Analytical solutions to the Navier Stokes equations

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1 JOURAL OF MATHEMATICAL PHYSICS 49, Analytical solutions to the avie Stokes equations Yuen Manwai a Depatment of Applied Mathematics, The Hong Kong Polytechnic Univesity, Hung Hom, Kowloon, Hong Kong Received 1 August 2008; accepted 14 Octobe 2008; published online 12 ovembe 2008 With the pevious esults fo the analytical blowup solutions of the -dimensional 2 Eule Poisson equations, we extend the same stuctue to constuct an analytical family of solutions fo the isothemal avie Stokes equations and pessueless avie Stokes equations with density-dependent viscosity Ameican Institute of Physics. DOI: / I. ITRODUCTIO The avie Stokes equations can be fomulated in the following fom: t + u =0, u t + u u + P = vis,u. As usual, =x,t and ux,t ae the density and the velocity, espectively. P= P is the pessue. We use a -law on the pessue, i.e., P = K, with K0, which is a univesal hypothesis. The constant =c P /c v 1, whee c p and c v ae the specific heats pe unit mass unde constant pessue and constant volume, espectively, is the atio of the specific heats. is the adiabatic exponent in 2. In paticula, the fluid is called isothemal if =1. It can be used fo constucting models with nondegeneate isothemal fluid. can be the constant 0 o 1. When =0, we call the system pessueless; when =1, we call that it is with pessue. Additionally, vis,u is the viscosity function. When vis,u=0, the system 1 becomes the Eule equations. Fo the detailed study of the Eule and avie Stokes equations, see Refs. 1 and 4. In the fist pat of this aticle, we study the solutions of the -dimensional 1 isothemal equations in adial symmety, t + u + u + 1 u =0, u t + uu + K = vis,u. Definition 1: Blowup We say a solution blows up if one of the following conditions is satisfied. 1 The solution becomes infinitely lage at some point x and some finite time T. 2 The deivative of the solution becomes infinitely lage at some point x and some finite time T. Fo the fomation of singulaity in the thee-dimensional case fo the Eule equations, please efe to the pape of Sideis. 10 In this aticle, we extend the esults fom the study of the blowup analytical solutions in the -dimensional 2 Eule Poisson equations, which descibes the a Electonic mail: nevetsyuen@hotmail.com /2008/4911/113102/10/$ , Ameican Institute of Physics

2 Yuen Manwai J. Math. Phys. 49, evolution of the gaseous stas in astophysics 2,3,7,12,13 to the avie Stokes equations. Fo the same kinds of blowup esults in the nonisothemal case of the Eule o avie Stokes equations, please efe to Refs. 5 and 12. Recently, in Yuen s esults, 13 thee exists a family of the blowup solution fo the Eule Poisson equations in the two-dimensional adial symmety case, t + u + u + 1 u =0, u t + uu + K = 2 t,ssds. 0 4 The solutions ae t, = 1 at 2ey/at, ut, = ȧt at, ät = at, a0 = a 0 0, ȧ0 = a 1, 5 ÿx + 1 x ẏx + 2 K eyx =, y0 =, ẏ0 =0, whee K0, =2/K with a sufficiently small, and ae constants. 1 When 0, the solutions blow up in a finite time T. 2 When =0, if a 1 0, the solutions blow up at t= a 0 /a 1. In this pape, we extend the above esult to the isothemal avie Stokes equations in adial symmety with the usual viscous function whee v is a positive constant, vis,u = vu, t + u + u + 1 u =0, 6a u t + uu + K = vu + 1 u 1 2 u. 6b Theoem 2: Fo the -dimensional isothemal avie Stokes equations in adial symmety (6a) and (6b), thee exists a family of solutions; those ae t, = 1 at ey/at, ut, = ȧt at, ät = at, a0 = a 0 0, ȧ0 = a 1, 7 yx = 2K x2 +,

