Summary_ (Исследование некоторых типов дифференциальных уравнений с сильной нелинейностью), страница 2

Sinai seminar of Institute forInformation Transmission Problems, RAS (2016).Personal contribution of the author. The results of the theoretical investigationsincluded in the dissertation are completely carried out by the author personally.Publications. On the topic of the dissertation, 35 main works are published, among them14 articles in peer-reviewed journals indexed in the databases Web of Science and Scopus, 8articles in peer-reviewed journals recommended by the Higher Attestation Commission ofthe Ministry of Education and Science of Russian Federation, and 2 monographs.CONTENTS OF THE WORK1. Some new types of axially symmetric solutions to the Navier-Stokes equations61.1.

Investigation of axially symmetric flows of a homogeneous incompressible viscousfluidConsider the Navier-Stokes equations describing a homogeneous incompressible viscousfluid. They have the form1vvvv1(1) v1 v2 v3  grad p  f  v ,txyzdiv v  0,  const,   const ,(2)where v=v(t,x,y,z) is the vector of velocity, p=p(t,x,y,z) is pressure, v1, v2 , v3 are theprojections of the vector of velocity on orthogonal axes x, y, z, t is time, f=f(t,x,y,z) is theforce acting on unit mass of the considered fluid,  is its density and  is its kinematicviscosity.The Navier-Stokes equations are basic equations of mechanics of fluids and a largenumber of analytical and numerical investigations are devoted to them.

However, because ofsubstantial nonlinearity of these equations, only a small number of classes of their exactsolutions have been found. Our aim is to consider and study some new analytical solutions tothe Navier-Stokes equations in the case of axial symmetry.Further the case in which the force f is potential will be considered.

Then for itspotential  we have the equality(3)f  grad  .In this case, Eqs. (1) and (3) can be represented in the formvvvv v1  v2 v3 grad q  v, q   p /    .txyz(4)It should be noted that the differential equations (2) and (4) describe the vector function vand, instead of the pressure p, the scalar function q. However, when the potential  isknown and the function q is found, the pressure p can be determined from the equalityp    (q  ) .Consider axially symmetric solutions to the Navier-Stokes equations.

Then we will seekthe components v1, v2 , v3 of the vector function v and the function q in the following form:v1  y  x, v2  x  y, v3   ,    (t , r , z ),   (t , r , z ),    (t , r , z ), q  q(t , r , z ), r  x 2  y 2 .(5)Here the function  presents the angular velocities of points of a rotating fluid and thefunctions  and  describe changing its shape.Substituting expressions (5) into Eq. (2), we findrr  2   z  0 ,1Landau L. D., Lifshitz Е.М. Hydrodynamics.

– Moscow: Nauka, 1986.(6)7where r   / r,  z   / z .Let us now substitute formulas (5) into the Navier-Stokes equations (4). Then we come tothe following three nonlinear partial differential equations:t   (r r  2 )   z  ( rr  3 r / r   zz )  0, t   / t ,(7)t   (rr   )   z   2  (rr  3r / r   zz )  qr / r,(8) t  r r   z  ( rr   r / r   zz )  qz .(9)Consider the obtained equations (7)-(9). At first, let us eliminate the function q in them.For this purpose, differentiating Eqs. (8) and (9) with respect to the variables z and r andusing the evident identity qr / z  qz / r and also Eq.

(6), we come to the system ofequations z  r r  2 , 2   t  r r   z  ( rr  3 r / r   zz ),(10)2 z  t  rr   z  ( rr  3 r / r   zz ),    z   r / r.1.2. Description of axially symmetric flows of a viscous fluid in the form of powerseries in the radial coordinateLet us seek solutions to Eqs. (10) in the following form: an (t , z)r 2n ,n 0 bn (t , z)r 2n ,n 0 cn (t , z)r 2n ,(11)n 0where an , bn , cn are some functions of the variables t and z in the region of convergence ofthese power series.Then we come to the following system of recurrence relations:an 1 d n 1 n (n  k  1)a c k1k n  k  (k  1)ak cn an  an  4 (n  1)(n  2) n  k 1k 0,n (n  k  1)(2a a k1k n  k  d k cn  k )  kdk cn d n  d n  4 (n  1)(n  2) n  k 1k 0bn  cncn 1  d n ,, cn1  2(n  1)2(n  1) 2(n  1) (12) , (13)(14)in which the three functions a0 (t , z ) , c0 (t , z ) and d 0 (t , z ) are arbitrary infinitelydifferentiable functions.1.3.

Particular solutions to the Navier-Stokes equations in the form of power seriesconvergent for arbitrary values of variables8Consider three cases in which relations (12)-(14) give solutions to the Navier-Stokesequations in the form of series (11) for any values of the variables t , z and r .1.3.1. A class of solutions of a closed formLet us turn to the case a0  a(t ), c0  g (t )  zh(t ), d0  0 .Then from (12)-(14) we findan 1 an  an (t ), cn 1(t )  0, dn  0, n  0, 1, 2, ... ,(15)an  (n  1)han, n  0, 1, 2, ....

