Critical thresholds in 1D pressureless Euler-Poisson systems with varying background
CCRITICAL THRESHOLDS IN 1D PRESSURELESS EULER-POISSONSYSTEMS WITH VARYING BACKGROUND
MANAS BHATNAGAR AND HAILIANG LIU
Abstract.
The Euler Poisson equations describe important physical phenomena inmany applications such as semiconductor modeling and plasma physics. This paper isto advance our understanding of critical threshold phenomena in such systems in thepresence of different forces. We identify critical thresholds in two damped Euler Poissonsystems, with and without alignment, both with attractive potential and spatially vary-ing background state. For both systems, we give respective bounds for subcritical andsupercritical regions in the space of initial configuration, thereby proving the existenceof a critical threshold for each scenario. Key tools include comparison with auxiliarysystems, phase space analysis of the transformed system. Introduction
We are concerned with the critical threshold phenomenon in Euler-Poisson equationssubject to local and nonlocal forces.1.1.
Euler Poisson equations.
Euler Poisson equations have been an area of intensivestudy due to their vast relevance in modeling physical phenomena [1, 15, 18, 27, 28, 29].The general system is composed of three sets of equations: the mass conservation equation,the momentum equations and the Poisson equation. The system ρ t + ∇ · ( ρ u ) = 0 , u t + u ∇ u + ∇ P ( ρ ) ρ = − ν u − k ∇ φ, − ∆ φ = ρ − c governs the unknown density ρ = ρ ( t, x ) and velocity u = u ( t, x ) for x ∈ R N (or a boundeddomain) and time t >
0, subject to initial conditions ρ (0 , x ) and u (0 , x ). P is the pressureand c = c ( x ) is the background state which varies with the space variable. The parameter k signifies the property of the underlying force, repulsive k > k < φ .Such systems are widely used in semiconductor modeling where the charge densityand current need to be modeled. φ then represents the electric potential and hence, −∇ φ is the electric field. c represents the background charge the semiconductor is dopedwith. This could be a constant or vary with position within the semiconductor; see [29].Another widely known application of this system is modeling plasmas dynamics [18].Here, the pressure forcing plays an important role, which is usually adiabatic of form P ( ρ ) = Aρ γ , γ ≥ Mathematics Subject Classification.
Primary, 35L65; Secondary, 35L67.
Key words and phrases.
Euler-Poisson system, critical threshold, global regularity, shock formation. a r X i v : . [ m a t h . A P ] A p r MANAS BHATNAGAR AND HAILIANG LIU
An addition of convolution terms on the right hand side of the momentum equation givesrise to a different class of systems with nonlocal forcing. Such systems have primarily beenstudied without pressure. It is then called the Euler alignment/Euler Poisson alignmentsystem for k = 0 and k (cid:54) = 0 respectively. The momentum equation reads u t + u ∇ u = − k ∇ φ + ψ ∗ ( ρ u ) − u ψ ∗ ρ. Euler alignment systems arise as macroscopic realization of agent-based dynamics [5, 6]which describes the collective motion of finite agents, each of which adjusts its velocityto a weighted average of velocities of its neighbors˙ x i = v i , ˙ v i = 1 N N (cid:88) j =1 ψ ( | x i − x j | )( v j − v i ) . Here ψ is often called influence potential. See [14] for realization of Euler alignmentsystem as a mean field limit of the above type finite agent model as N → ∞ . It is knownthat global-in-time strong solutions for the hydrodynamic alignment system will flock.Global regularity or critical thresholds for such systems have been analyzed extensivelyduring the recent years, see [16, 32]. Further relevant literature is discussed in Section1.4.1.2. Critical threshold phenomena.
It is well known that the finite-time breakdownof the systems of Euler equations for compressible flows is generic in the sense that finite-time shock formation occurs for all but a “small” set of initial data. Lax [21] showedthat for pairs of conservation laws, C -smoothness of solutions can be lost unless its twoRiemann invariants are non decreasing. With the additional Poisson forcing the system ofEuler–Poisson equations admits a “large” set of initial configurations which yield globalsmooth solutions, see, e.g. [8, 22, 23, 24, 33]. Indeed, for a class of pressureless Euler–Poisson equations, the question addressed in [8] is whether there is a critical thresholdfor the initial data such that the persistence of the C solution regularity depends onlyon crossing such a critical threshold. For example, for system of Euler–Poisson equationswith only electric force, ρ t + ( ρu ) x = 0 ,u t + uu x = − kφ x , − φ xx = ρ − c. It was shown in [8, Theorem 3.2] that the system with k < u x ( x ) ≥ (cid:114) − kc ( ρ ( x ) − c ) ∀ x ∈ R , and for k >
0, the critical threshold condition becomes | u x ( x ) | < (cid:112) k (2 ρ ( x ) − c ) c = const > . It is evident that in the attractive forcing case ( k < k >
RITICAL THRESHOLDS IN EULER-POISSON SYSTEMS 3 momentum equation, an enlarged subcritical region may be observed due the furtherbalancing effect from damping. A novel phase plane analysis method was introduced in[2] to identify sharp critical thresholds in various scenarios.Even amidst the vast study of critical thresholds in Euler-Poisson systems, the variablebackground case has not been studied much. In fact, to our best knowledge, there is noknown critical threshold result for Euler-Poisson equations with a background state thatvaries in space. This paper is devoted to the study of such scenario.1.3.
Present investigation.
