CC BY
32
Серия «Математика»
2010. Т. 3, № 1. С. 61-69
Онлайн-доступ к журналу: http://isu.ru/izvestia
УДК 517.946+532.517
Nonlinear diffusion and exact solutions to the Navier-Stokes equations *
V. V. Pukhnachev
Lavrentyev Institute of Hydrodynamics
Abstract. There are considered a number of invariant or partially invariant solutions to the Navier-Stokes equations (NSE) of rank two. These solutions are determined from onedimensional linear or quasi-linear diffusion equations. Explicit solution, which describes smoothing of initial velocity discontinuity in a liquid with initial uniform vorticity, is constructed. This problem is reduced to a linear equation with coefficients depending on time. The global existence and non-existence theorems in the problem of a longitudinal strip deformation with free boundaries are formulated. In this case, the governing quasilinear equation is turned out to be integro-differential one. Third example demonstrates process of axially symmetric spreading of a layer on a solid plane. The corresponding free boundary problem is reduced to the Cauchy problem for the second-order degenerate quasi-linear parabolic equation. It allows us to prove the global-in-time solvability of this problem.
Keywords: linear and nonlinear diffusion, Navier-Stokes equations, free boundary problems, invariant and partially invariant solutions.
describing the motion of a viscous incompressible liquid in the absence of external forces. Here t is time, V and A are gradient and Laplacian in variables x = (x1,x2,x3), respectively; v = (v^v2,v3) is velocity vector, p is pressure. Without loss of generality, the viscosity coefficient and the liquid density are taken to be equal to zero.
From the physical point of view, the first equation (1) describes the diffusive-convective process of momentum transport. The second equation
* Author is grateful to Professor S.I. Shmarev for the fruitful discussion.
1. Group properties of NSE.
Let us consider NSE
vt + v ■ Vv = -Vp + Av, V ■ v = 0,
(1)
(the incompressibility condition) assigns a kinematic constraint, while the pressure gradient characterizes the constraint reaction. The purpose of this communication is to determine situations, in which the above mentioned process is described in terms of solutions of a diffusive-type scalar equation. This is achieved with the help of methods of group analysis of differential equations [1,2].
System (1) admits the infinite-dimensional Lie group Gœ. The basis of corresponding Lie algebra is formed by operators
X0 = d/dt, = ^(t)d/dxi + ,tp(t)d/dvi — xi'ip(t)d/dp (i = 1,2,3),
$ = <p(t)d/dp,
Xjj = Xjd/dxi — xid/dxj + vjd/dvi — vid/dvj (i = 1,2,3; j = 1, 2; j < i),
3
Z = 2td/dt + ^2(xi9/dxi — vid/dvi) — 2pd/dp.
i=1
Here <£, ^i are arbitrary (of C^ class) functions of time; dot denotes
differentiation with respect to t. Group G^ contains the 10-parameter Galilei group generated by operators Xo, Xj, = d/dxi, Yi = td/dxi + d/dvj, Xij = xjd/dxi — xid/dxj + vjd/dvi — vid/dvj where i = 1,2,3; j = 1,2; j < i.
2. Free boundary problems for NSE.
Let us suppose that the flow domain is bounded (partially or entirely) by a free surface rt, which is unknown a priori. It means that the following conditions are fulfilled:
-pn +
Vv + (Vv)T =2aKn, v ■ n = Vn, x € rt, 0 < t < T. (2)
Here n is a unit vector of external normal to rt, Vn is displacement velocity of the surface rt in n direction, a = const > 0 is the surface tension coefficient, K is the mean curvature of rt.
Theorem 1. If the free surface rt is invariant under a subgroup H of Gio then conditions (2) on this surface will be also invariant under H.
This theorem allows us to obtain invariant and partially invariant solutions of NSE, which are compatible in advance with conditions on the free boundary. A number of such solutions are given in [2]. One more example is presented in Section 5.