3 Analytical solutions to avie Stokes equations J. Math. Phys. 49, whee and ae abitay constants. In paticula, fo 0, the solutions blow up in finite time T. In the last pat, the coesponding solutions to the pessueless avie Stokes equations with density-dependent viscosity ae also studied. II. THE ISOTHERMAL =1 CASES Befoe we pesent the poof of Theoem 2, Lemma 6 of Ref. 13 could be needed to futhe extend to the -dimensional space. Lemma 3: The Extension of Lemma 6 of Ref. 13 Fo the equation of consevation of mass in adial symmety, thee exist solutions t + u + u + 1 u =0, 8 t, = f/at ȧt, ut, = at at, with the fom f 0C 1 and at0c 1. Poof: We just plug 9 into 8. Then 9 t + u + u + 1 u = ȧtf/at at +1 ȧtḟ/at at +2 + ȧt ḟ/at at at f/at ȧt at at =0. + f/at ȧt at at The poof is completed. Besides, Lemma 7 of Ref. 13 is also useful. Fo the bette undestanding of the lemma, the poof is given hee. Lemma 4: Lemma 7 of Ref. 13 Fo the Emden equation, ät = at, a0 = a 0 0, ȧ0 = a 1, we have that, if 0, thee exists a finite time T + such that at =0. Poof: By integating 10, we have ȧt2 = ln at +, 11 whee = ln a a 1 2. Fom 11, weget at e /. If the statement is not tue, we have Howeve, since 0 at e / fo all t 0.

4 Yuen Manwai J. Math. Phys. 49, we integate this twice to deduce ät = at e /, t 0 at 0 By taking t lage enough, we get e /dsd + C 1t + C 0 = t2 2e / + C 1t + C 0. at 0. As a contadiction is met, the statement of the lemma is tue. By extending the stuctue of the solutions 5 to the two-dimensional isothemal Eule Poisson equations 4 in Ref. 13, it is a natual esult to get the poof of Theoem 2. Poof of Theoem 2: By using Lemma 3, we can get that 7 satisfy 6a. Fo the momentum equation, we have By choosing u t + u u + K vu + 1 u u ẏ 1 2 = ät at + K at at = at at + Kẏ at. yx = 2K x2 +, we have veified that 7 satisfies the above 6b. If0, by Lemma 4, thee exists a finite time T fo such that at =0. Thus, thee exist blowup solutions in finite time T. The poof is completed. With the assistance of the blowup ate esults of the Eule Poisson equations, i.e., Theoem 3 in Ref. 13, it is tivial to have the following theoem. Theoem 5: With 0, the blowup ate of the solutions (7) is lim t,0t t O1, t T whee the blowup time T and ae constants. Remak 6: If we ae inteested in the mass of the solutions, the mass of the solutions can be calculated by Mt t,sds = t,ss =R 0 1 ds, whee denotes some constant elated to the unit ball in R : 1=1; 2=2; fo 3, = 2V = 2 /2+1, whee V is the volume of the unit ball in R and is the gamma function. We obseve the following fo the mass of the initial time 0: 1 Fo 0, + /2

5 Analytical solutions to avie Stokes equations J. Math. Phys. 49, M0 = + e a 0 0 /2Ks2 + s 1 ds. 2 The mass is infinitive. The vey lage density comes fom the ends outside the oigin O. Fo 0, M0 = + e a 0 0 /2Ks2 + s 1 ds = e + e a 0 0 /2Ks2 s 1 ds. The mass of the solution can be abitaily small but without compact suppot if is taken to be a vey small negative numbe. Remak 7: Ou esults can be easily extended to the isothemal Eule/avie Stokes equations with fictional damping tem with the assistance of Lemma 7 in Ref. 12, t + u + u + 1 u =0, whee 0 and v0. The solutions ae u t + u u + K + u = vu + 1 t, = ey/at ȧt, ut, = at at, u 1 2 u, ät + ȧt = at, a0 = a 0 0, ȧ0 = a 1, yx = 2K x2 +. Remak 8: Ou esults can be easily extended to the isothemal Eule/avie Stokes equations with fictional damping tem with the assistance of Lemma 7 in Ref. 12, t + u + u + 1 u =0, whee 0 and v0. The solutions ae u t + u u + K + u = vu + 1 t, = ey/at ȧt, ut, = at at, u 1 2 u, ät + ȧt = at, a0 = a 0 0, ȧ0 = a 1, yx = 2K x2 +.