In this work we focus on the pressureless case, in one di-mensional periodic setting. Without loss of generality, we can set T := [ − / , /
2] tobe the domain of the spatial variable. More precisely, we consider the following dampedEuler–Poisson system with potential induced by a background which is a function of thespace variable, ρ t + ( ρu ) x = 0 ,u t + uu x = − νu − kφ x , − φ xx = ρ − c ( x ) , (1.1a)on (0 , ∞ ) × T subject to periodic initial conditions, ρ (0 , x ) = ρ ( x ) ≥ , ρ ∈ C ( T ) ,u (0 , x ) = u ( x ) , u ∈ C ( T ) , (1.1b)where c ( x ) is the periodic background term which is Lipschitz continuous and satisfies0 < c ≤ c ≤ c , ν ≥ k < ρ t + ( ρu ) x = 0 ,u t + uu x = − νu + ψ ∗ ( ρu ) − uψ ∗ ρ − kφ x , − φ xx = ρ − c ( x ) , (1.2a)on (0 , ∞ ) × T , subject to periodic initial conditions, ρ (0 , x ) = ρ ( x ) ≥ , ρ ∈ C ( T ) ,u (0 , x ) = u ( x ) , u ∈ C ( T ) , (1.2b)where ψ : R → [0 , ∞ ) is assumed to have the following properties, • ψ ( x ) = ψ ( − x ) , ∀ x > • ψ ( x + 1) = ψ ( x ) , ∀ x ∈ R (1-periodic), • | ψ ( x ) − ψ ( y ) | ≤ K | x − y | , x, y ∈ R and some K > ψ = ψ m and max ψ = ψ M . Also, without loss of generality, (cid:90) T ρ ( x ) dx = (cid:90) T ρ ( t, x ) dx = 1 and (cid:90) T c ( x ) dx = 1 . In this paper, we obtain bounds for supercritical as well as subcritical regions in theconfiguration of initial data for the aforementioned cases, thereby proving the existenceof a critical threshold for each system.
MANAS BHATNAGAR AND HAILIANG LIU
Related work.
There is a considerable amount of literature available on the so-lution behavior of Euler–Poisson equations. [9, 34] gives results for nonexistence andsingularity formation; [4, 35] for global existence of weak solutions with geometrical sym-metry; [26] for isentropic case, and [30] for isothermal case. For 3-D irrotational flowconsult [10, 12, 13]. Smooth irrotational solutions for the two dimensional Euler–Poissonsystem are constructed independently in [17, 25]. See also [19, 20] for related results ontwo dimensional case. In the one-dimensional Euler–Poisson system with both adiabaticpressure and a nonzero background, the authors in [11] showed the persistence of globalsolutions for initial data which is a small perturbation about the equilibrium. Yet theexistence of a critical threshold for such setting is still open.For results on critical thresholds in restricted Euler-Poisson systems, we refer to [24]for sharp conditions on global regularity vs finite time breakdown for the 2-D restrictedEuler–Poisson system, and [23] for sufficient conditions on finite time breakdown for thegeneral n-dimensional restricted Euler–Poisson systems. A relative complete analysisof critical thresholds in multi-dimensional restricted Euler–Poisson systems is given in[22] for both attractive and repulsive forcing. For multidimensional Euler-Poisson withspherically symmetric solutions, see [8, 36].During recent years, Euler alignment systems have been studied by several researchers,see [3, 32] for alignment forces dictated by bounded kernels, [7, 31] by singular kernels. Theauthors in [32] give bounds on subcritical and supercritical regions for the Euler alignmentsystem, i.e., k = 0 , ν = 0 in (1.2a) with bounded kernel in one and two dimensions. Thecritical threshold condition for one dimensional Euler alignment system was further madeprecise in [3]. The authors also studied undamped Euler Poisson alignment system, i.e. k (cid:54) = 0, ν = 0, c = 0 in (1.2a) where they showed that for such a system with k < k > Plan of the paper.
Our work analyses two classes of Euler Poisson systems:(1) Pressureless Euler Poisson with background, and(2) Pressureless Euler Poisson alignment with background.As a result all the further components of this paper are divided into two parts, eachpertaining to a system. Section 2 contains the main results along with some necessarypreliminary analysis. It has three subsections. The first one is devoted to the preliminarycalculations. The other two contain the main results for each of the aforementionedsystems. Section 3 contains the analysis/tools/proof to the theorems pertaining to thefirst system and Section 4 contains the same for Euler Poisson alignment system.2.
Main results
Preliminaries.
The critical threshold analysis to be carried out is the a priori es-timate on smooth solutions as long as they exist. For the one-dimensional Euler-Poissonproblem, local existence of smooth solutions was long known, it can be justified by usingthe characteristic method in the pressureless case.
