Formation of singularities in cold ion dynamics
FFORMATION OF SINGULARITIES IN COLD ION DYNAMICS
JUNSIK BAE, JUNHO CHOI, AND BONGSUK KWON
Abstract.
We propose a criterion for the singularity formation of the pressureless Euler-Poisson system equipped with the Boltzmann relation, which describes the dynamics ofcold ions in an electrostatic plasma. Under the proposed criterion, we prove that thesmooth solutions develop a C blow-up in a finite time and obtain their temporal blow-uprates. In general, it is known that smooth solutions to nonlinear hyperbolic equations failto exist globally in time when the gradient of initial velocity is negatively large. In contrast,our blow-up condition does not require the largeness of gradient of velocity, and our resultparticularly implies that the smooth solutions can break down even if the gradient of initialvelocity is trivial. Keywords : Euler-Poisson system; Boltzmann relation; cold ion; singularity Introduction
We consider the pressureless Euler-Poisson system in a non-dimensional form:(1.1a)(1.1b)(1.1c) ρ t + ( ρu ) x = 0 ,u t + uu x = − φ x , − φ xx = ρ − e φ , where ρ > u and φ are the unknown functions of ( x, t ) ∈ R × R + representing the iondensity, the fluid velocity for ions, and the electric potential, respectively.The Euler-Poisson system (1.1) is a fundamental fluid model describing the dynamicsof cold ions in an electrostatic plasma. We briefly discuss important physical assumptionsimposed on (1.1). In the one-fluid model (1.1), the electron density ρ e is assumed to satisfythe Boltzmann relation (1.2) ρ e = e φ . Based on the physical fact that the electron mass m e is much smaller than the ion mass m i , i.e, m e /m i (cid:28)
1, the relation (1.2) can be formally derived from the two-fluid modelof ions and electrons by neglecting the momentum of electrons under the massless electron assumption. We refer to [9] for a rigorous justification of the massless electron limit. Onthe other hand, taking account of the fact that the ion temperature T i is much smaller thanthe electron temperature T e , the ion pressure is often neglected to simplify the model. Inother words, the pressureless Euler-Poisson system (1.1) is an ideal model for cold ions (aplasma with T i /T e (cid:28) Date : December 18, 2020.2020
Mathematics Subject Classification.
Primary: 35Q35, 35Q53 Secondary: 35Q31, 76B25. a r X i v : . [ m a t h . A P ] D ec J. BAE, J. CHOI, AND B. KWON mathematically justify the phenomena of plasma solitons by showing existence [5, 15, 21]and linear stability of traveling solitary waves [2, 12].A question of global existence or finite time blow-up of smooth solutions naturally arises inthe study of large-time dynamics of the Euler-Poisson system. To the best of our knowledge,no global well-posedness of smooth solutions is known except that of [10] for 3D case withthe isothermal pressure law showing that smooth irrotational flow can persist globally. Onthe other hand, (1.1) is shown to have the smooth solutions leaving the class of C in afinite time when the initial velocity has negatively large gradient at some point [16].In the present work, we establish a criterion for singularity formation of (1.1), underwhich we show the smooth solutions develop a C blow-up in a finite time along with thetemporal blow-up rates. In general, it is known that smooth solutions to nonlinear hyper-bolic equations fail to exist globally in time when the gradient of initial velocity is negativelylarge. Roughly speaking, this means that if the given initial data is near the shock waves,then the corresponding solutions develops into the shock waves. In contrast, our blow-upcondition does not require the largeness of gradient of the initial velocity. In particular,our results demonstrate that C norm of velocity blows up even if the initial velocity hastrivial gradient. From a physical point of view, this phenomenon is caused by the effectof the electrostatic repulsive force. For instance, when the initial density is locally lowerthan the background density, i.e., ion density is locally rarefied, the electrostatic potential isdetermined in a way that the fluid momentum with negative gradient is generated at latertimes, resulting in the finite-time singularity formation. (See the numerical simulations ofspecific examples in Section 3.)1.1. Main result.
