Blowup phenomena of solutions to the Euler equations for compressible fluid flow

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1 J. Differential Equations Blowup phenomena of solutions to the Euler equations for compressible fluid flow Tianhong Li a,, Dehua Wang b a Department of Mathematics, Stanford University, Stanford, CA 94305, USA b Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 1560, USA Received 5 October 004 Available online 5 January 005 Abstract The blowup phenomena of solutions is investigated for the Euler equations of compressible fluid flow. The approach is to construct special explicit solutions with spherical symmetry to study certain blowup behavior of multi-dimensional solutions. In particular, the special solutions with velocity of the form ctx are constructed to show the expanding and blowup properties. The solution with velocity of the form ȧtx/at for γ 1 and for any space dimensions is obtained as a corollary. Another conclusion is that there is only trivial solution with velocity of the form ct x α 1 x for α = 1 and multi-space dimensions. 005 Elsevier Inc. All rights reserved. Keywords: Euler equations; Compressible fluid; Spherical symmetry; Special solution; Blowup 1. Introduction In fluid dynamics, blowup is an interesting and important phenomena. Here, we are concerned with this phenomena for an inviscid compressible fluid flow in the polytropic case, which is governed by the following Euler equations in N-space dimensions [14]: ρ t + ρu = 0, ρu t + ρu u + p = 0, 1 Corresponding author. addresses: thli@math.stanford.edu T. Li, dwang@math.pitt.edu D. Wang /$ - see front matter 005 Elsevier Inc. All rights reserved. doi: /j.jde

2 9 T. Li, D. Wang / J. Differential Equations where x R N is the space variable, t [0, is the time, ρ is the density of the fluid, u R N is the fluid velocity, p is the scalar pressure, and the symbol denotes the Kronecker tensor product. For the polytropic gas, p = pρ = ρ γ /γ with γ 1 the adiabatic exponent. The fluid is called isothermal if γ = 1, and isentropic if γ > 1. The blowup phenomena for the multi-dimensional compressible flows has attracted lots of interests and attentions because of its physical importance and mathematical challenge. It is a difficult problem to understand the blowup behavior of the general solutions of the multi-dimensional compressible Euler equations. There have been many studies on this kind of phenomena by attempting to construct explicit examples in some special setting. Courant and Friedrichs see their book [9] and references therein studied the special solutions called progressing waves. For non-isentropic gas, Taylor [1,] described the point blast. For isentropic gas, the Euler equations can be transformed into a system of ordinary differential equations. For this dynamical system, the critical lines exist and thus the center of the gas cannot be reached, that is, global solutions cannot be obtained. Taylor [1,] succeeded in finding the wave motion produced by an expanding sphere and preceded by a shock front. It is similar to the one-dimensional gas flow produced by a piston with constant speed. The progressing waves were also succeeded in finding some other types of spherical waves like detonation, deflagration, combustion and reflected shocks. See [,19] for more related discussions. Recently, Li [15] obtained two types of special solutions to Euler equations with spherical symmetry. The first type solutions are the only possible C solutions satisfying ρ x = 0 where x = x is the radius. Outward gas expands eventually while inward gas blows up in finite time. The second type solutions were obtained by assuming u =ȧtx/at for γ = 4 3 and N = 3. There have been many papers on the special solutions of the Euler Poisson equations for gaseous star with self-gravitation, see [18,11 13] and the references cited therein. In [13], the authors found that there exists critical mass M 0 for the special solutions in [18] for γ = 4 3 and N = 3. If the mass of the gaseous star is less than M 0, it is impossible for the solution to blowup. Otherwise, with suitable initial velocity, the solutions blowup in a finite time. In [11], the authors excluded the possibility of δ function or bigger blowup for γ > 4 3 for spherically symmetric or non-spherically symmetric gas by the total potential energy control and Fourier transformation. In [15], the critical mass is infinite for the Euler equations. In this paper, we generalize the previous results in this direction to more general solutions of spherical symmetry. The spherically symmetric flow, along with other flows with symmetry-like transonic nozzle flow, cylindrically symmetric rotating flow, and axisymmetric flow, are motivated by many physical problems, such as supernovae formation in stellar dynamics, inertial confinement fusion, and explosion waves in water, air, and other media [9]. An important issue for the spherically symmetric flow is whether the solution blows up to infinity in a finite time, especially whether the density blows up near the origin for the flow moving radially inward. This question is not easily understood even in physical experiments and numerical simulations. The difficulties of this spherically symmetric problem in unbounded domains include the singularity at the origin, the reflection of waves from infinity and their strengthening as they move radially inward, the reflection of spherical shock fronts at the center, the associated steady-state equations changing type from