RITICAL THRESHOLDS IN EULER-POISSON SYSTEMS 5
Theorem 2.1. ( Local existence ) If ρ ∈ C and u ∈ C , then there exists T > ,depending on the initial data, such that the initial value problem (1.1a), (1.1b) admits aunique solution ( ρ, u ) ∈ C ([0 , T ) × T ) . Moreover, if the maximum life span T ∗ < ∞ , then lim t ↑ T ∗ ∂ x u ( t, x ∗ ) = −∞ for some x ∗ ∈ T . To our knowledge, such local existence theorem has been known for a constant back-ground case ( c =const in (1.1a)). However, we will formally justify that the dependenceof c on the space variable does not change the result of the theorem for (1.1a) as well as(1.2a) as long as c ( x ( t )) is well-defined and bounded. We will show this by analyzing aset of equations obtained along the characteristic curve.We proceed to derive the characteristic system which is essential to our critical thresholdanalysis. Differentiate the second equation in (1.1a) with respect to x , and set d := u x toobtain: ρ (cid:48) + ρd = 0 , (2.1a) d (cid:48) + d + νd = k ( ρ − c ( x ( t ))) , (2.1b)where we have used the Poisson equation in (1.1a) for φ , and {} (cid:48) = ∂∂t + u ∂∂x denotes the differentiation along the particle path,Γ = { ( t, x ) | x (cid:48) ( t ) = u ( t, x ( t )) , x (0) = α ∈ T } . Here, we employ the method of characteristics to convert the PDE system (1.1a) to ODEsystem (2.1) along the particle path which is fixed for a fixed value of the parameter α . Consequently, the initial conditions to the above equations are ρ (0) = ρ ( α ) and d (0) = d ( α ) = u x ( α ) for each α ∈ T . Note that this in itself is not a closed system.However, we can obtain a complete system with additional ODEs. Setting E := − φ x = (cid:90) x − / ρ ( t, y ) − c ( y ) dy − (cid:90) t ( ρu )( s, − / ds, we obtain, x (cid:48) = u,u (cid:48) + νu = kE,E (cid:48) = − cu. (2.2)For E to be periodic, necessarily (cid:90) T ρ ( t, y ) − c ( y ) dy = 0for any t >
0. This combined with the conservation of mass requires (cid:82) T ρ ( y ) − c ( y ) dy = 0.Note that this subsystem is a closed system, which allows us to independently analyze MANAS BHATNAGAR AND HAILIANG LIU (2.1) with c obtained from system (2.2). Since c ( x ) is Lipschitz continuous, the systemfor ( x, u, E ) admits a unique solution for each given initial data. Moreover,12 ( x + u + E ) (cid:48) = xu + kuE − νu − cuE ≤ (1 + | k | + 2 ν + max c )( x + u + E ) . On integrating we get x + u + E ≤ ( α + u + E (0 , α )) e | k | +2 ν +max c ) t ∀ t > , which says that x, u, E remain bounded for all time. Hence, we can solely analyze (2.1)to conclude the long time behavior of the solution. The system with alignment (1.2a) isno different. Indeed, u remains bounded because at any time, the alignment force is amere weighted average of the relative speed.The above discussion shows that we still lie in the purview of Theorem 2.1. That is,if u x remains bounded for all the characteristics then we ensure global-in-time solutionfrom Theorem 2.1. Likewise, if for any characteristic, | u x | → ∞ in finite time, there isfinite time breakdown. This allows us to analyze (2.1) as a system for our purpose usingthe bounds of c ( x ).We are now in a position to establish our critical threshold theory which includes resultson both the global-in-time solution and finite time breakdown for (1.1a) and (1.2a).In order to conveniently present our main results and their proofs, we here introducetwo functions from R + × R + → R ,Ω( γ, β ) := β + (cid:112) β − kγ , (2.3a) Θ( γ, β ) := − β + (cid:112) β − kγ , (2.3b)where these two functions are both non-negative, and − Ω( γ, β ) ≤ ≤ Θ( γ, β ) . Euler-Poisson with variable background.
We first state our main results for(1.1a).
Theorem 2.2 (Global Solution) . Consider the system (1.1a) with initial conditions (1.1b) .Let k < , λ = Ω( c , ν ) , < c = min x ∈ T c ( x ) and c = max x ∈ T c ( x ) . If ( ρ ( x ) , u x ( x )) ∈ (cid:26) ( ρ, d ) : d > λ c ( ρ − c ) (cid:27) ∀ x, then there exists a unique global-in-time solution, ρ, u ∈ C ((0 , ∞ ) × T ) . In particular, || ρ ( t, · ) || ∞ ≤ || ρ || ∞ e λ t , and − λ ≤ u x ( t, · ) ≤ max { max u x , Θ( c , ν ) } . Theorem 2.3 (Finite Time Breakdown) . Consider the system (1.1a) with initial condi-tions (1.1b) . Let k < and c = max x ∈ T c ( x ) . If ∃ x ∈ T such that u x ( x ) < Ω( c , ν ) c ( ρ ( x ) − c ) , RITICAL THRESHOLDS IN EULER-POISSON SYSTEMS 7 then ∃ ( t ∗ , x ∗ ) such that lim t ↑ t ∗ u x ( t, x ∗ ) = −∞ .Remark . We would like to point out that essentially the same threshold results holdfor the case when the domain is all of R . We work on the periodic case to avoid atechnical discussion at far fields ( x → ±∞ ), especially in the alignment with backgroundcase (Section 4), where talking about far fields is physically less meaningful as the totalmass is infinite because of the following imperative neutrality condition (cid:90) ∞−∞ ( ρ ( y ) − c ( y )) dy = 0 . Theorem 2.5.
Consider the system (1.1a) with initial conditions (1.1b) and assume c ( x ) ≡ c . Then there exists unique global solution ρ, u ∈ C ((0 , ∞ ) × T ) iff ( ρ ( x ) , u x ( x )) ∈ (cid:26) ( ρ, d ) : d ≥ Ω( c, ν ) c ( ρ − c ) (cid:27) ∀ x ∈ T . Euler-Poisson alignment with variable background.
We now state the mainresults for (1.2a).
Theorem 2.6.
