws with the discontinuous flux functions arise in many areas, such as the continuous sedimentation of solid particles in a liquid, the two-phase flow in porous media, and the traffic flow theory.System (3) has been studied very extensively; for example, see [6][7][8] and the references cited therein.It has been shown that deltashocks and vacuum states do occur in the Riemann solutions of (3).Let us recall some knowledge with respect to deltashocks and vacuum states.Delta-shocks are an important kind of nonclassical wave for systems of conservation laws.Mathematically, they are characterized by the delta functions appearing in the state variables.Physically, they can describe the concentration phenomenon.As for delta-shocks, see [9][10][11][12][13][14][15][16][17][18].The other extreme situation is the vacuum state.It describes the cavitation phenomenon.Recently, the phenomena of concentration and cavitation and the formation of delta-shock and vacuum state have attracted wide attention from researchers.For example, Li [19] and Chen and Liu [20,21] discussed this topic by considering the vanishing pressure limits of solutions of the isentropic and nonisentropic Euler equations.With respect to this topic, also see [22][23][24].
The second task of this paper is to study the behaviors of solutions of system (1) as the flux V vanishes (i.e.,  → 0 + ) by the Riemann problem.We are especially concerned with the phenomena of concentration and cavitation and the formation of delta-shocks and vacuum states in the limit.As a result, we rigorously show that as  → 0 + , the Riemann solutions of (1) just converge to the Riemann solutions of (3) with the same initial data.Especially, when  + ⩽ 0 ⩽  − , the two-shock solution of ( 1) and ( 2) tends to the deltashock solution of ( 3) and (2), where the intermediate density between the two shocks tends to a weighted -measure which forms the delta-shock; by contrast, when  + ⩾ 0 ⩾  − , the two-rarefaction-wave solution of (1) and (2) tends to the twocontact-discontinuity solution of (3) and (2), in which the nonvacuum intermediate state between the two rarefaction waves tends to a vacuum state.It can also be seen that such a flux-function limit may be very singular: the limit functions of solutions are no longer in the spaces of functions  or  ∞ , and the space of Radon measures is a natural space in order to deal with such a limit.
The rest of the paper is organized as follows.In Section 2, we recall the Riemann problem for system (3).In Section 3, we solve the Riemann problem for (1) by the analysis method in phase plane.Sections 4 and 5 are devoted to the studies of the limits of solutions of the Riemann problem for (1) as  → 0 + .In Section 6, we present some numerical results to examine the formation processes of delta-shocks and vacuum states as  decreases.Finally, we give the conclusions in Section 7.
Since the equations and the Riemann initial data are invariant under uniform stretching of coordinates (, ) → (, ) ( > 0), we consider the self-similar solutions (, V)(, ) = (, V)(), where  = /.Then this Riemann problem turns into (5) This is a two-point boundary value problem of first-order ordinary differential equations with the boundary values in the infinity.
Besides the constant states and the vacuum states (V ≡ 0), the self-similar wave (, V)() ( = /) of the first family is a standing wave discontinuity SW: and that of the second family is a contact discontinuity where the indices  and  denote the left and right states, respectively.
Using the above waves, by the analysis in phase plane, one can construct the solutions of Riemann problem (3) and ( 2) in the following cases.
(1) When  − > 0,  + > 0, the solution is SW + : (2) When  − < 0,  + < 0, the solution is  + SW: (3) When  − ⩽ 0 ⩽  + , the solution is  + Vac + : However, for the case  + ⩽ 0 ⩽  − , the singularity cannot be a jump with finite amplitude; that is, there is no solution which is piecewise smooth and bounded.Hence a solution containing a weighted -measure (i.e., delta-shock) supported on a line should be introduced in order to establish the existence in a space of measures from the mathematical point of view.
Denote by BM(R) the space of bounded Borel measures on R, then the definition of a measure solution of (3) in BM(R) can be given as follows.
And ( 3) is satisfied in the measure and distributional senses; that is, Remark 2. The continuity conditions in ([0, ∞),  − (R)) may be used to give an interpretation for (, V)(, ) to take on initial values [25].
We propose to find a solution of (3) of the form where   (), (), () ∈  1 [0, ∞), and ( 1 , V 1 )(, ) and ( 2 , V 2 )(, ) are respective bounded smooth solutions of (3).We assert that ( 13) is a measure solution of (3) if the relation is satisfied, where [] =  1 −  2 is the jump of  across the discontinuity with  being the limit value of  on the discontinuity.The proof is similar to that in [11,17], and we omit it.System ( 14) is called the generalized Rankine-Hugoniot relation.
In addition, to guarantee uniqueness of solution, we propose the following admissible condition: which means that all characteristics on both sides of discontinuity line are not outgoing.
A discontinuity in the form ( 13) satisfying ( 14) and ( 15) will be called a delta-shock, symbolized by .