3 T. Li, D. Wang / J. Differential Equations elliptic to hyperbolic at the sonic state, and so on. There have been many studies about the spherically symmetric flows which are outgoing or outside a solid ball, see the discussions and review in [6,8] and the references cited therein, as well as the recent paper [7] for γ = 1. The difficulty arising from the singularity at the origin is still unsolved. This singularity at the origin makes the spherically symmetric problem really multi-dimensional. The methods useful for the one-dimensional problem do not work for the multi-dimensional problem [10]. Many explosion phenomena e.g., atomic bomb and underwater explosion can be idealized as the sudden release of an amount of energy concentrated at a point [3]. It would be interesting to know whether global entropy solutions exist or there are explosions in finite time if the singularity x = 0 is included. A piston problem with such singularity initially was studied in [5]. For a spherically symmetric flow modeling outgoing blast, a large entropy solution was constructed in [4]. Our interest in this paper is to understand the blowup phenomena of solutions when the flow moves inward radially. For the discussions on the finite-time formation of singularities and shock waves in solutions of 1, see [1,3,17,0]. Our approach in this paper is to construct special explicit solutions with spherical symmetry to study the blowup property. In particular, we generalize the second type of special solutions in [15] to u = ctx for any γ 1 and any N. In[16] the particular solutions of the form u = ctx was constructed for the compressible flow with damping near vacuum. In this paper, we obtain an autonomous second-order ordinary differential equation for ct and then consider the corresponding first-order dynamical system. The solution curve c + c = 0 divides c c phase plane into two regions. In one region, any point corresponds to a finite total mass; in another region, any point corresponds to infinite total mass. By analysis of direction field, it turns out that if the total mass is finite, then the fluid flow e.g., gas is impossible to blowup and it will expand. If the total mass is infinite, with suitable initial velocity, the solutions blowup in finite time. As a corollary, we have the solution of the form u =ȧtx/at for any γ 1 and for any N, which has the same behavior as that for γ = 4 3 and N = 3. Surprisely, the solutions which satisfy c + c = 0 when c =ȧt/at are the first type solutions in [15], which are the only C solutions with ρ x = 0, where x = x. Finally, we will conclude that there is only trivial solution of the form u = ct x α 1 x, α = 1, N>1. In the next section, we state and prove our results on the blowup and non-blowup solutions.. Blowup and non-blowup phenomenon We now construct a family of solutions showing certain blowup phenomenon when the flow is spherically symmetric. The solution of 1 with spherical symmetry ρ = ρx, t, u = ux, t x x, where x = x, satisfies the following equations: ρ t + uρ x + ρu x + N 1 ρu = 0, x ρu t + uu x + p x = 0, where p = ρ γ /γ with γ 1, and N = 1,, 3,.... Below is our main result.