Consider the system (1.2a) with initial conditions (1.2b) . Let k < , < c = min x ∈ T c ( x ) , c = max x ∈ T c ( x ) , ψ M = max x ∈ R ψ ( x ) , ψ m = min x ∈ R ψ ( x ) and λ M = Ω( c , ν + ψ M ) . If u x ( x ) > λ M ρ ( x ) c − Θ( c , ν + ψ M ) − ν − ψ ∗ ρ ( x ) ∀ x ∈ T , then there exists a unique global solution ρ, u ∈ C ((0 , ∞ ) × T ) . Furthermore, we havethe following bounds, || ρ ( t, · ) || ∞ ≤ || ρ || ∞ e λ M t , − λ M ≤ u x ( t, · ) ≤ max { max u x , Θ( c , ν + ψ M ) } + ψ M − ψ m . Theorem 2.7.
Consider the system (1.2a) with initial conditions (1.2b) . Let k < , c = max x ∈ T c ( x ) and ψ m = min x ∈ R ψ ( x ) . If ∃ x ∈ T such that, u x ( x ) < Ω( c , ν + ψ m ) ρ ( x ) c − Θ( c , ν + ψ m ) − ν − ψ ∗ ρ ( x ) , then inf lim t ↑ t c u x ( t, · ) = −∞ for some finite t c .Remark . We would like to point out that if ψ ≡ ψ ( x ) ≡ ψ is a constant, then we obtain thefollowing two corollaries. Corollary 2.9.
Consider the system (1.2a) , initial conditions (1.2b) with ψ ( x ) ≡ ψ (constant). Let k < , < c = min x ∈ T c ( x ) , c = max x ∈ T c ( x ) and λ = Ω( c , ν + ψ ) . If u x ( x ) > λ c ( ρ ( x ) − c ) ∀ x ∈ T , then there exists a unique global-in-time solution ρ, u ∈ C ((0 , ∞ ) × T ) . Furthermore, wehave the following bounds, || ρ ( t, · ) || ∞ ≤ || ρ || ∞ e λ t , MANAS BHATNAGAR AND HAILIANG LIU − λ ≤ u x ( t, · ) ≤ max { max u x , Θ( c , ν + ψ ) } , Corollary 2.10.
Consider the system (1.2a) , initial conditions (1.2b) with ψ ( x ) ≡ ψ (constant). Let k < , c = max x ∈ T c ( x ) . If ∃ x ∈ T such that, u x ( x ) < Ω( c , ν + ψ ) c ( ρ ( x ) − c ) , then inf lim t ↑ t c u x ( t, · ) = −∞ for some finite t c . Euler-Poisson systems with variable background
Critical thresholds for an auxiliary system.
The main tool in dealing with thevariable background is the use of comparison. To this end, we introduce an auxiliaryODE system corresponding to (2.1), η (cid:48) = − ηξ, (3.1a) ξ (cid:48) = − ξ − νξ + kη − kγ. (3.1b)where γ ≥ η, ξ are functions of time as well as the parameter γ . However, we will omit the latter dependence on the parameter whenever it is clearfrom context. We will make use of the phase plane analysis technique introduced in [2]to prove a proposition for this auxiliary problem which will play a crucial role in provingthe theorems stated. Proposition 3.1.
Consider the ODE system (3.1) with initial conditions ( η (0) ≥ , ξ (0)) ,then ≤ η ( t ) and ξ ( t ) ≤ max { ξ (0) , Θ( γ, ν ) } for all t > . The solution exists globally forall t > with η ( t ) ≤ η (0) e λt , and − λ ≤ ξ ( t ) , if and only if ξ (0) ≥ λγ ( η (0) − γ ) . Here λ = Ω( γ, ν ) . Moreover, if ξ (0) < λγ ( η (0) − γ ) , then lim t → t − c η ( t ) = − lim t → t − c ξ ( t ) = ∞ for some t c > .Proof. Note that from (3.1a), we have η ( t ) = η (0) e − (cid:82) t ξ dτ and hence, if η (0) > η ( t ) > t > η (0) = 0 = ⇒ η ( t ) ≡
0. Hence, η maintains sign. For a uniform upper bound on ξ , note that since η ≥
0, from (3.1b), ξ (cid:48) ≤ − ξ − νξ − kγ = − ( ξ + λ )( ξ − µ ) . with λ = Ω( γ, ν ) and µ = Θ( γ, ν ) satisfying − λ ≤ ≤ µ . Comparing the above inequalitywith (3.2), we obtain ξ ( t ) ≤ max { ξ (0) , µ } for all t > η (0) = 0 ≡ η ( t ). Then from (3.1b), ξ (cid:48) = − ξ − νξ − kγ =: − ( ξ + λ )( ξ − µ ) . (3.2) RITICAL THRESHOLDS IN EULER-POISSON SYSTEMS 9
Using phase line analysis, we have that ξ ( t ) exists for all time if and only if ξ (0) ≥ − λ .In the case of global solution, we have − λ ≤ ξ ( t ) ≤ max { ξ (0) , µ } . If ξ (0) < − λ , ξ ( t ) tends to −∞ at a finite time t c . Such time may be determined fromthe solution formula of form ξ − µξ + λ = ξ (0) − µξ (0) + λ e ( λ + µ ) t , we have t c = 1 µ + λ log (cid:12)(cid:12)(cid:12)(cid:12) ξ (0) + λξ (0) − µ (cid:12)(cid:12)(cid:12)(cid:12) . As a result of the above discussion, we can now assume η (0) > η ( t ) > t >
0. We proceed to introduce the transformation r := ξ/η, s := 1 /η, so that (3.1) is transformed to the following linear system, r (cid:48) = − νr + k − kγs, (3.3a) s (cid:48) = r (3.3b)with initial data r (0) := ξ (0) /η (0) and s (0) := 1 /η (0) >
0. This is a linear ODE system,its solution ( r, s ) will remain bounded for all time. This fact when combined with thetransformation says that ( ξ, η ) exists globally if and only if s ( t ) > s ( t ) > , /γ ) beingthe saddle point. Written in matrix form, the system is: (cid:20) rs − /γ (cid:21) (cid:48) = (cid:20) − ν − kγ (cid:21) (cid:20) rs − /γ (cid:21) The coefficient matrix has eigenvalues − λ and µ . Hence the general solution to thissystem is, (cid:20) rs − γ (cid:21) = A (cid:20) − λ (cid:21) e − λt + B (cid:20) µ (cid:21) e µt . (3.4)From the flow of solution trajectories we see that the separatrix with incoming trajectoriesserves to divide the upper half plane( s >
0) into two invariant regions, one of which hasthe property that if s (0) >
0, then s ( t ) > t > B = 0, i.e., (cid:20) rs − γ (cid:21) = A (cid:20) − λ (cid:21) e − λt . Consequently, this trajectory equation is, γr = λ (1 − γs ) . Figure 1.
