Asymptotic solutions of a parabolic equation near singular points of a and b types Текст научной статьи по специальности «Математика»

URAL MATHEMATICAL JOURNAL, Vol. 5, No. 1, 2019, pp. 101-108

DOI: 10.15826/umj.2019.1.010

ASYMPTOTIC SOLUTIONS OF A PARABOLIC EQUATION NEAR SINGULAR POINTS OF A AND B TYPES

Sergey V. Zakharov

Krasovskii Institute of Mathematics and Mechanics,

Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya str., Ekaterinburg, Russia, 620990 svz@imm.uran.ru

Abstract: The Cauchy problem for a quasi-linear parabolic equation with a small parameter multiplying a higher derivative is considered in two cases when the solution of the limit problem has a point of gradient catastrophe. Asymptotic solutions are found by using the Cole—Hopf transform. The integrals determining the asymptotic solutions correspond to the Lagrange singularities of type A and the boundary singularities of type B. The behavior of the asymptotic solutions is described in terms of the weighted Sobolev spaces.

Keywords: Quasi-linear parabolic equation, Cole—Hopf transform, Singular points, Asymptotic solutions, Whitney fold singularity, Il'in's universal solution, Weighted Sobolev spaces.

Introduction

Consider the Cauchy problem for a quasi-linear parabolic equation

du d<p(u) d2u dt dx dx2' u(x, t0) = q(x), x € R

with a small parameter e > 0 in such cases that the solution of the limit problem (for e = 0) has a point of gradient catastrophe. The interest of studying the behavior of solutions near singular points is explained, in particular, by the fact that such singular events themselves occupy very small time, however, in many respects determining all subsequent behavior of the system. The asymptotic behavior of solutions in neighborhoods of singular points is directly connected with constructing an approximation in neighborhoods of shock waves in physical media with a small nonzero viscosity.

Although the types of singular points of solutions are classified in detail [1, ch. 2] and processes of the shock waves formation are studied [2], constructing asymptotic series in the small viscosity parameter e for an equation of the general form (0.1) is a separate problem in every specific case. First substantial results for several types of singularities, including the Whitney fold singularity A3, were obtained by A.M. Il'in [3, ch. VI], who constructed a complete asymptotics and derived a universal solution near the fold singularity; see formulas (1.1)-(1.2) in the next section.

In the present paper, the statements about asymptotic solutions of the preliminary notice [4] for Lagrange singularities of types A2n+1 and boundary singularities of types B2n+1 with any n ^ 2 are proved.

1. Fold singularity

In Il'in's pioneer paper [5], problem (0.1)-(0.2) was first studied in the case when in the strip {t0 ^ t ^ T, x € R} the solution of the limit problem is a function which is smooth everywhere

(0.1) (0.2)

except for one smooth line of jump discontinuity x = s(t), t ^ t*. It is supposed that ^ € C0 ^>''(u) > 0, ^>(0) = ^>'(0) = 0, and to = -1. For an appropriate choice of the bounded initial function q € C°(R), the singular point (s(t*),t*) coincides with the origin and in its neighborhood the stretched variables £ = e—3/4x and t = e—1/2t are introduced. An asymptotic expansion of the solution was obtained in the form of the series

o k—1

u(x, t, e) = ^ ek/4 ^ wkj(£, t) lnj e1/4, e ^ +0. (1.1)

k=1 j=0

Observe properties of the leading term of the expansion e1/4w1;0(£,T). The coefficient w1)0(£,T) is found using the Cole-Hopf transform

2 dA(£,T )

wi,q(c,t ) = -

where

p"(0)A(£,r ) d£ 1

A(£, r) = j exp ( - ^(z4 - 2z2t + 4^)) dz.

(1.2)

The argument of the exponent is a generating family of the Lagrange singularity A3, see [1, 6].