4 94 T. Li, D. Wang / J. Differential Equations Theorem.1. There exists a family of solutions ρ, ux, t with u = ctx for with γ 1 satisfying the following properties: 1 If c0 +c 0 >0, then for any t, ct +c t > 0, and ct 0, ρ, ux, t 0 as t ; if c0 + c 0 <0, then for any t, ct + c t < 0. Furthermore, in the region where ct +c t < 0, if c 0+ +N c0 0 and c 0, then ct 0 as t and there is no blowup; if otherwise, ct and ρx, t as t. Proof. When u = ctx, becomes ρ t + cxρ x + Nρc = 0, 3 From 4, when γ > 1, we have c x + c x + ρ γ ρ x = 0. 4 Therefore, ρ γ 1 x = c + c x. ρ γ 1 = c + c x + Dt 5 for some function Dt. Multiplying 3 byρ γ, then Using 5, ρ γ 1 t ρ γ 1 t ρ + cx + Nγ 1 ρ γ 1 x γ 1 c = 0. cxc + c x Nγ 1c + c x c + Nγ 1Dc = 0. Thus, D t + Nγ 1Dc = 0, 6 c Nγ 1cc + Nγ 1 + c 3 = 0. 7 Observe that c is a solution to 7 ifc + c = 0, since 7 can be also written as c + c + + Nγ 1cc + c = 0. 8

5 T. Li, D. Wang / J. Differential Equations I 1 c 0 I III -1 +c =0 - II +0.6 c =0-0 4 Fig. 1. The phase portrait for c and. Consider the dynamical system obtained from 7 { = 4 + Nγ 1 + Nγ 1c c, c =. 9 The curve c + c = 0 divides the c phase plane into two invariant regions. Fig. 1 demonstrates the direction fields and phase portrait for c and when N = 3, γ = 4 3, +N 4+N = 0.6. From the direction field in Fig. 1, it can be easily seen a Solution curves starting from region I and the third quadrant approach,. b Solution curves starting from region I approach 0, 0. c Solution curves starting from region III enter I and then approach 0, 0. The case where solution curves start from region II is less obvious. Consider solution curves starting from region II, then c = + c c + Nγ 1 + c c 3 < 0.

6 96 T. Li, D. Wang / J. Differential Equations The last inequality is due to the fact that c<0 and + c > 0 in II. Moreover, since 1 + c c is negative and decreasing in II, 1 + is increasing in II, where hence 1 + c c is decreasing in II. From this observation, in II we have c + c c = 1 + c c 1 + c0 c0 <0. 0 This implies that any solution curve starting from II has to enter region III, then I, and approaches 0, 0 as t approaches infinity. Finally, we consider the case where solution curves starting region I and the second quadrant. From 9, we have d dc = Let = z 0 for z 0. Then we have 4 + Nγ 1 + Nγ 1 c c. dz dc = 4 + Nγ 1c z + Nγ 1c Let z be of the form z = βc α. From 10, we get α = 4, and β = 1 Then we get two solutions of 9: + c = 0 and + +N c = 0. The first solution curve is known, and the second solution curve is inside the region I. Case 1: It is easy to see that, if c 0 + +N c0 0, then the solution curves or β = +N 8. will stay in the region bounded by + c = 0 and + +N c = 0, and ct approaches 0 when t goes to. Case : If c 0 + +N c0 < 0, we will prove by contradiction that there exists some t 0 > 0 such that ct 0 = 0, and t 0 = 0, therefore the solution curve ct, t will enter the third quadrant after t 0 and approaches, when t goes to infinity. Indeed, if this is not true, then ct, t 0ast. From 9, Then + c = + Nγ 1 + c c. t+ ct 0 + c0 = t e +N 0 cs ds. 11 Since ct 0, t 0ast,11 implies that 0 cs ds =. 1

7 Observe that, for wt = t ct T. Li, D. Wang / J. Differential Equations and b = N, w t = cw + 1w b < Therefore w decreases with t from 1 + b and 1 + b w + 1 = b 1 + e b t 0 cs ds. w0 + 1 From 1, it follows that wt 1ast. It contradicts with that w decreases from 1 + b < 1. Therefore, the solution curves will enter the third quadrant and approach. From 5 and 6, when γ > 1, ρ γ 1 = c + c x + t de N 0 cs ds 14 for some constant d>0. Following a similar procedure, when γ = 1, ρ = de c +c x e N t 0 cs ds 15 for some constant d>0. This completes the proof. Remark.. From 14 and 15, the total mass is finite when c0 + c 0 >0, and is infinite when c0 +c 0 0. The density of the solutions described in Theorem.1 approaches zero if the mass is finite, and blowup can occur only when the mass is infinite. Corollary.3. Let ρ,u be a solution for with u = ȧt at x and at > 0. Then at satisfies λ ä = a 1+N, for some λ R. Furthermore, 1 Suppose λ < 0. Then ρ is bounded from above and it approaches 0 as t. Suppose λ 0. Then λ v e = Nγ 1a0 N 16 is the escape velocity. If ȧ0 v e, then ρ approaches 0 as t. If ȧ0 <v e, then ρ blows up in finite time.