Direction field for reduced linear system along with the invariantregion.( ν = 3 , k = − , γ = 1)Thus the above mentioned region can be characterized byΣ γ := { ( r, s ) : γr ≥ λ (1 − γs ) , s > } . In order to see this is an invariant region, we only need to show that on ∂ Σ γ ∩ { s = 0 } thetrajectories go into the region. Note that the r − intercept of the separatrix is ( λ/γ ) > r ≥ λ/γ and s = 0 we have s (cid:48) = r > η, ξ ), Σ γ transforms to (cid:101) Σ γ := (cid:26) ( η, ξ ) : ξ ≥ λγ ( η − γ ) , η > (cid:27) . Likewise, if ( η (0) , ξ (0)) ∈ (cid:101) Σ γ , then η ( t ) < ∞ and ξ ( t ) ≥ λγ ( η ( t ) − γ ) ≥ − λ for all t > r (0) , s (0) > / ∈ Σ γ . Since, the linear ODE system has only one criticalpoint, we have that lim t →∞ ( | r ( t ) | , s ( t )) = ( ∞ , −∞ ). Hence, the solution crosses s = 0line at some finite time, t c .We will now derive an upper bound on t c . Using the general solution (3.4) we can findthe solution formula, s ( t ) = 1 γ + µ ( λ + µ ) (cid:18) s (0) − γ − r (0) µ (cid:19) e − λt + λ ( λ + µ ) (cid:18) s (0) − γ + r (0) λ (cid:19) e µt . Assuming the finite time breakdown condition, i.e., s (0) − γ + r (0) λ <
0, then s ( t ) ≤ γ + (cid:18) s (0) + 1 γ + | r (0) | µ (cid:19) − λ ( λ + µ ) (cid:12)(cid:12)(cid:12)(cid:12) s (0) − γ + r (0) λ (cid:12)(cid:12)(cid:12)(cid:12) e µt . Consequently, s ( t c ) = 0 for some t c ≤ µ ln ( λ + µ ) λ s (0) + γ + | r (0) | µ (cid:12)(cid:12)(cid:12) s (0) − γ + r (0) λ (cid:12)(cid:12)(cid:12) . RITICAL THRESHOLDS IN EULER-POISSON SYSTEMS 11
And therefore, if ( η (0) , ξ (0)) / ∈ (cid:101) Σ γ , then lim t → t − c η ( t ) = ∞ . And from (3.1a), we obtainlim t → t − c ξ ( t ) = −∞ . This completes the proof to the proposition. (cid:3) For the case when c ( x ) ≡ c in (1.1a), we can apply Propostion 3.1 with γ = c to (2.1)and immediately obtain Theorem 2.5.3.2. Comparison lemma.
We are now in position to present the comparison result.
Lemma 3.2 (Comparison lemma) . Let ( ρ, d ) be the solution to (2.1) , and ( η, ξ ) be solutionof (3.1) . Then as long as these solutions exist, we have(i) For γ = min T c ( x ) : If ρ (0) < η (0) and d (0) > ξ (0) , then ρ ( t ) < η ( t ) , d ( t ) > ξ ( t ) . (ii) For γ = max T c ( x ) : If ρ (0) > η (0) and d (0) < ξ (0) , then ρ ( t ) > η ( t ) , d ( t ) < ξ ( t ) . Proof.
We will show ( i ). Similar arguments follow for ( ii ). We will argue by contradiction.To this end, let t be the first time when the result is violated. Integrate (2.1a) and (3.1a)respectivley to get ρ ( t ) = ρ (0) e − (cid:82) t d dτ and η ( t ) = η (0) e − (cid:82) t ξ dτ . Since (cid:82) t d dτ > (cid:82) t ξ dτ and ρ (0) < η (0), we obtain ρ ( t ) = ρ (0) e − (cid:82) t d dτ < η (0) e − (cid:82) t ξ dτ = η ( t ) . We can then conclude that d ( t ) = ξ ( t ) along with ρ ( t ) < η ( t ). Subtracting (3.1b)from (2.1b), we obtain,( d − ξ ) (cid:48) = − ( d + ξ + ν )( d − ξ ) + k ( ρ − η ) − k ( c − γ ) . Plugging in t = t and taking γ = min c , we obtain( d − ξ ) (cid:48) ( t ) = k ( ρ ( t ) − η ( t )) − k ( c − min c ) > . This is a contradiction because this implies that for all t < t sufficiently close, d ( t ) < ξ ( t )and hence, t cannot be the first time of violation. (cid:3) Proofs of Theorems 2.2 and 2.3.