3. Linear diffusion in viscous flows.
A simple example of exact solution to NSE is the so called parallel flow, v1 = u(y,t), v2 = v3 = 0, p = 0, y = x2. Function u satisfies the linear diffusion equation ut = uyy. We remark that this solution is invariant under the group G {X\, X3) c G10. Let us consider another two parameter subgroup of G10,H = H {X1 + wY2,X3) where w is constant. There is the following solution among the invariant solutions of system (1) with respect to group H:
v\ = q((, t), v2 = ux\ + utq(Z, t), v3 = 0, p = h((, t), ( = x2 — utx\. Function q satisfies the equation
d_
dt
1 + u2t2) q
= ( 1 + u2t2
dZ 2
1 + u2t2) q
(3)
Equation (3) is also a linear diffusion equation but its diffusivity coefficient strongly depends on time.
Equation (3) has the particular solution
q
= V ^1 + u2^2^ erf((r 1/2), t = t ^1 + u2t2/3^
where V = const. The corresponding velocity field is given by formulas
v\ = V ^1 + w2t2^J erf((r 1/2), v2 = wx2 + Vwt ^1 + w2t2^J erf((r 1/2).
In the limit t ^ +0, we have v1 = V as x2 > 0, v1 = —V as x2 < 0,v2 = wx1. Thus, we obtain solution, which describes smoothing of the initial velocity discontinuity on the plane x2 = 0, while the initial vorticity is constant outside of this plane. If w = 0, we arrive to the well-known solution describing the vortex sheet diffusion in a parallel flow.
4. Longitudinal deformation of a strip with free boundaries.
In 2D case, the analog of Th. 1 takes place with replacement of group G\0 by group G6 formed by operators X0,Xk, Yk,X12 (k = 1,2). Let us take the subgroup G (X2, Y2) c G6. There is no invariant solution of system (1) with respect to this subgroup. However, we can look for its partially invariant solution in the form
X
u = — J f (z, t)dz, v = yf (x, t), p = p(x, t) (4)
0
where x = x\, y = x2, u = v\, v = v2.
We note that any curve x = s(t) is an invariant manifold of group G. Therefore, we can utilize the form (4) for constructing NSE solutions, which are conjugated with free boundary conditions on the surface x = s(t), y eR. Substitution of (4) into the system (1) leads to equations
X X
Px = J ftdz - f J fxdz - fx, (5)
0 0
x
ft + f2 - fx J f (z, t)dz = fxx- (6)
0
The problem is to find the positive function s(t) and the solution f (x,t) of the equation (6) in the domain ST = {x,t : 0 < x < s(t), 0 <t <T} satisfying initial and boundary conditions
s(0) = 1, f (x, 0) = fo(x), 0 < x < 1, (7)
S(t)
ds
fx(0, t) = 0, fx(s(t),t) = 0, — = -y f (x, t)dx, 0 <t<T. (8)
0
The solution of problem (5)-(8) describes the symmetric deformation of a viscous strip, both boundaries |x| = s(t) of which are free. First condition (8) together with (4) provides the motion symmetry. Other conditions (8) guarantee the fulfillment of two out of three scalar conditions (2). To satisfy the third condition (2), we use the arbitrariness in the determination of function p from equation (5). We note also that the assumption s(0) = 1 does not restrict generality because of the scaling transform admitted by relations (6), (8). Below f0 denotes the mean value of function f0(x) over interval [0, 1], prime denotes differentiation of f0 with respect to x.
Proposition 1. [3] Let us assume that f0 e C2+a [0,1], 0 < a < 1; f0(0) = f'(1) = 0. If f0(x) > 5 > 0 for x e [0,1] or f0 > 0 and
2 0
J [fo(x) - f0]2 dx < min^9^2 > tt) >
then problem (6) - (8) has a unique solution f € C‘2+a,l+a/‘2 (St),s G G C 2+a/‘2[0,T ] for an arbitrary T > 0. If f0 < 0, then the lifespan of problem (6)-(8) solution is less than or equal to -l/f0.