We consider the Euler-Poisson system (1.1) around a constant state, i.e.,( ρ, u, φ ) → (1 , ,
0) as | x | → ∞ . It is known that the system (1.1) admits a unique smoothsolution locally in time for the smooth initial data, for instance, ( ρ − , u ) ∈ H ( R ) × H ( R ).Furthermore, as long as the smooth solution exists, the energy H ( t ) := (cid:90) R ρu + 12 | ∂ x φ | + ( φ − e φ + 1 dx is conserved, that is,(1.3) H ( t ) = H (0) . We refer to [14] for more details.To state our main theorem, let us define a function f − : ( −∞ , → [0 , ∞ ) by f − ( z ) := (cid:90) z (cid:112) s − e s + 1) ds for z ∈ ( −∞ , . By inspection, we see that f − is well-defined since ( s − e s + 1 is nonnegative, it is strictlydecreasing in ( −∞ , f − − : [0 , + ∞ ) → ( −∞ , Theorem 1.1.
For the initial data satisfying (1.4) exp (cid:0) f − − ( H (0)) (cid:1) > ρ ( α ) for some α ∈ R , the maximal existence time T ∗ for the smooth solution to the Euler-Poisson system (1.1) isfinite. In particular, lim t (cid:37) T ∗ sup x ∈ R ρ ( x, t ) = + ∞ and inf x ∈ R u x ( x, t ) ≈ t − T ∗ BLOW UP FOR THE EULER-POISSON SYSTEM 3 for all t < T ∗ sufficiently close to T ∗ . Our main theorem demonstrates that singularities in solutions to (1.1) can occur in afinite time if the initial density at some point is small compared to the initial energy. Infact, the negativity of the initial velocity gradient is not required.We remark that there is a fairly wide class of the initial data satisfying the condition(1.4). From the elliptic estimates for the Poisson equation (1.1c), we have (Appendix 4.1)(1.5) 0 ≤ H (0) ≤ sup x ∈ R ρ (cid:90) R | u | dx + 1inf x ∈ R ρ (cid:90) R | ρ − | dx =: C ( ρ , u ) . On the other hand, since lim ζ (cid:38) f − − ( ζ ) = 0, for any given constant 0 < c < /
2, thereis δ c > ζ < δ c implies exp( f − − ( ζ )) > c . Thus, (1.4) holds for all initialdata satisfying inf ρ = c ∈ (0 , /
2) and C ( ρ , u ) < δ c (cid:28)
1. In particular, one can take u ≡
0. We shall give more specific examples and their numerical experiments exhibitingthe singularities in Section 3.We discuss some difficulties of our problem and related results. Along the characteristiccurve x ( α, t ) associated with the fluid velocity u , issuing from an initial point α ∈ R (see(2.1)), one can easily obtain from (1.1) that(1.6) ˙ ρ = − u x ρ, ˙ u x = − u x + ρ − e φ , where ˙ := ∂ t + u∂ x . The behavior of ρ and u x depends not only on the initial data, butthe potential φ along the characteristic curve due to the nonlocal nature of the system(1.1). This makes the problem challenging, and also distinguishes the aforementioned blow-up mechanism for (1.1) from those for different types of the Euler-Poisson systems, forinstance, the one with constant background density [7]. The most relevant study to ourresult is that of [16], where smooth solutions to (1.1) are shown to blow up when the initialdata satisfies ∂ x u ≤ −√ ρ at some point, i.e., the gradient of velocity is large negativelycompared to the density. In fact, the result is obtained by discarding e φ in (1.6) and solvingthe resulting (closed) system of differential inequalities for ρ and u x . For this reason, thecriterion, which is described only by the