Using the tools developed above, we are nowready to prove our main results.
Proof of Theorem 2.2:
Consider (2.1) along a fixed characteristic and (3.1) for γ = c .From hypothesis of theorem, we see that d (0) > λ c ( ρ (0) − c ). As a result we can choose η (0) > ρ (0) and ξ (0) < d (0) such that d (0) > ξ (0) ≥ λ c ( η (0) − c ) > λ c ( ρ (0) − c ) . Applying Lemma 3.2, we obtain ρ ( t ) < η ( t ) , and d ( t ) > ξ ( t ) , for as long as these functions exist. Using Propostion 3.1, we obtain that ρ ( t ) < η (0) e λ t and d ( t ) > − λ for all t >
0. However, note that η (0) can be chosen to be greater thanbut arbitrarily close to ρ (0) and all the above arguments still hold. Therefore, in the limit, we have ρ ( t ) ≤ ρ (0) e λ t . Also, a uniform upper bound on d can be obtained in a similarfashion as in Proposition 3.1. From (2.1b), d (cid:48) ≤ − d − νd − kc = − ( d + Ω( c , ν ))( d − Θ( c , ν )) . Hence, d ( t ) ≤ max { d (0) , Θ( c , ν ) } .Collecting all the characterstics, we finally obtain, || ρ ( t, · ) || ∞ ≤ || ρ || ∞ e λ t , − λ ≤ u x ( t, · ) ≤ max {|| u x || ∞ , Θ( c , ν ) } . This concludes the proof of Theorem 2.2. (cid:3)
Proof of Theorem 2.3:
Consider (2.1) with ρ (0) = ρ ( x ), d (0) = u x ( x ) for x as inthe statement of the theorem. In (3.1), let γ = c . From the hypothesis of the theorem,we see that d (0) < λ c ( ρ (0) − c ). We can then choose η (0) < ρ (0) and ξ (0) > d (0) suchthat, d (0) < ξ (0) < λ c ( η (0) − c ) < λ c ( ρ (0) − c ) . Applying Lemma 3.2, we obtain ρ ( t ) > η ( t ) , and d ( t ) < ξ ( t ) . Using Propostion 3.1, we obtain that for some t c > ρ ( t ) → ∞ as t → t − c . And from(2.1a), lim t → t − c d ( t ) = −∞ and the solution ceases to be C . (cid:3) EPA systems with variable background
Reformulations.
Set G := u x + ν + ψ ∗ ρ . Taking derivative of G along (cid:48) = ∂∂t + u ∂∂x , we have, G (cid:48) = ( u t ) x + ψ ∗ ρ t + u ( u x + ψ ∗ ρ ) x = ( − uu x − νu + ψ ∗ ( ρu ) − uψ ∗ ρ − kφ x ) x − ( ψ ∗ ( ρu )) x + u ( u x + ψ ∗ ρ ) x = − Gu x + k ( ρ − c )= − G ( G − ν − ψ ∗ ρ ) + k ( ρ − c ) . We used (1.2a) to obtain the second and third equations. Consequently, along the particlepath given by, Γ = { ( t, x ) | x (cid:48) ( t ) = u ( t, x ( t )) , x (0) = α ∈ T } , we get the following ODE system, ρ (cid:48) = − ρ ( G − ν − ψ ∗ ρ ) , (4.1a) G (cid:48) = − G ( G − ν − ψ ∗ ρ ) + k ( ρ − c ( x ( t ))) , (4.1b)with initial condition, ρ (0) = ρ ( α ) , G (0) = u x ( α ) + ν + ψ ∗ ρ ( α ) . RITICAL THRESHOLDS IN EULER-POISSON SYSTEMS 13
The roadmap to the proofs of the main theorems will be similar to the previous section.However, due to the addition of the non local term, here we first transform the ODEsystem (4.1) into a simple system, and then introduce an auxiliary ODE system whichcan be used for comparison. And eventually we use these tools to prove our main results.Note that the transformation will require ρ ( t ) > ρ (0) >
0. In fact, from (4.1a), we have that ρ maintains sign, hencethe zero case can be handled separately.Next, we use the following transformation of variables for the case ρ > w = Gρ , s = 1 ρ , (4.2)to derive an ODE system for w and s . Differentiating w , (cid:18) Gρ (cid:19) (cid:48) = − ρ (cid:48) ρ + G (cid:48) ρ = ( G − ν − ψ ∗ ρ ) Gρ − ( G − ν − ψ ∗ ρ ) Gρ + k ( ρ − c ) ρ = k − kcs. Likewise, we differentiate s , (cid:18) ρ (cid:19) (cid:48) = ( G − ν − ψ ∗ ρ ) ρ = w − νs − sψ ∗ ρ. We then obtain the following ODE system, w (cid:48) = k − kcs, (4.3a) s (cid:48) = w − ( ν + ψ ∗ ρ ) s, (4.3b)with initial conditions w (0) := G (0) ρ (0) and s (0) := 1 ρ (0) . Threshold analysis for the auxiliary system.
Corresponding to (4.3), we intro-duce the following auxiliary system, p (cid:48) = k − kγq, (4.4a) q (cid:48) = p − βq, (4.4b)where γ ≥ β ≥ ν are parameters and initial conditions ( p (0) , q (0) > p, q are functions of time as well as the parameters γ, β . However, we will omit the latterdependence on the parameters whenever it is clear from context. We have the followingproposition. Proposition 4.1.