Theorem 1. The function w1)0 satisfies the asymptotic relations

0

w1,0(£, T) = b''(0)] —1H(£, T) + ^ h1—41 (£, T), 3[H(£, T)]2 - t

1=1

(£,t) € Q1 = r2 \{|£| <ty1—1/2, t > 0, 0 <71 < 2}, where H(£,t) is the Whitney fold function, H3 — tH + £ = 0, r) are homogeneous functions of power 1 — 4I, relative to r), (r)1/2

and v/3[ii(C,T)]2 — r, which are polynomials in H(£,t), t and (3[ii(£, r)]2 — r)

k=1

(£,t) € i22 = {ICI1"1^2 < r72> 7i < 72 < 2}, where z = ¿¡ n/t/2, and the coefficients of the series for k ^ 1 satisfy the estimates |qk(z)| ^ Mk(1 + |z|k).

0 C

Q1

Fig. 1. Domains Q and Q2.

—o

oo

Proof of the theorem is based on the calculation of the asymptotics of the integral A(£, t) by Laplace's method. In the domain Q1, see Figure 1, the essential contribution into the asymptotics

is given by one local maximum and in the domain Q2 by two local maxima; see Theorem 4.1 and Lemma 6.1 in [3, ch. VI]. □

Il'in investigated the problem under condition that

f'(q(x)) = —x + x3 + O(x4), x — 0.

Now consider the following condition

f'(q(x)) = —x + x2n+1 + O(x2n+2), x - 0. (1.3)

Let us clear out scales of the inner variables, which are introduced using the change

x = ^, t = u(x,t,e) = eKu*(n,0,e). (1.4)

Since all terms in equation (0.1) should be of the same order, we obtain the relations

—^ = k — a = 1 — 2a. (1.5)

From the characteristic equation x = y + (t + 1)f'(q(y)) with y € R being an arbitrary parameter and condition (1.3) we have x « —yt — y2n+1, whence by change (1.4) and the relation u ~ —y, we obtain a = k + ^ = (2n + 1)k. From these equalities and relations (1.5) we find

'2n +1 n 1 n

(7 = -, 11 = -, X = -. (1-6)

2n + 2 ' ^ n + 1 ' 2n + 2

Since u = equation (0.1) becomes the Burgers equation, whose solution can be

written in the form of the Cole-Hopf transform. Moreover, the coefficient at s2n+2 is determined from the condition of matching the inner asymptotics and the outer expansion

U?+1-0Uo + T, = 0. f ' (0) 0

To give some description of the behavior of asymptotic solutions, we will use the weighted Sobolev spaces Wpq(R) with the norm

/ f Cf U(X) p \ VP

«llw^(K) = E ( J (! + M)~* dx) (1J)

1=0

and also the proposed in [7] approach to the definition of an asymptotic solution of an evolutionary differential equation.

Theorem 2. In the domain Qe = {(x,t) : |xe-K| + |t| < K > 0} for any natural n ^ 2

the function

uin(x,t,e) = -2s&"(0)V(x,t,s)]-ldV{^£), (1.8)

where

V tM,e)=/exp( -

22n s2n+2 ts2 xs\

+ --t (1-9)

n + 1 eV e V

2n + 1 n 1

a =-, a =-, K = ,

2n + 2 n + 1 2n + 2

is an asymptotic solution of equation (0.1) in the following sense: duin/di + d^(uin)/dx — ed2-uin/dx2

sup {|duin/dt| + |cd^>(uin )/dx| + |ed2uin/dx2|}

(x,i)en£

2 an =°(£X)> e +0- (1-10)

For (m + 1)p < (2n + 1)(q — 1) and any fixed t, uin(-, t, e) € Wpq(R) and ||uin(-, t, e)||Wm (r) ~ e

™(R) and ||uin(-,i,£)||w™ f "

as e ^ +0.

Proof. Formula (1.9) implies that the function V(x,t, e) satisfies the heat equation

dV _ _d2V_ dt dx2'

it follows that function (1.8) is the exact solution of the Burgers equation:

dt in dx " dx'2 '

since uin has the form of the Cole-Hopf transform (it can also be established by direct substitition). Then using Taylor's formula (u) = (0)u+O(u2), u ^ 0, we easily obtain the following estimate:

duin , d^(u

in) d uin // \ duin Z//o\ duin ( 2 duin\

~df + " ^ = * " * = ° ) ■ (L11)

It is also elementary checked that

2 ^ 4£2-2a (Vj/}2 du.D ^ 2£l-2a / y^ ^ du^ = f V^ V^Vg

u'm b"(o)]2 v2 ' dx <p"(o) v^ /' <p"(o) V^ f2

x t

in terms of the inner variables r? = — and 0 = —, where the functional factors of the powers of e

1 ea e^

have a finite order in the domain Qe, since

|xe-K | + |t| = |neCT-K | + |0e^| = + |0|).