local quantities of the initial data, is not sharp,i.e., it only provides a sufficient condition resulting from the local structure of (1.6) alonedropping e φ . Our analysis takes account of the non-local structure. As such, the blow upcriterion (1.4) involves the non-local quantity.To overcome the non-locality issue of the problem, by making use of the energy conser-vation, we first show that the amplitude of φ is bounded uniformly in x and t as long as thesmooth solution exists (Lemma 2.1) and that this uniform bound can be controlled only bythe size of initial energy H (0).Next, we define w ( α, t ) := ∂x∂α ( α, t )and derive a second-order ODE (2.7) for w . Using Lemma 2.1, we find that w vanishes ata finite time T ∗ if and only if the solution blows up in the C topology, i.e., u x (cid:38) −∞ as t (cid:37) T ∗ at a non-integrable order in time t (Lemma 2.2). Our goal is then to find somesufficient conditions guaranteeing w vanishes in a finite time. By applying Lemma 2.1, weemploy a comparison argument for the differential inequality to study the behavior of w .The derivation of (2.7) is motivated by the well-known fact that the Riccati equation canbe reduced to a second-order linear ODE ([13], pp.23–25). This formulation is also employedin [7] for the 1D pressureless Euler-Poisson system with constant doping profile. In this J. BAE, J. CHOI, AND B. KWON case, the associated ODE is explicitly solvable, and thus, the so-called critical threshold isobserved. We refer to [17, 18] for 2D case. We also refer to [3] for singularity formation ofthe Euler-Poisson system with nonlocal alignment forces. An interesting open question iswhether such critical threshold is available for the Euler-Poisson system with the Boltzmannrelation (1.2). 2.
Proof of Theorem 1.1
In this section, we present all the necessary lemmas, and prove Theorem 1.1 at the endof the section. -6 -4 -2 0 2012345678 (a) -1 f -1- (H(0)) 0 f -1+ (H(0)) 10H(0)0.40.6(b)
Figure 1. (a): The graph of U ( s ) = ( s − e s + 1. (b): The graph of f ( z ).By Lemma 2.1, φ is confined in the interval [ f − − ( H (0)) , f − ( H (0))].2.1. Uniform bound of φ . We first show the uniform boundedness of φ ( x, t ) in x and t .Let us define the functions f ( z ) := f + ( z ) := (cid:90) z (cid:112) U ( s ) ds for z ≥ ,f − ( z ) := (cid:90) z (cid:112) U ( s ) ds for z ≤ , where U ( s ) := ( s − e s + 1 is nonnegative for all s ∈ R and satisfies U ( s ) → + ∞ as s → + ∞ , U ( s ) → s → −∞ (see Figure 1). Hence, f + and f − have the inverse functions f − : [0 , + ∞ ) → [0 , + ∞ ) and f − − : [0 , + ∞ ) → ( −∞ , f is of C ( R ). Lemma 2.1.
As long as the smooth solution to (1.1) exists for t ∈ [0 , T ] , f − − ( H (0)) ≤ φ ( x, t ) ≤ f − ( H (0)) for all ( x, t ) ∈ R × [0 , T ] . BLOW UP FOR THE EULER-POISSON SYSTEM 5
Proof.
Since f ∈ C ( R ) and f ≥
0, we have that for all t ≥ x ∈ R ,0 ≤ f ( φ ( x, t )) = (cid:90) x −∞ f (cid:48) ( φ ( y, t )) φ y dy ≤ (cid:90) x −∞ | f (cid:48) ( φ ( y, t )) || φ y | dy ≤ (cid:90) ∞−∞ U ( φ ) dy + 12 (cid:90) ∞−∞ | φ y | dy ≤ H ( t ) = H (0) , where the last equality holds due to the energy conservation (1.3). This completes theproof. (cid:3) Second order ODE along characteristic curves.