For the system (4.4) , with initial conditions ( p (0) , q (0) > , we havethat q ( t ) > for all t > if and only if p (0) ≥ λγ − µq (0) , where λ = Ω( γ, β ) and µ = Θ( γ, β ) . Additionally, if the above inequality holds, then itholds for all times, i.e., p ( t ) ≥ λγ − µq ( t ) , ∀ t > . We will once again make use of the phase plane analysis technique developed in [2].
Proof. (4.4) is a linear system with critical point ( β/γ, /γ ). Written in matrix form, thesystem is: (cid:34) p − βγ q − γ (cid:35) (cid:48) = (cid:20) − kγ − β (cid:21) (cid:34) p − βγ q − γ (cid:35) . The eigenvalues of the coefficient matrix are − λ and µ and the general solution to thesystem is, (cid:34) p − βγ q − γ (cid:35) = A (cid:20) kγλ (cid:21) e − λt + B (cid:20) − kγµ (cid:21) e µt . (4.5)From the flow of solution trajectories we see that the separatrix with incoming trajectoriesserves to divide the upper half plane ( q >
0) into two invariant regions, one of which hasthe property that if q (0) >
0, then q ( t ) > t > B = 0, i.e., (cid:34) p − βγ q − γ (cid:35) = A (cid:20) kγλ (cid:21) e − λt . Consequently, this trajectory equation is, λp = λβγ − k + kγq. Note that λ = − µkγ and β + µ = λ , the above equation becomes p = λγ − µq. RITICAL THRESHOLDS IN EULER-POISSON SYSTEMS 15
Figure 2.
Direction field for linear system along with the invariantregion.( β = 1 . , k = − , γ = 1)Thus the above mentioned region can be characterized byΣ γ,β := (cid:26) ( p, q ) : p ≥ λγ − µq, q > (cid:27) . Now suppose ( p (0) , q (0) > / ∈ Σ γ,β . Since, the linear ODE system has only one criticalpoint, we have that lim t →∞ ( | p ( t ) | , q ( t )) = ( ∞ , −∞ ). Hence, the solution crosses q = 0line at some finite time, t c . This by itself concludes the proof but we will, however, derivean upper bound on t c using the general solution (4.5), q ( t ) = 1 γ + (cid:32) − p (0) + λq (0) − µγ λ + µ (cid:33) e − λt + (cid:32) p (0) + µq (0) − λγ λ + µ (cid:33) e µt . Assuming ( p (0) , q (0)) / ∈ Σ γ,β , we have η ( t ) = 1 γ + (cid:32) − p (0) + λq (0) − µγ λ + µ (cid:33) e − λt − (cid:12)(cid:12)(cid:12) p (0) + µq (0) − λγ (cid:12)(cid:12)(cid:12) λ + µ e µt ≤ γ + | p (0) | λ + µ + q (0) − (cid:12)(cid:12)(cid:12) p (0) + µq (0) − λγ (cid:12)(cid:12)(cid:12) λ + µ e µt . Hence, q ( t c ) = 0 for some t c ≤ µ ln (cid:18) ( λ + µ )( q (0) + γ − ) + | p (0) || p (0) + µq (0) − λγ − | (cid:19) . (cid:3) Comparison lemma.
We will now derive the comparison lemma.
Lemma 4.2 (Comparison Lemma) . Let ( w, s ) be solution to (4.3) and ( p, q ) be solutionto (4.4) . Then as long s ≥ , we have:(i) For c = c , β = ν + ψ M : If s (0) > q (0) and w (0) > p (0) , then s ( t ) > q ( t ) , w ( t ) > p ( t ) . (ii) For c = c , β = ν + ψ m : If s (0) < q (0) and w (0) < p (0) , then s ( t ) < q ( t ) , w ( t ) < p ( t ) . Proof.
We only prove the first assertion. Second assertion can be proved by similar ar-guments. We argue by contradiction: let t be the first time at which statement (i) isviolated. Subtracting (4.3a) from (4.4a), and integrating we obtain, w ( t ) − p ( t ) = w (0) − p (0) − k (cid:90) t ( cs − γq ) dτ = w (0) − p (0) − kγ (cid:90) t ( s − q ) dτ − k (cid:90) t s ( c − γ ) dτ. Taking γ = c = min T c ( x ) and plugging in t = t in the equation obtained, we have that w ( t ) − p ( t ) ≥ w (0) − p (0) − kγ (cid:90) t ( s − q ) dτ > . Therefore, the only possibility left is that s ( t ) = q ( t ).Subtracting (4.3b) from (4.4b), we obtain( s − q ) (cid:48) = ( w − p ) + βη − s ( ν + ψ ∗ ρ )= ( w − p ) + β ( q − s ) + s ( β − ν − ψ ∗ ρ ) . Note that ψ ∗ ρ ∈ [min R ψ, max R ψ ] = [ ψ m , ψ M ]. Taking β = ν + ψ M and plugging in t = t in the above equation, we get( s − q ) (cid:48) ( t ) ≥ w ( t ) − p ( t ) > . This means that for t < t sufficiently close, we must have s ( t ) < q ( t ), which is a contra-diction. (cid:3) Proofs of Theorems 2.6 and 2.7.