8 98 T. Li, D. Wang / J. Differential Equations Proof. If ct = ȧt at, then c + c äa =.By8, Therefore From 17, clearly ȧt λ Nγ 1 is a constant. Moreover, ä a + + Nγ 1ȧ a a 3 ä ä a = 0, = 1 + Nγ 1ȧ a, log ä = 1 + Nγ 1 log a +const. ä = λ a 1+N for some λ R ȧ0 = at N t ȧt =ȧ0 + 0 λ Nγ 1 1 =: h 18 a0 N λ ds. 19 a1+n If λ < 0, then a is a convex function. By 18, we have ȧt < h and at N λ Nγ 1h > 0. In particular, this implies that ρ is bounded from above. If we can show that ȧt > 0 for some T, then clearly at approaches infinity, and hence ρ goes to 0, as t goes to infinity. This is quite obvious because if ȧt 0, for all t, at would be bounded from above and, by 19, ȧt would be unbounded, contradicting the fact that ȧt is bounded. Suppose λ 0, then at is concave and ȧt is decreasing in t. IfȧT 0 for some T, then at goes to 0 in finite time. This is the moment when the solution blows up. If ȧt > 0 for all t and, by 19, at increases to infinity as t goes to infinity. The escape velocity that characterizes these two cases is the initial velocity that leads to lim t ȧt = 0, in which case at goes to infinity according to 19. By 18 the escape velocity is v e defined in 16. This completes the proof. A natural question arises as whether there are solutions of the form u = ctx α with α = 1.

9 T. Li, D. Wang / J. Differential Equations Theorem.4. Assume N>1. Then there are only trivial solutions to of the form u = ctx α with α = 1. Proof. Let u = ctx α. First we consider when γ > 1. By the second equation of, we get ρ = c x α + c αx α 1. 0 γ 1 Therefore if α = 1 or 0, then where Dt is an arbitrary function of t. By the first equation of, ρ γ 1 By 0 and, we get t x ρ γ 1 = c t α + 1 xα+1 + c xα + Dt, 1 + ctx α ρ + N 1 + αγ 1 ρ γ 1 x γ 1 ctxα 1 = 0. c α + 1 xα+1 α + + γ 1α + N 1 γ 1α + N 1 α + 1 c 3 x 3α 1 c cx α + D +γ 1α + N 1cDx α 1 = 0. 3 If α = 1, 0, 1, or 1 3, then all the terms in above equation have different power of x. Therefore we can get a set of equalities: c = 0, 4 + α + γ 1α + N 1 c c = 0, 5 α + 1 D = 0, 6 γ 1α + N 1 c 3 = 0, 7 α + N 1cD = 0. 8

10 100 T. Li, D. Wang / J. Differential Equations From 4 and 6, we get c = At + B, D = C 1. 9 There are two cases: Case 1: If α + N 1 = 0, by 7, then we get c = 0. Therefore c = 0 and D = C 1. By 1, this is the trivial solution u = 0 and ρ = C. Case : If α + N 1 = 0, by 8, then we get cd = 0. Either c = 0orD = 0. a c = 0 and D = C 1. i.e. u = 0 and ρ = C. b D = 0. If α + α+n 1 = 0, by 7, then we get c = 0. This is u = 0 and ρ = 0. If α + α+n 1 = 0, then + α+n 1 α+1 = 0. It implies c = B. By expression of ρ 1, we get ρ = B x 0. It implies B = 0. Therefore u = 0 and ρ = 0. Next we consider α = 1 3, α = 1 and α = 0. If α = 1 3, then from 3, we get + D = c = 0, γ 1α + N 1 c c = 0, α + 1 α + γ 1α + N 1 c 3, α + N 1cD = 0. Since N 1 = 0, it implies either c = 0orD = 0 from the last equality. By the third equality, if c = 0, then D = C 1.IfD = 0, then c = 0. For α = 1, we follow the same procedure, and get there are only trivial solutions. For α = 0, we follow the same procedure, and get for N>1, there are only trivial solutions. The case γ = 1 is same. Remark.5. If N = 1, then the only non-trivial solutions with form u = ctx α to are for γ > 1, u = At + B, ρ γ 1 = A t + ABt + C 1 Ax, for γ = 1, u = At + B, ln ρ = A t + ABt + C 1 Ax, where A, B and C 1 are arbitrary constants.