For u ∈ C , the characteristiccurves x ( α, t ) are defined as the solution to the ODE(2.1) x (cid:48) = u ( x ( α, t ) , t ) , x ( α,
0) = α ∈ R , t ≥ , where (cid:48) := d/dt and the initial position α is considered as a parameter. Since x ( α, t ) isdifferentiable in α , we obtain from (2.1) that(2.2) w (cid:48) = u x ( x ( α, t ) , t ) w, w ( α,
0) = 1 , t ≥ , where w = w ( α, t ) := ∂x∂α ( α, t ) . We show that w satisfies a certain second-order ordinary differential equation. By inte-grating (1.1b) along x ( α, t ), we obtain that(2.3) x (cid:48) = u ( x ( α, t ) , t ) = u ( α ) − (cid:90) t φ x ( x ( α, s ) , s ) ds. Differentiating (2.3) in α ,(2.4) w (cid:48) = ∂ α u ( α ) − (cid:90) t φ xx ( x ( α, s ) , s ) w ( α, s ) ds. Since the RHS of (2.4) is differentiable in t , so is the LHS. Hence, we get(2.5) w (cid:48)(cid:48) = − φ xx w = ( ρ − e φ ) w, where we have used (1.1c). On the other hand, using (1.1a) and (2.2), we obtain that( ρ ( x ( α, t ) , t ) w ( α, t )) (cid:48) = − ρu x w + ρu x w = 0 , which yields(2.6) ρ ( x ( α, t ) , t ) w ( α, t ) = ρ ( α ) . Finally, combining (2.2), (2.5), (2.6), we see that w ( α, t ) satisfies the second-order non-homogeneous equation(2.7) w (cid:48)(cid:48) + e φ ( x ( α,t ) ,t ) w = ρ ( α ) , w ( α,
0) = 1 , w (cid:48) ( α,
0) = u x ( α ) . J. BAE, J. CHOI, AND B. KWON
Blow-up criterion.
From (2.6), it is obvious that for each α ∈ R ,0 < w ( α, t ) < + ∞ ⇐⇒ < ρ ( x ( α, t ) , t ) < + ∞ , lim t (cid:37) T ∗ w ( α, t ) = 0 ⇐⇒ lim t (cid:37) T ∗ ρ ( x ( α, t ) , t ) = + ∞ . Using Lemma 2.1, we show that sup x ∈ R | ρ ( x, t ) | and sup x ∈ R | u x ( x, t ) | blow up at the sametime, if one of them blows up at a finite time T ∗ . Lemma 2.2.
Suppose that the smooth solution to (1.1) exists for all ≤ t < T ∗ < + ∞ .Then the following statements hold.(1) For each α ∈ R , the following holds true: (2.8) lim t (cid:37) T ∗ w ( α, t ) = 0 if and only if (2.9) lim inf t (cid:37) T ∗ u x ( x ( α, t ) , t ) = −∞ . (2) If one of (2.8) – (2.9) holds for some ˜ α ∈ R , then there are uniform constants c , c > such that (2.10) c t − T ∗ < u x ( x ( ˜ α, t ) , t ) < c t − T ∗ for all t < T ∗ sufficiently close to T ∗ .Remark . (1) By integrating (2.2), we obtain(2.11) w ( α, t ) = exp (cid:18)(cid:90) t u x ( x ( α, s ) , s ) ds (cid:19) . While it is easy by (2.11) to see that (2.8) implies (2.9), the converse is not obvioussince one cannot exclude the possibility that u x diverge in some other earlier time,say T < T ∗ with an integrable order in t , for which we still have w ( α, T ) >
0. Forthe proof of the converse and obtaining the blow-up rate (2.10), Lemma 2.1, theuniform boundedness of φ , will be crucially used.(2) From (2.15) and (2.16), we see that if w (cid:48) ( T ∗ ) <
0, the vanishing (or blow-up) orderof w (or ρ ) is ( t − T ∗ ) (or ( t − T ∗ ) − ) and if w (cid:48) ( T ∗ ) = 0, the vanishing (or blow-up)order of w (or ρ ) is ( t − T ∗ ) (or ( t − T ∗ ) − ). Proof.