Remark 4.
Here the delta-shock is defined as a measure solution just as in [10,17,18,25].In fact, like [14,15], it can also be defined as a solution in the sense of distributions, which gives a natural generalization of the classical definition of the weak  ∞ -solution and specifies the definition of measure solution.
Remark 5.Here in a delta-shock, we assign  to be   () on the discontinuity line.Physically, for instance, in adhesion particle dynamics, the formation of delta-shock can describe the process of the concentration of particles, and such an assignment can be interpreted as the velocity of colliding particles [17].Now we consider the Riemann problem (3) and (2) for the case  + ⩽ 0 ⩽  − .At this time, the solution is a delta-shock in the form (  () ,  ()  ( −  ())) ,  =  () , Solving the generalized Rankine-Hugoniot relation (14) under the admissible condition (15) gives

Solutions of the Riemann Problem for (1)
In this section, we solve the Riemann problem for system (1) with initial data (2) and examine the dependence of the Riemann solutions on the parameter  > 0. For (1), the characteristic roots and corresponding right characteristic vectors are It is easy to calculate that So ( 1) is nonstrictly hyperbolic because of   1 =   2 as  = V = 0, and both characteristic fields are genuinely nonlinear in virtue of (19).
As usual, we seek the self-similar solutions (, V)(, ) = (, V)(), where  = /.Then the Riemann problem turns into the two-point boundary value problem For any smooth solution, (20) becomes Besides the constant states, the smooth solutions are composed of the backward rarefaction waves and the forward rarefaction waves For them, we have Let (  , V  ) and (  , V  ) denote the states connected by a rarefaction wave on the left and right sides, respectively.Then the conditions   1 (  , V  ) >   1 (  , V  ) and   2 (  , V  ) >   2 (  , V  ) are required for the forward and backward rarefaction waves, respectively.From (25), we have that both forward and backward rarefaction wave should satisfy By solving the differential equations in ( 23) and ( 24), we further have the backward and forward rarefaction waves as follows: For a given state (  , V  ), all possible states which can connect to (  , V  ) on the right by a backward rarefaction wave must be located on the curve For a given state (  , V  ), all possible states which can connect to (  , V  ) on the left by a forward rarefaction wave must be located on the curve It is easy to check that the backward shock curve ←  (  , V  ) is monotonously decreasing and convex, and the forward shock curve  → (  , V  ) is monotonously increasing and convex.
In order to identity the admissible solutions, the discontinuity (36) associating with  1 should satisfy while the discontinuity (37) associating with  2 should satisfy (41) Then one can check that both inequalities (40) and (41) are equivalent to The discontinuity (36) with ( 42) is called a backward shock and symbolized by ←   , and (37) with ( 42) is called a forward shock and symbolized by  →  .For a given state (  , V  ), all possible states which can connect to (  , V  ) on the right by a backward shock must be located on the curve For a given state (  , V  ), all possible states which can connect to (  , V  ) on the left by a forward shock must be located on the curve One can check that the backward shock curve ←   (  , V  ) is monotonously decreasing, convex, and lim →−∞ V = +∞, and the forward shock curve  →  (  , V  ) is monotonously increasing, convex, and lim →+∞ V = +∞.
Then the curve ←  (  , V  ) is monotonously decreasing, convex, and lim →−∞ V = +∞; besides, it has the -axis as the asymptote as   > 0 and interacts with the -axis at  = − 3 √ 2  /2 as   ⩽ 0. The curve  → (  , V  ) is monotonously increasing, convex, and lim →+∞ V = +∞; besides, it has the -axis as the asymptote as   > 0 and interacts with the -axis at  = 3 √ 2  /2 as   ⩽ 0. We can construct the solutions of the Riemann problem by using the standard analysis method in the phase plane [26,27] The conclusion can be stated in the following theorem.
Theorem 6.The Riemann problem for (1) with initial data (2) has a unique piecewise smooth solution consisting of waves of constant states, vacuums, shocks, and rarefaction waves.

Limits of Solution of (1) and (2) for
In this section, we study the limits of solution of ( 1) and (2) as  → 0 + for the case  − >  + ,  − V − >  + V + .We especially pay more attention on the phenomenon of concentration and the formation of delta-shocks in the limit.

Lemma 7. If 𝑢
or If V + = V − ,  0 may be taken as any real positive number.If V + ̸ = V − , we have the conclusion by taking The proof is finished.
For fixed  <  0 , let   () denote the two-shock Riemann solution for (1) and (2): where ( − , V − ) and (  In this paper, we consider (93) for the case  = 0, that is, model (1), which arose in the context of the true self-repelling motion constructed by Toth and Werner [1].Firstly, we solve the Riemann problem, which is very useful for the understanding of equations because all properties, such as shocks and rarefaction waves, appear as characteristics in the solution.By the analysis method in phase plane, we obtain five kinds of structures of solutions containing shock(s) and/or rarefaction wave(s).Secondly, we consider the flux-function limits of solutions of system (1).We prove that the Riemann solutions of system (1) just converge to the Riemann solutions of the limit system (3).We especially identify and analyze the formation of delta-shocks and vacuum states in the limit.
For (93), the parameter  is of crucial importance: different values of  lead to completely different behaviors.One can carry out the investigation similar to that in this paper for some other cases, such as  = 1 (Leroux equation) and  = 1/2 (shallow water equation).