Then estimate (1.11) becomes

dUin | dipjUin) jj2Uin =

dt dx " cte2 ' '

and, taking into account that 1 — ^ — a = 2 — 3a by (1.6), we find the exact order e2-3°" of derivatives entering into the denominator in formula (1.10). Thus, we obtain the necessary estimate in the right-hand side of (1.10), since 1 — a = k by (1.6).

Further, differentiating (1.8) with respect to x and passing to the inner variable n, we derive the relation

Qlu.n £ gmoy gmiy ^ £l-(i+l)<r gmoy grmy

~ yl+l «mo.-mi ■ ■ ■ J}- yl+1 Z^ «mo.-mi • • •

mo+-----+mj=Z+1 mo+-----+mj=Z+1

with some constant coefficients amo...mi. Using Laplace's method for the integral V(x, t, e), we write the equation

4ra(5(r?, e))2n+l - es(r], e) + ^ = o,

for critical points by equating the derivative of the argument of the exponent in (1.9) to zero and taking into account the change of variables (1.4). For 0 = const, as n ^ —to we obviously obtain a unique critical point s = S(n, 0) > 0, while as n ^ we obtain a unique critical point s = S(n, 0) < 0. Then according to the standard formula [8, ch. II, sect. 2.4], we find the following asymptotics:

Since in the domain Q£ the inner variables n, 0 are finite and the function V (together with its derivatives) does not depend on e in the leading order, by the definition (1.7) of the norm in the weighted Sobolev spaces Wpq(R), the above relations imply the asymptotic equivalence ||uin(-,t, e)||wpq(R) ~ e-m as e ^ +0 for any p, q, and m satisfying the inequality (m + 1)p < (2n + 1)(q — 1). Theorem 2 is proved. □

2. Transition of a weak discontinuity into a strong one

In papers [9, 10] the solution of problem (0.1)-(0.2) is studied in the case when the initial function is smooth everywhere except for one point, at which it is continuous and has a jump discontinuity of the first derivative:

q(x) = —(x + ax2) ©(—x) (1 + gi(x)), t0 = —1,

where a > 0, g1 € Crc(R), g1(x) = 0 in some neighborhood of zero, 0 denotes the Heaviside function. Then (q(x)) = —(x + bx2) 0(—x) (1 + g2(x)), where b = a — ^>/"(0)/2 > 0, and such a weak discontinuity in the limit problem propagates a finite time along a characteristic, and then turns into a shock wave.

In a neighborhood of the singular point (x = 0, t = 0) we introduce the stretched inner variables

C = e-2/3x, t = e-1/3t.

The asymptotics of the solution has the form of the series

rc [p/2] — 1

u(x,t,e) = J^ ep/6 ^

Wp,s(C,T) lns e, e ^ +0.

p=2 s=0

The leading term of the expansion e1/3w2;0({,T) is found using the Cole-Hopf transform (under the assumption that '(0) = 1)

, 2 d $(£,t )

t) = j exp ( - Js3 + rs2 - ds. (2.1)

0

The argument of the exponent corresponds to a versal deformation for the boundary singularity B3, whose general form is s3 + A1s2 + A2s + A3, see [11], however, the factor eAs does not play a role in this case due to the form of the Cole-Hopf transform.

To describe the behavior of the functions wP)S(£, t) as £2 + t2 ^ to, in the plane of independent variables the following overlapping domains are introduced:

X0 = {(c,t ): |C| < |t |1—y , t< 0}, Xs = {(c,t ): |C — 3t 2/16b| <t 2—v , t> 0}.

X- = {(£,T) : e < 0, -|£|2-a < T} u {(£,r) : t > 0, 166£ < 3t2 - Ta-1},

x + = {(e, t): e > o, -e2-a < t < 0} u {(e, t): t > 0, i6be > 3t2+Ta-1},

where 0 < 7 < 1/2, parameters a and v are chosen so that 0 < v < a < (1 — 27 )/(1 — 7) < 1. Then the union X + u X0 u X- u Xs is a neighborhood of infinity in R2, see Figure 2.