The structure of blowing up solution to problem (6)-(8) is investigated in [4]. If the initial function f0(x) monotonously decreasing and satisfies certain additional conditions (including a “steepness condition”), then function f has asymptotics [4]
f (x,t)-------(t* — t)
-1cos2
x\JY(t* — t)
as t t*
for 0 < x < s(t) and s(t) ~ 7(t* — t) where 7 = 7(/o) = const > 0. In
a "favourable"case /0(x) > 5 > 0, a universal asymptotics is valid [3]
f (x,t) = t-1 1 + Olt-1')
as t
uniformly in x € [0, s(t)], moreover, s(t) = Ct-1[1 + O(t )] with some positive constant C = C(f0). Both possibilities of an evolution of the solution to general problem (6)-(8) are well demonstrated on the example of exact solutions to the equation (6)
f (x,t) = l(t) + m(t) cos[nnx/s(t)]
where n is natural and I, m, s satisfy a certain dynamical system [3].
Effective investigation of the free boundary problem (6)-(8) is reached by transition to the Lagrangian coordinate £ instead of x, which is introduced
X
by relations xt(£,t) = — / f (z,t)dz for t > 0, x(£, 0) = £. Function
0
f [x(£,t),t] = A(£,t) is a solution of the second initial boundary value problem for the equation
OX
~dt
exp
X(£,t)dr
d_
exp
X(£,r)dr
%— X’
(9)
in a fixed domain nT = {(,t : 0 < £ < 1, 0 <t<T}. Equation (9) is similar to a nonlinear diffusion equation with the source term, but the diffusivity coefficient is not constant now; moreover, it depends on prehistory of the process.
5. Axially symmetric spreading of a layer on a solid plane.
In this section we consider a class of solutions of system (1), in which
z
vr = rg(z,t), vg = 0, vz = —:2 Jg(x,t)dx, p = p(z,t) (10)
where function g satisfies the equation
t
t
z
gt + g2 - 2gz g(x, t)dx = gzz.
I
0
Here vr,v$,vz are projections of the velocity vector on axes r,9,z of a cylindrical system of coordinates, respectively.
Solution (10) belongs to a wide class of partially invariant solutions of system (1) with respect to subgroup G5 C Gi0 with generators Xi, X2, Yi,
Y2, Xi2. Any plane z = s(t) is an invariant manifold of G5. This allows us to rewrite free boundary conditions (2) in terms of invariant functions g,s as follows
we complete formulation of the free boundary problem: to find the positive function s(t) and the solution of equation (11) in domain ST = {z,t : 0 < z < s(t), 0 <t<T} satisfying conditions (12)-(14). In view of (10), (13), the plane z = 0 can be taken as a solid plane since the no slip condition is fulfilled on this plane.
Further we suppose that function g0 satisfies the smoothness and compatibility conditions,
go(z) € C3+a[0,1], go(0)= g0(0) = 0, g!0(1) = g0"(1) = 0. (15)
The global solvability of problem (11)-(14) takes place if g0 is monotonously increasing,
Under conditions (15), (16), functions g and gz are strictly positive according to the strong maximum principle [5]. This gives a possibility to reformulate problem (11)-(14) introducing the new space variable and unknown function x so that
0 <t<T.
(12)
0
Besides, we demand that
g(0, t) =0, 0 < t < T.
(13)
Posing the initial condition
g(z, 0) = go(z), 0 < x < 1
(14)
g0(z) > 0 as 0 < z < 1.
(16)
P = g(z,t), x(P,t)= gl-
(17)
In consequence of (11), (17), function x is the solution of the following equation:
which is similar to nonlinear diffusion equation.
The reduction of integro-differential equation (11) to a second-order differential equation (18) has a group-theoretic origin. In fact, equation (11) is equivalent to the system
which admits the infinite Lie group with operator ^ = 0(t)d/dz+0(t)d/dw (0 is an arbitrary smooth function of t). It turns out that transform (17) realizes the so called group stratification [1] of this system on the basis of the above mentioned infinite group.
Let us denote Po = go(1) and define xo(P) for P € [0,Po] by relations P = go(z), Xo(P) = [go(z)]2; then let xo = 0 for P > Po. Joining to (18) conditions
x(P, 0) = Xo(P) as P > 0, xp(0,t) = 0 as 0 < t < T (19)
(second of them is the corollary of (11), (13)), we arrive to the initial boundary value problem for a degenerate parabolic equation in a semistrip = {P,t : P > 0, 0 <t< T}. According to (12), (17), the free boundary in plane z, t corresponds to the line of degeneration of equation (18) (or the interface) in plane P,t. Dual setting (18), (19) of free boundary problem (11)-(14) will be used for obtaining an a priori estimate of its solution.