As usual, we will analyze the solution on asingle characteristic and since the inequality in the statement of the theorem holds for all x , we can then collect all the characteristics to conclude the result.Therefore, it suffices to obtain the thresholds results for (4.1) using Proposition 4.1 andLemma 4.2.First we show that G is always bounded form above irrespective of the choice of theinitial data. From (4.1b), we have G (cid:48) ≤ − G ( G − ν − ψ ∗ ρ ) − kc = − ( G − ( ν + ψ ∗ ρ ) G + kc )= − ( G − G + )( G − G − ) , (4.6)where G + = Ω( c, ν + ψ ∗ ρ ) , G − = − Θ( c, ν + ψ ∗ ρ ) RITICAL THRESHOLDS IN EULER-POISSON SYSTEMS 17 depend on c and ψ ∗ ρ , therefore changing in time.Note that G + ≤ Ω( c , ν + ψ M )and the fact that G is non-increasing in the regime where G ≥ G + , hence G ≤ max { G (0) , sup G + } ≤ max { u x (0) + ν + ψ M , Ω( c , ν + ψ M ) } . Hence, u x ≤ sup G − ν − min ψ ∗ ρ ≤ max { u x (0) , Ω( c , ν + ψ M ) − ν − ψ M } + ψ M − ψ m = max { u x (0) , Θ( c , ν + ψ M ) } + ψ M − ψ m Note that this upper bound holds irrespective of the hypothesis of the theorem. u x beingupper bounded is a result of the dynamics of the system (4.1).We now handle the ρ (0) = 0 ≡ ρ case before dealing with the case ρ > ρ ≡
0. Consider (4.1) with ρ (0) = ρ ( α ), G (0) = u x ( α ) + ν + ψ ∗ ρ ( α ) with a fixed α ∈ T . Hence along the characteristics starting from α we have G (cid:48) = − G ( G − ν − ψ ∗ ρ ) − kc = − ( G − G + )( G − G − ) , (4.7)where G ± are same as above. From phase line analysis, we have that G ( t ) ≥ sup G − for all t > G (0) ≥ sup G − = − Θ( c , ν + ψ M ) . We will show that this indeed satisfies the threshold inequality in the theorem. u x (0) = G (0) − ν − ψ ∗ ρ (0) > − Θ( c , ν + ψ M ) − ν − ψ ∗ ρ (0)= sup G − − ν − ψ ∗ ρ (0) , then u x ( t ) = G ( t ) − ν − ψ ∗ ρ ( t ) ≥ − Θ( c , ν + ψ M ) − ν − ψ M = − λ M for all t > α = x as in the statement of the Theorem 2.7.Then from (4.1b), G (cid:48) = − ( G − G + )( G − G − ) . From phase line analysis, we have that G → −∞ in finite time if G (0) < inf G − = − Θ( c , ν + ψ m ) . Hence, if u x (0) = G (0) − ν − ψ ∗ ρ (0) < − Θ( c , ν + ψ m ) − ν − ψ ∗ ρ (0)then lim t → t − c G = lim t → t − c u x = −∞ for some time t c and this is indeed the statement ofTheorem 2.7.Now we deal with the case when ρ > Proof of Theorem 2.6:
Along the a fixed characteristics from α , we rewrite the initialthreshold condition in the theorem as, G (0) ρ (0) > λ M c − µ M ρ (0) , µ M := Θ( c , ν + ψ M ) , and this when transformed by (4.2), reads w (0) > λ M c − µ M s (0) . We can then choose p (0) < w (0) and q (0) < s (0) in (4.4) such that the following holds, w (0) > p (0) ≥ λ M c − µ M q (0) > λ M c − µ M s (0) . Applying Lemma 4.2 and Proposition 4.1 for γ = c , β = ν + ψ M , we have that w ( t ) > p ( t ) ≥ λ M c − µ M q ( t ) > λ M c − µ M s ( t ) . for all t > s , i.e. s ( t ) >
0. Hence, w ( t ) > λ M c − µ M s ( t ) , ∀ t > . Transforming back to ( ρ, G ), we have G ( t ) > λ M ρ ( t ) c − µ M . From this we can obtain a lower bound on u x . u x = G − ν − ψ ∗ ρ > − µ M − ν − ψ M = − λ M . Integrating (4.1a), ρ ( t ) = ρ (0) e − (cid:82) t u x dτ < ρ (0) e λ M t . Collecting all the characteristics finishes the proof of the theorem. (cid:3)
Proof of Theorem 2.7:
Under the transformation (4.2), the initial threshold conditionfrom the theorem reads, w (0) < λ m c − µ m s (0) , where λ m := Ω( c , ν + ψ m ) and µ m := Θ( c , ν + ψ m ). Consequently, in (4.4), we canchoose p (0) > w (0) and q (0) > s (0) such that the following holds, w (0) < p (0) < λ m c − µ m q (0) < λ m c − µ m s (0) . From Lemma 4.2, we have that w ( t ) < p ( t ) , s ( t ) < q ( t )as long as s ≥
0. Applying Proposition 4.1 with γ = c and β = ν + ψ m , we have theexistence of a finite time t ∗ such that, q ( t ∗ ) = 0 . RITICAL THRESHOLDS IN EULER-POISSON SYSTEMS 19
Therefore s ( t ) must touch zero before t ∗ , say at t c < t ∗ . Consequently, lim t → t − c ρ ( t ) = ∞ and therefore, from (4.1a), lim t → t − c u x ( t, x ( t, x )) = −∞ . This concludes the proof. (cid:3)
Acknowledgement
This work was partially supported by the National Science Foundation under GrantDMS181266.
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