We suppress the parameter α for notational simplicity. We first make a few basicobservations. By the assumption, we have that w ( t ) > t ∈ [0 , T ∗ ). From (2.7) andthe fact that e φ w ( t ) >
0, we obtain that(2.12) w (cid:48)(cid:48) ( t ) < w (cid:48)(cid:48) ( t ) + e φ ( x ( α,t ) ,t ) w ( t ) = ρ , for which we integrate (2.12) in t twice to deduce that w ( t ) is bounded above on [0 , T ∗ ).This together with (2.7) and Lemma 2.1 implies that | w (cid:48)(cid:48) ( t ) | is bounded on the interval[0 , T ∗ ). Using this for w (cid:48) ( t ) − w (cid:48) ( s ) = (cid:90) ts w (cid:48)(cid:48) ( τ ) dτ, we see that w (cid:48) ( t ) is uniformly continuous on [0 , T ∗ ). Hence, we see that the following limit w (cid:48) ( T ∗ ) := lim t (cid:37) T ∗ w (cid:48) ( t ) ∈ ( −∞ , + ∞ ) BLOW UP FOR THE EULER-POISSON SYSTEM 7 exists. In a similar fashion, one can check that w ( T ∗ ) := lim t (cid:37) T ∗ w ( t ) ∈ [0 , + ∞ ) . We prove the first statement. It is obvious from (2.11) that (2.8) implies (2.9). To showthat (2.9) implies (2.8), we suppose lim t (cid:37) T ∗ w ( t ) >
0. Then, since w (0) = 1, w ( t ) has astrictly positive lower bound on [0 , T ∗ ). From (2.9), we may choose a sequence t k such that u x ( t k ) → −∞ as t k (cid:37) T ∗ . Now using (2.2), we obtain that u x ( t k ) w ( t k ) − u x ( s ) w ( s ) = w (cid:48) ( t k ) − w (cid:48) ( s ) = (cid:90) t k s w (cid:48)(cid:48) ( τ ) dτ, which leads a contradiction by letting t k (cid:37) T ∗ . Hence, (2.8) holds.Now we prove the second statement. Due to the first statement, it is enough to assumethat (2.8) holds for some ˜ α ∈ R . From (2.7) and Lemma 2.1, we see that (2.8) implies(2.13) lim t (cid:37) T ∗ w (cid:48)(cid:48) ( t ) = ρ > . Since w ( t ) > , T ∗ ), (2.8) also implies that(2.14) w (cid:48) ( T ∗ ) = lim t (cid:37) T ∗ w (cid:48) ( t ) ≤ . By the fundamental theorem of calculus, one has w (cid:48) ( t ) = w (cid:48) ( τ ) + (cid:82) tτ w (cid:48)(cid:48) ( s ) ds for all t, τ ∈ [0 , T ∗ ). Then taking the limit τ (cid:37) T ∗ and integrating once more, we obtain that for t < T ∗ ,(2.15) w (cid:48) ( t ) = w (cid:48) ( T ∗ ) + (cid:90) tT ∗ w (cid:48)(cid:48) ( s ) ds,w ( t ) = w (cid:48) ( T ∗ )( t − T ∗ ) + (cid:90) tT ∗ w (cid:48)(cid:48) ( s )( t − s ) ds. Using (2.13), we have that for all t < T ∗ sufficiently close to T ∗ ,(2.16) 2 ρ ( t − T ∗ ) < (cid:90) tT ∗ w (cid:48)(cid:48) ( s ) ds < ρ t − T ∗ ) ,ρ T ∗ − t ) < (cid:90) tT ∗ w (cid:48)(cid:48) ( s )( t − s ) ds < ρ ( T ∗ − t ) . Thanks to (2.14), we note that either w (cid:48) ( T ∗ ) < w (cid:48) ( T ∗ ) = 0 holds. Combining (2.15)–(2.16), we conclude that if w (cid:48) ( T ∗ ) <
0, then1 / < ( t − T ∗ ) u x = ( t − T ∗ ) w (cid:48) w < , and if w (cid:48) ( T ∗ ) = 0, then 1 < ( t − T ∗ ) u x = ( t − T ∗ ) w (cid:48) w < . This completes the proof of (2.10). (cid:3)
J. BAE, J. CHOI, AND B. KWON
Proof of main theorem.