Proposition 2. Under conditions (15), (16), problem (11)-(14) has a unique solution g € C3Ta,s/2Ta/2(ST), s € Cb/2+a/2[0,T] for any T > 0.
Proof. To prove the solvability of problem (11)-(14) for small T > 0, we pass to Lagrangian coordinates and apply methods developed in [5] to initial boundary value problem in a fixed domain. A local behavior of x(P, t) near the interface P = n(t) is governed by the corresponding properties of the function g(z,t). If g is the solution of problem (11)-(14) and |g(z,t)| < M < to in ST then , gzt, gzzz are Holder-continuous in ST and gzzz(s(t),t) = 0. Exploiting these properties of function g and relations (17), we derive that functions x, Xp are continuous in the image QT of the domain ST under the map (17); besides, xl/2xpp ^ 0 as P ^ n(t) — 0, t € [0,T]. That makes possible to get the identity
(18)
2
gt + wgz + g = gzz, wz = -2g,
0
0
for solution of (18), (19). In turn, this identity together with inequalities g > 0, gz > 0 in ST, s < 1 if t € [0,T] leads to an estimate g(z,t) < \\g'o\\L3(o i) in ST for an arbitrary T > 0. This allows us to prove the statement of Proposition 2 for any T > 0.
Let us assume now that function go satisfies conditions (15) and g'o(z) < 0 for 0 < x < 1. In this case, passing to new variables P = g(z,t), gz = — [x(P, t)]1/2 results to the problem (19) for equation (18) in the domain J2— = {P,t : P < 0, 0 <t <T} where the first condition is replaced by x(P, 0) = xo(P) as P < 0. Arising problem has a unique solution at least for small T > 0. Its important property is presence of the full set of self-similar solutions, x = (t + c)-3w(^), ^ = P(t + c) where c = const > 0 and w(^) is compactly supported, w(0) = k > 0. (We note that problem (18), (19) has no nontrivial self-similar solutions). In view of results of [6], we can suppose that the new problem for equation (18) has also a global in time solution.
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References
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4. Galaktionov V. A. Blow-up for a class of solutions with free boundary for the Navier-Stokes equations / V. A. Galaktionov, J. L. Vazquez // Adv. Diff. Eq. -1999. - T. 4 - P. 297-321.
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В. В. Пухначев
Нелинейная диффузия и точные решения уравнений Навье-Стокса
Аннотация. Рассматриваются примеры инвариантных или частично инвариантных решений уравнений Навье-Стокса ранга два. Эти решения определяются из одномерных линейных или квазилинейных уравнений диффузии. Построено точное решение, описывающее сглаживание начального разрыва поля скоростей в жидкости, которая в начальный момент имеет равномерную завихренность. Эта задача
сводится к линейному уравнению диффузии с коэффициентами, зависящими от времени. Сформулированы теоремы существования и несуществования в целом по времени решения задачи о продольной деформации полосы со свободными границами. В этом случае основное квазилинейное уравнение диффузии оказывается интегро-дифференциальным. Третье решение описывает осесимметричный процесс растекания жидкого слоя на твердой плоскости. Здесь соответствующая задача со свободной границей редуцируется к задаче Коши для квазилинейного вырождающегося параболического уравнения второго порядка. Это позволяет доказать ее глобальную разрешимость.
Ключевые слова: линейная и нелинейная диффузия, уравнения Навье-Стокса, задачи со свободной границей, инвариантные и частично инвариантные решения.
Пухначев Владислав Васильевич, доктор физико-математических наук, профессор, член-корр. РАН. Институт гидродинамики им. М.А.Лав-рентьева СО РАН, пр-т Лаврентьева, 15, Новосибирск, Россия, 630090, тел./факс (383) 333-16-12 ([email protected])
Vladislav Pukhnachev, professor, Lavrentyev Institute of Hydrodynamics, 15, Lavrentyev St., Novosibirsk, Russia, 630090, Phone: (383)333-16-12 ([email protected])