11 Acknowledgments T. Li, D. Wang / J. Differential Equations Dehua Wang s research was supported in part by the National Science Foundation and the Office of Naval Research. References [1] S. Alinhac, Blowup for Nonlinear Hyperbolic Equations, Birkhauser, Boston, [] G.I. Barenblatt, Scaling, Self-similarity, and Intermediate Asymptotics, Cambridge Texts in Applied Mathematics, vol. 14, Cambridge University Press, Cambridge, [3] J.Y. Chemin, Remarques sur l apparition de-singularités dans les ecoulements Euleriens compressibles, Comm. Math. Phys [4] G.-Q. Chen, Remarks on spherically symmetric solutions of the compressible Euler equations, Proc. Roy. Soc. Edinburgh 17A [5] G.-Q. Chen, S.-X. Chen, D. Wang, Z. Wang, A multidimensional piston problem for the Euler equations for compressible flow, Discrete and Continuous Dynamical Systems [6] G.-Q. Chen, J. Glimm, Global solution to the compressible Euler equations with geometrical structure, Comm. Math. Phys [7] G.-Q. Chen, T.H. Li, Global entropy solutions in L to the Euler equations and Euler Poisson equations for isothermal fluids with spherical symmetry, Methods Appl. Anal [8] G.-Q. Chen, D. Wang, The Cauchy problem for the Euler equations for compressible fluids, Handbook of Mathematical Fluid Dynamics, vol. I, North-Holland, Amsterdam, 00, pp [9] R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Springer, New York, 196. [10] C.M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer, Berlin, 000. [11] Y. Deng, T.-P. Liu, T. Yang, Z.-A. Yao, Solutions of Euler Poisson equations for gaseous stars, Arch. Rational Mech. Anal [1] Y. Deng, J. Xiang, T. Yang, Blowup phenomena of solutions to Euler Poisson equations, J. Math. Anal. Appl [13] C.-C. Fu, S.-S. Lin, On the critical mass of the collapse of a gaseous star in spherically symmetric and isentropic motion, Japan J. Ind. Appl. Math [14] L.D. Landau, E.M. Lifshitz, Fluid Mechanics, second ed., Butterworth-Heinemann, Oxford, [15] T.H. Li, Some special solutions of the Euler equation in R N, Comm. Pure Appl. Anal [16] T.P. Liu, Compressible flow with damping and vacuum, Japan J. Indust. Appl. Math [17] A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space ariables Applied Mathematical Science, vol. 53, Springer, New York, [18] T. Makino, Blowing up solutions of the Euler Poisson equation for the evolution of gaseous stars, Proceedings of the Fourth International Workshop on Mathematical Aspects of Fluid and Plasma Dynamics Kyoto, 1991, Transport Theory Statist. Phys [19] L.I. Sedov, Similarity and Dimensional Methods in Mechanics, Transl. by M. Friedman transl. M. Holt Ed.,, Academic Press, New York, London, [0] T.C. Sideris, Formation of singularities in three-dimensional compressible fluids, Comm. Math. Phys [1] G.I. Taylor, The Formation of a Blast Wave by a ery Intense Explosion, Ministry of Home Security RC 10 II [] G.I. Taylor, The Propagation and Decay of Blast Waves, British Civilian Defense Research Committee, [3] G.B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, New York, 1973.