Now we are ready to prove our main theorem.
Proof of Theorem 1.1.
We consider the equation (2.7) with α ∈ R , for which (1.4) holds.Suppose that the smooth solution to (1.1) exists for all t ∈ [0 , + ∞ ). Then, thanks toLemma 2.2, we must have(2.17) w ( α, t ) > t ∈ [0 , + ∞ ) . Combining (2.7) and Lemma 2.1, we have that for all t ∈ [0 , + ∞ ),(2.18) w (cid:48)(cid:48) ( t ) + aw ( t ) ≤ b, w (0) ≥ , where we let w ( t ) = w ( α, t ) , a := exp (cid:0) f − − ( H (0)) (cid:1) , b := ρ ( α )for notational simplicity. We notice that the inequality w (0) ≥ T ∗ > t (cid:37) T ∗ w ( α, t ) = 0.This contradicts to (2.17), and hence finishes the proof of Theorem 1.1.We consider two disjoint cases, call them Case A and
Case B for w (cid:48) (0) ≤ w (cid:48) (0) > Case A : We first consider the case w (cid:48) (0) ≤
0. We claim that b − aw ( t ) = 0 for some t .Suppose to the contrary that b − aw ( t ) (cid:54) = 0 for all t ≥
0. Since b − aw (0) < w (0) ≥
1, we have(2.19) b − aw ( t ) < t ∈ [0 , + ∞ ) . Combining (2.18)–(2.19), we see that w (cid:48)(cid:48) ( t ) < t . From this and w (cid:48) (0) ≤
0, we havethat w (cid:48) ( t ) → c ∈ [ −∞ ,
0) as t → + ∞ , which implies that w ( t ) → −∞ as t → + ∞ . This isa contradiction to (2.19). This proves the claim.Then, by the continuity of w , we can choose the minimal T > b = aw ( T ) . Hence there holds w (cid:48)(cid:48) ( t ) ≤ b − aw ( t ) < t ∈ (0 , T ), which in turn implies w (cid:48) ( t ) = (cid:90) t w (cid:48)(cid:48) ( s ) ds + w (cid:48) (0) < t ∈ (0 , T ] . Now we split the proof further into two cases:(2.21a)(2.21b) (i) w (cid:48) ( t ) < , T ] and w (cid:48) ( t ) has a zero on ( T , + ∞ ) , (ii) w (cid:48) ( t ) < t > . Case (i) : We choose the minimal T > T satisfying(2.22) w (cid:48) ( T ) = 0 . Then, w (cid:48) ( t ) < t ∈ (0 , T ). It suffices to show that w ( T ) ≤ w ( t ) = 0 for some t ∈ (0 , T ] as desired.We shall show that w ( T ) ≤ w ( T ) >
0. Then since w decreases on [ T , T ], we have(2.23) 0 < w ( T ) < w ( T ) = b/a, BLOW UP FOR THE EULER-POISSON SYSTEM 9 where the equality is from (2.20). Multiplying (2.18) by w (cid:48) ≤
0, and then integrating over[0 , t ], we obtain that for t ∈ [0 , T ] , (2.24) | w (cid:48) ( t ) | ≥ − a (cid:18) w ( t ) − | w (0) | (cid:19) + b ( w ( t ) − w (0)) + | w (cid:48) (0) | . Here we define a function g ( w ) := − a (cid:16) w −| w (0) | (cid:17) + b ( w − w (0)) + | w (cid:48) (0) | . We see that(2.25) g (0) = a | w (0) | − w (0) b + | w (cid:48) (0) | ≥ a − b + | w (cid:48) (0) | > , where we have used the assumption w (0) ≥ g ( w ) is strictly increasing on [0 , b/a ].Using this together with (2.25), we have(2.26) g ( w ) ≥ g (0) > w ∈ [0 , b/a ] . Combining (2.22)–(2.26), we have0 = | w (cid:48) ( T ) | ≥ g ( w ( T )) > , which is a contradiction. Case (ii) : We first claim that lim sup t →∞ w (cid:48) ( t ) = 0. If not, i.e., lim sup t →∞ w (cid:48) ( t ) (cid:54) = 0,then thanks to (2.21b), we have lim sup t →∞ w (cid:48) ( t ) <