6 Yuen Manwai J. Math. Phys. 49, Remak 9: The solutions (5) to the Eule Poisson equations only wok fo the twodimensional case. Howeve, the solutions (7) to the avie Stokes equations wok fo the -dimensional 1 case. Remak 10: We may extend the solutions to the two-dimensional Eule/avie Stokes equations with a solid coe, 6 t + u + u + 1 u =0, u t + uu + K + u = M 0 + vu + 1 u 1 2u, whee M 0 0; thee is a unit stationay solid coe locating 0, 0, whee 0 is a positive constant, suounded by the distibution density. The coesponding solutions ae t, = ey/at ȧt 2, ut, = at at fo 0, ät + ȧt = at, a0 = a 0 0, ȧ0 = a 1, whee /2K is a constant. yx = 2K x2 + M 0 ln x +, III. PRESSURELESS AVIER STOKES EQUATIOS WITH DESITY-DEPEDET VISCOSITY ow we conside the pessueless avie Stokes equations with density-dependent viscosity, in adial symmety, vis,u u, t + u + u + 1 u =0, u t + uu = u + u + 1 u 1 2 u, whee is a density-dependent viscosity function, which is usually witten as with the constants, 0. Fo the study of this kind of the above system, the eades may efe to Refs. 8, 9, and 11. We can obtain the same estimate about Lemma 4 to the following odinay diffeential equation ODE: 12 ät = ȧt at 2, a0 = a 0 0, ȧ0 = a 1 a 0. 13

7 Analytical solutions to avie Stokes equations J. Math. Phys. 49, Lemma 11: Fo the ODE (13), with 0, thee exists a finite time T + such that at =0. Poof: If at0 and ȧ0=a 1 /a 0 fo all time t, by integating 13, we have Take the integation fo 14, ȧt = at + a 1 a 0 at t t asȧsds ds, 0 When t is vey lage, we have 1 2 at2 t a at2 1. A contadiction is met. The poof is completed. Hee we pesent anothe lemma befoe poceeding to the next theoem. Lemma 12: Fo the ODE, ẏxyx n x =0, y0 = 0, n 1, whee and n ae constants, we have the solution 15 yx = n n +1x2 + n+1. Poof: The above ODE 15 may be solved by the sepaation method, ẏxyx n x =0, By taking the integation with espect to x, ẏxyx n = x. 0 x x ẏxyx n dx xdx, =0 we have 0 x yx n dyx = 1 2 x2 + C 1, 16 whee C 1 is a constant. By integation by pat, then the identity becomes x yx n+1 yx n0 n 1 ẏxyxdx = 1 2 x2 + C 1,

8 Yuen Manwai J. Math. Phys. 49, yx n+1 ẏxyx n0 n dx = 1 2 x2 + C 1. Fom Eq. 16, we can have the simple expession fo yx, x yx n+1 n 1 2 x2 + C 1 = 1 2 x2 + C 1, yx n+1 = 1 2 n +1x2 + C 2, whee C 2 =n+1c 1. By plugging into the initial condition fo y0, we have Thus, the solution is y0 n+1 = n+1 = C 2. yx = n n +1x2 + n+1. The poof is completed. The family of the solution to the pessueless avie Stokes equations with density-dependent viscosity, t + u + u + 1 u =0, 17a u t + uu = u + u + 1 u 1 2 u, 17b is pesented as the following. Theoem 13: Fo the pessueless avie Stokes equations with density-dependent viscosity (17a) and (17b) in adial symmety, thee exists a family of solutions. Fo =1, t, = ey/at ȧt, ut, = at at, ät = ȧt at 2, a0 = a 0 0, ȧ0 = a 1, yx = 2 x2 +, whee and ae abitay constants. In paticula, fo 0 and a 1 /a 0, the solutions blow up in finite time. Fo 1, t, =y/at fo y at 0; ȧt 0 fo y at ut, = at 0,, ät = ȧt at 2+2, a0 = a 0 0, ȧ0 = a 1, 18