Note that the domain Xs contains the parabola 16be — 3t2 = 0, where the inner local maximum of the integrand in the right-hand side of (2.1) equals the boundary maximum. In each of these four domains the functions wp,s(e,T) have specific asymptotic behavior; corresponding series for the leading term w2,0(e,T) are given: in the domain X0 by [10, Theorem 1], in the domain X- by [10, Theorem 3], in the domain Xs by [10, Theorem 4], in the domain X + by [10, Theorem 5].

Now, let us consider the following case, where the initial function is such that ^'(q(x)) = —(x + bx2n) ©(—x) (1 + g2n(x)), n ^ 1.

From the characteristic equation x = y + (t + 1)^>'(q(y)) with y being a negative parameter and the relation u ~ —y we have x « —yt — by2n. Making a new change of variables (1.4), we obtain the equalities a = k + ^ = 2nx. Taking into account relations (1.5), we find

2n

a=

2n + 1'

^ =

2n - 1

2 n + 1:

1

k =

2n + 1

(2.2)

X-

e

/ x°

Fig. 2. Domains X+, X0, X-, and Xs. By analogy with Theorem 2 we obtain the following result.

Theorem 3. In the domain Q's = {(x,t) : |xe-K| + |t| < Ke^, K > 0} the function

uux,t,£) = -2 e[W(x,t,e)]-ldW{X,t,S)

dx

+^0

W (x,t,e)=/exp( —

2n + 1 e^ ea

any natural n ^ 2

(2.3)

(2.4)

where numbers a, and k are defined by formulas (2.2), is an asymptotic solution of equation (0.1) in the following sense:

-dnjdt + d^/dx-ecfu /dx> = Q {£X) £ ^ +Q_ (2 5)

sup ||duas/dt| + |d^(uas)/dx| + |ed uas/dx | }

(x,t)ews

For (m + 1)p < 2n(q — 1) and any fixed t, uas(-,t,e) € Wpq(R) and ||uas(-,t, e)||Wm (R) ~ e-m as e —> +0.

Proof. Formula (2.4) implies that the function W(x,t,e) satisfies the heat equation

dW d 2W - = £--— •

dt dx2 '

it follows that function (2.3) is the exact solution of the Burgers equation:

dUas g-Uas _ ,92ttas

M. "T" Uas M — t M 9 •

dt dx dx2

Then, using the relation (u) = u + O(u2), u ^ 0, we easily obtain the following estimate:

d-Q-as + dtfijUas) _&2u, dt dx

It is also elementary checked that

d2 Uas ', AdUas dUas ~ / 2 dUas \

u2 = 4.-2-2(7 W;)2 9uas = 2-1"20" (^;)2N\ dUas = _^ 1—CT

W2 ' dx V W W2 J' dt V W W2

in terms of the inner variable n = x/eCT and 0 = t/eM, where the functional factors of the powers of e have a finite order in the domain Q^, since |xe-K| + |t| = |neCT-K| + |0eM| = eM(|n| + |0|). Then, estimate (2.6) becomes

dtias + dyjUzs) _ £&U1s = Q, 3-4,7) _ ^ +Q dt dx " cte2 ' '

and, taking into account that 1 — ^ — a = 2 — 3a by (2.2), we find the exact order e2-3°" of derivatives entering into the denominator in formula (2.5). Thus, we obtain the necessary estimate in the right-hand side of (2.5), since 1 — a = k by (2.2).

The estimate of the weighted norm 11uas(-,t,e)||Wmq(R) ~ e-m as e ^ +0 is obtained almost exactly in the same way as in the proof of Theorem 2. □

In conclusion, it should be said that by using a priori estimates of solutions of the parabolic equation (0.1) one can prove that the asymptotic solutions defined by formulas (1.8), (1.9) and (2.3), (2.4) are close to some exact solutions; however, such a proof needs uniform formal asymptotic solutions of equation (0.1) in a whole strip (t0 ^ t ^ T, x € R} containing a singular point to be constructed.

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