0. This implies w ( t ) = 0 for some t > w is monotonically decreasing on (0 , ∞ ) thanks to (2.21b), wesee that w ∞ := lim t →∞ w ( t ) exists and w ∞ ∈ [0 , b/a ] by (2.20). Similarly as in obtaining(2.24), we multiply (2.18) by w (cid:48) ( t ) ≤ t ∈ [0 , ∞ ), and then integrate the resultant over [0 , t ]to obtain that (2.24) holds for t ∈ [0 , ∞ ). Since 0 = lim sup t →∞ w (cid:48) ( t ) = lim inf t →∞ | w (cid:48) ( t ) | ,we arrive at 0 = lim inf t →∞ | w (cid:48) ( t ) | / ≥ lim inf t →∞ g ( w ( t )) = g ( w ∞ ) ≥ g (0) > , where we have used (2.26) for the last inequality. This is absurd, which completes the prooffor Case A . Case B : Now we consider the case w (cid:48) (0) >
0. We claim that w (cid:48) ( t ) = 0 for some t >
0. Ifnot, i.e., w (cid:48) ( t ) > t ≥
0, we have w (cid:48)(cid:48) ( t ) ≤ b − aw ( t ) ≤ b − aw (0) < . This implies that w (cid:48) ( t ) → −∞ as t → + ∞ , which is a contradiction to the assumption that w (cid:48) ( t ) > t ≥ w (cid:48) ( t ), there is a minimal number T > w (cid:48) ( T ) = 0.Since w (cid:48) ( t ) > t ∈ [0 , T ), we see that w ( T ) ≥ w (0) ≥
1. Now one can apply thesame argument as
Case A to conclude that w ( t ) has a zero on the interval [ T , + ∞ ). Thiscompletes the proof of Theorem 1.1. (cid:3) We remark that, following the proof of Theorem 1.1, one obtains an interesting lemmaconcerning the existence of zeros of second-order linear differential inequality (see Appendix4.2). Numerical experiments
In this section, we present numerical examples concerning our theoretical result in The-orem 1.1. Referring to [15], the implicit pseudo-spectral scheme is employed to solve (1.1)numerically on periodic domains for numerical convenience.We demonstrate three cases (see Table 1). In the case (a), the condition (1.4) holds, and ρ and u x are expected to blow up after t = 2 . t = 2 . t ∈ [0 ,
20] in Figure 4.(a) (b) (c) ρ ( x ) 1 − . sech (3 x ) 1 − . sech (2 x ) 1 − . sech (2 x ) H (0) 0 . . . f − − ( H (0))) 0 . . . Table 1. ρ is the initial density function. The initial velocity u are givenas identically zero function for all cases. H (0) is the energy defined in (1.3)for the initial data. 4. Appendix
Proof of inequality (1.5) . Proof.
We use the following elliptic estimates (see [14]):(4.1a)(4.1b) K − := inf x ∈ R ρ ≤ e φ ≤ sup x ∈ R ρ =: K + for all x ∈ R , (cid:90) R | φ x | + K − | φ | dx ≤ K − (cid:90) R | ρ − | dx. Using the Poisson equation (1.1c), we have that(4.2) (cid:90) ( ρ − φ dx = (cid:90) | φ x | + ( e φ − φ dx = (cid:90) | φ x | + ( φ − e φ + 1 + ( e φ − − φ ) dx ≥ (cid:90) | φ x | + ( φ − e φ + 1 dx since e φ − − φ ≥
0. Using Young’s inequality and (4.1b), we obtain(4.3) (cid:90) ( ρ − φ dx ≤ K − (cid:90) | φ | dx + 12 K − (cid:90) | ρ − | dx ≤ K − (cid:90) | ρ − | dx. Combining (4.1a), (4.2) and (4.3), we obtain the inequality (1.5). (cid:3) BLOW UP FOR THE EULER-POISSON SYSTEM 11 -10 -8 -6 -4 -2 0 2 4 6 8 100.511.522.5-10 -8 -6 -4 -2 0 2 4 6 8 10-0.2-0.100.10.2-10 -8 -6 -4 -2 0 2 4 6 8 10-0.25-0.2-0.15-0.1-0.05 t=0t=1t=2t=2.30 1 2123456 0 1 20510
Figure 2.