9 Analytical solutions to avie Stokes equations J. Math. Phys. 49, yx = x2 + 1, whee 0. Poof of Theoem 13: To 17a, we may use Lemma 3 to check it. Fo =1, 17b becomes u t + u u u u + 1 u 1 2 = u t + u u u = ät ey/at at at = ȧt at2 ẏ at at, whee we use u ȧt at = ȧt at 3 ẏ ey/at at at +1 ȧt at 19 By choosing ät = ȧt at 2. y at yx = 2 x2 +, 19 is equal to zeo. Fo the case of 1, 17b can be calculated, u t + u u u u + 1 at y at u u 1 2 = ät ȧt at at ȧt = at at 2+2 y at 1 ẏ at ȧt at 1 at at ȧt = at at 2+2 y aty at 2 ẏ atȧt at at 2+2 ȧt = at at 2+2 y at 2 ẏ at 2+2 atȧt 20 y = ȧt at 2+2 at at ẏ + 2 at. Define x/at, n 2; it follows = ȧt at 2+2x + yxn ẏx = ȧt at 2+2x + yxn ẏx, and / in Lemma 12, and choose 23

10 Yuen Manwai J. Math. Phys. 49, y at 1 yx = x Moeove this is easy to check that ẏ0 =0. Equation 22 is equal to zeo. The poof is completed. Remak 14: By contolling the initial conditions in some solutions 18, we may get the blowup solutions. Additionally the modified solutions can be extended to the system in adial symmety with fictional damping, t + u + u + 1 u =0, u t + uu + u = u + u + 1 whee 0. With the assistance of the ODE, u 1 2 u, ät + ȧt = ȧt at S, whee S is a constant. a0 = a 0 0, ȧ0 = a 1, 1 Chen, G. Q. and Wang, D. H., The Cauchy Poblem fo the Eule Equations fo Compessible Fluids, Handbook of Mathematical Fluid Dynamics, Vol. I oth-holland, Amstedam, 2002, pp Deng, Y. B., Xiang, J. L., and Yang, T., Blowup phenomena of solutions to Eule-Poisson equations, J. Math. Anal. Appl. 286, Goldeich, P. and Webe, S., Homologously collapsing stella coes, Astophys. J. 238, Lions, P. L., Mathematical Topics in Fluid Mechanics Claendon, Oxfod, 1998, Vols. 1 and 2. 5 Li, T. H., Some special solutions of the multidimensional Eule equations in R, Commun. Pue Appl. Anal. 4, Lin, S. S., Stability of gaseous stas in adially symmetic motions, SIAM J. Math. Anal. 28, Makino, T., Blowing up solutions of the Eule-Poisson equation fo the evolution of the gaseous stas, Tansp. Theoy Stat. Phys. 21, Makino, T., in Pattens and Wave-Qualitative Analysis of onlinea Diffeential Equations, edited by T. ishida, M. Mimua, and H. Fujii oth-holland, 1986, pp ishida, T., Equations of fluid dynamics-fee suface poblems, Commun. Pue Appl. Math. 39, S T. C. Sideis, Fomation of singulaities in thee-dimensional compessible fluids, Commun. Math. Phys. 101, Yang, T., Yao, Z.-A., and Zhu, C., Changjiang compessible avie-stokes equations with density-dependent viscosity and vacuum, Commun. Patial Diffe. Equ. 26, Yuen, M. W., Blowup solutions fo a class of fluid dynamical equations in R, J. Math. Anal. Appl. 329, Yuen, M. W., Analytical blowup solutions to the 2-dimensional isothemal Eule-Poisson equations of gaseous stas, J. Math. Anal. Appl. 341,

d 2 x 0a d d =0. Relative to an arbitrary (accelerating frame) specified by x a = x a (x 0b ), the latter becomes: d 2 x a d 2 + a dx b dx c

d 2 x 0a d d =0. Relative to an arbitrary (accelerating frame) specified by x a = x a (x 0b ), the latter becomes: d 2 x a d 2 + a dx b dx c Chapte 6 Geneal Relativity 6.1 Towads the Einstein equations Thee ae seveal ways of motivating the Einstein equations. The most natual is pehaps though consideations involving the Equivalence Pinciple.