, h  h(t ), an  an (t ).4 (n  1)(n  2)(16)Consider now the case aN 1  0 , where N is some nonnegative integer. Then from (16)we obtainan  0, n  N  1(17)and successively find aN , aN 1, ... , a0 from the recurrence relation t an 1 ( )an  4 (n  1)(n  2) An (t ) d  Cn  , An ( )0tAn (t )  exp (n  1)  h( )d  , (18)0where n  N , N  1, ... , 0 , aN 1  0 and Cn are arbitrary constants.As a result, we obtain the following solutions to Eqs. (10):N an (t )r 2n ,n 0   12 h(t ),   g (t )  zh(t ) ,(19)where an (t ) are determined by formulas (17)-(18), N is an arbitrary nonnegative integer,g (t ) and h(t ) are arbitrary differentiable functions and the expressions for  and  containN+1 arbitrary constants С0 , C1, ..., CN .1.3.2. A class of solutions independent of zConsider the case a0  a(t ), c0  c(t ), d0  d (t ) .

Then from (12)-(14) we findan 1 an  an (t ), cn  cn (t ), dn  dn (t ), n  0 ,(20)andndn., d n 1 , cn 1  4 (n  1)(n  2)4 (n  1)(n  2)2(n  1)(21)Formulas (21) givean a ( n) (t )d ( n) (t )1d ( n) (t ),d,c, n  0 , (22)nn 12 (4 )n (n  1)!2(4 ) n n!(n  1)!(4 )n n!(n  1)!where a(0) (t )  a(t ) .As a result we obtaina( n) (t ) r 2n1  d ( n) (t ) r 2( n 1)  a(t )  ,   0,   c(t )  .n2 n 0 (4 )n (n  1)!2n 1 (4 ) n!(n  1)!(23)9Consider now the particular casem1m1a(t )  A0   Am exp   m t , d (t )  D0   Dm exp   m t  ,whereAm , Dm , m ,marem  0, m  0, t  0constants,(24)andinfiniteconsequences Am и Dm are absolutely summable.Then from formulas (23) we find, using the Bessel functions of the first kindJ 0 ( x) and J1 ( x) :  A0 2 rm 1AmmJ 1 ( m / )r exp   m t ,  c(t )  12 D0 r 2  2 Dmm 1  mJ 0  0, (25)( m / ) r  1 exp   m t  .Consider now a fluid occupying the cylindrical region 0  r  r0 , where r0 is some finiteradius, and let the following boundary conditions be given for it: (t , r0 )  const,  (t , r0 )  0 .(26)It should be noted that the particular case   0 is considered in the book2 .When  (t , r )  0 from (25) and (26) we obtain(m / ) r0  km , m  1, 2, 3, ...(27)where k1, k2 , k3 , ...

is the infinite consequence of the positive zeros of the Bessel functionJ1( x) : 0  k1  k2  k3  ..., J1(km )  0, m  1, 2, 3, ... .Therefore, the function  (t , r ) acquires the form  A0 2r0r k m J1 (k m r / r0 ) exp k m2 t / r02 .m1A(28)mAs is well-known, the Bessel functions J  (k m(  ) s) , where   1 и J  (k m(  ) )  0 ,0  k1(  )  k 2(  )  ...  k m(  )  ... , present an orthogonal system in the interval 0  s  1 and anarbitrary continuous function f (s) , defined in this interval, can be represented in the form ofthe seriesf ( s)   f m(  ) J  (k m(  ) s), 0  s  1,(29)m1wheref m(  ) 12sf ( s) J  (k m(  ) s)ds, m  1, 2, 3, ...

. .2( ) J  1 (k m ) 0(30)That is why the coefficients Am in (28) can be chosen so as to satisfy the initial conditionat t  0 :  (0, r )   0 (r ) , where  0 (r ) is an arbitrary continuous function.2Loiciansky L.G. Mechanics of Liquids and Gases. – Moscow: Nauka, 1978., p. 403.10Analogously, in order to satisfy the initial condition of a general form for the function at t  0 , we should choose numbers  m in the form(m / ) r0  lm , m  1, 2, 3, ... ,(31)where lm are different positive zeros of the Bessel function J 0 ( x) : 0  l1  l2  l3  ...

,J 0 (lm )  0 .Then we findc(t )  12 D0 r02  2r02 Dm2m1 l m  12 D0 (r02  r 2 )  2r02 Dm2m1 l mexp  l m2 t / r02 ,(32)J 0 l m r / r0  exp  l m2 t / r02 .(33)Thus, we obtain a class of solutions to the Navier-Stokes equations satisfying theboundary conditions (26).1.3.3. A class of solutions depending on one function of the arguments t and zConsider the case a0  0, c0  c(t , z ), d0  D  const ,where c(t , z ) is some differentiable function. Then from formulas (12)-(14) we findan  0, d n 1  0, cn  2  cn 11c ,n0,cD1224(n  2)2(34)and therefore,cn  (1) ncz( 2n)4n (n!) 2, n  2,(35)where cz( k )   k c / z k .As a result we obtain the following solution to Eqs.

(10):  0,  ( 2 n 1) 2n( 2n) 2n1 rDr 2 n 1 czn cz r(1),c(1). (36)2 n 024n (n  1)!n!4n (n!) 2n 1Consider the particular casec(t , z )  Cm (t ) sinm (t ) z   m (t ) ,(37)m 1where Сm (t ), m (t ) and  m (t ) are some differentiable functions of t  0 and the infiniteconsequence Cm (t ) is absolutely summable for any t  0 .Then from (36) we obtain, using the modified Bessel functions of the first kindI 0 ( x) and I1 ( x) : 1  Cm (t ) cosm (t ) z   m (t ) I1m (t )r ,r m 1(38)11   Dr 2   Cm (t ) sin m (t ) z   m (t )  I 0 m (t )r .12(39)m 11.4.