Numerical solution to (1.1) for the case (a). ρ (0 , t ) and u x (0 , t )are getting larger as time t goes by.4.2. Zeros of second-order differential inequality.
Following the proof of Theorem1.1, one obtains the following lemma:
Figure 3.
Numerical plots of sup x | ρ ( · , t ) − | (left) and sup x | u x ( · , t ) | (right)for the case (b). Although the condition (1.4) does not hold, ρ and u x areexpected to eventually blow up at a finite time. Figure 4.
Numerical plots of sup x | ρ ( · , t ) − | (left) and sup x | u x ( · , t ) | (right)for the case (c). sup x | ρ − | and sup x | u x | keep oscillating and decreasing astimes t goes by. Lemma 4.1.
Let a and b be positive constants. Suppose w ( t ) satisfies w (cid:48)(cid:48) + aw ≤ b for all t ≥ T and w ( T ) ≥ . If a/ > b and (4.4) a | w ( T ) | − w ( T ) b + | w (cid:48) ( T ) | > , then w ( t ) has a zero on the interval ( T , + ∞ ) . The authors are not aware of any literature addressing the existence of zeros of second-order linear differential inequality with the coefficient a > constant nonhomogeneous term b . We finish this subsection with some remarks regarding Lemma 2.3. BLOW UP FOR THE EULER-POISSON SYSTEM 13
Remark . (1) For the case of the differential equation w (cid:48)(cid:48) + aw = b ,(4.5) a | w (0) | − w (0) b + | w (cid:48) (0) | ≥ w to have a zero on [0 , + ∞ ).(2) One needs the restriction a/ > b (or a/ ≥ b ) in Lemma 4.1. If a/ < b , then thesolution to w (cid:48)(cid:48) + aw = b with w (0) = 1 and w (cid:48) (0) = 0 has no zero since (4.5) is notsatisfied. For another example, we consider the equation(4.6) w (cid:48)(cid:48) + aw = b − e − t , t ∈ [0 , + ∞ ) , where a, b > w ( t ) = α cos √ at + β sin √ at + ba − e − t a + 1 , we have w (0) = α + ba − a + 1 , w (cid:48) (0) = √ aβ + 1 a + 1 . Since min t ≥ w ( t ) ≥ min t ≥ (cid:0) α cos √ at + β sin √ at (cid:1) + min t ≥ (cid:18) ba − e − t a + 1 (cid:19) = − (cid:112) α + β + ba − a + 1 ,w ( t ) has no zero on [0 , + ∞ ) provided that(4.7) − (cid:112) α + β + ba − a + 1 > . We choose b = 1 / a > a + 1) > b − a > aa + 1 . For w (0) = 1 and w (cid:48) (0) = a +1 , the first inequality of (4.8) is equivalent to (4.4) andthe second inequality of (4.8) is equivalent to (4.7). On the other hand, b > a/ Acknowledgments.
B.K. was supported by Basic Science Research Program through the National ResearchFoundation of Korea (NRF) funded by the Ministry of science, ICT and future plan-ning (NRF-2020R1A2C1A01009184). The authors thank Shih-Hsin Chen and Yung-HsiangHuang for suggesting the example (4.6).
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Email address : [email protected] (JC) Department of Mathematical Sciences, Ulsan National Institute of Science and Tech-nology, Ulsan, 44919, Korea
Email address : [email protected] (BK) Department of Mathematical Sciences, Ulsan National Institute of Science and Tech-nology, Ulsan, 44919, Korea
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