EXISTENCE OF GLOBAL CLASSICAL SOLUTIONS FOR THE SAINT-VENANT EQUATIONS Текст научной статьи по специальности «Математика»

Vladikavkaz Mathematical Journal 2022, Volume 24, Issue 3, P. 21-36

YAK 517.95

DOI 10.46698/x4972-4013-9236-n

EXISTENCE OF GLOBAL CLASSICAL SOLUTIONS FOR THE SAINT-VENANT EQUATIONS

R. Azib1, S. Georgiev2, A. Kheloufi1 and K. Mebarki1

1 Bejaia University, Bejaia 06000, Algeria;

2 Sofia University "St. Kliment Ohridski",

15 Tzar Osvoboditel Blvd., Sofia 1504, Bulgaria

E-mail: [email protected], [email protected], [email protected], [email protected], [email protected], [email protected], [email protected]

2

1

Abstract. Nowadays, investigations of the existence of global classical solutions for non linear evolution equations is a topic of active mathematical research. In this article, we are concerned with a classical system of shallow water equations which describes long surface waves in a fluid of variable depth. This system was proposed in 1871 by Adhemar Jean-Claude Barre de Saint-Venant. Namely, we investigate an initial value problem for the one dimensional Saint-Venant equations. We are especially interested in question of what sufficient conditions the initial data and the topography of the bottom must verify in order that the considered system has global classical solutions. In order to prove our main results we use a new topological approach based on the fixed point abstract theory of the sum of two operators in Banach spaces. This basic and new idea yields global existence theorems for many of the interesting equations of mathematical physics.

Key words: Saint-Venant equations, classical solution, fixed point, initial value problem. AMS Subject Classification: 35Q35, 35A09, 35E15.

For citation: Azib, R., Georgiev, S., Kheloufi, A. and Mebarki, K. Existence of Global Classical Solutions for the Saint-Venant Equations, Vladikavkaz Math. J., 2022, vol. 24, no. 3, pp. 21-36. DOI: 10.46698/x4972-4013-9236-n.

One of the most important works in the study of nonlinear science is the investigation of the existence of global classical solutions. In this paper, we investigate the Cauchy problem for a classical system of shallow water equations which describes long surface waves in a fluid of variable depth. This system was proposed in 1871 by Adhemar Jean-Claude Barre de Saint-Venant [1]. Namely, we consider the following initial value problem for the Saint-Venant equations:

1. Introduction

dtu + dx(uv) = 0, t € (0, œ), x € R,

(1.1)

u(0, x) = uo(x), x € R, v(0,x) = v0(x), x € R,

© 2022 Azib, R., Georgiev, S., Kheloufi, A. and Mebarki, K.

where k € R+ represents the gravitational constant, the initial conditions u0,v0 and the topography of the bottom f are given functions. Here the unknowns u = u(t, x) and v = v(t, x) denote respectively, the depth and the average horizontal velocity of the fluid.

There are many papers devoted to the study of shallow water flows especially by numerical methods, we can cite [2-7] and the references therein. Let us also mention some works related to the mathematical analysis of Saint-Venant equations. The correct formulation of the boundary conditions for the Saint-Venant system for simulating closed and open basins is investigated in [8]. Local and global temporal existence of classical solutions for the dissipative shallow water equations is considered in [9] and [10], respectively. In [11], the uniqueness of the classical solution of the system of 1-D Saint-Venant equations is proved. In [12], an initial-boundary value problem for a system of Saint-Venant equations is solved by two methods. In [13], Li et al. proved the local-in-time well-posedness of classical solutions with finite total mass and/or energy, allowing the surface height goes to zero in the far field. In [14] and [15], Cheng et al. proved long-time and global existence and asymptotic behavior of classical solutions for two dimensional rotating shallow water system. In [16] classical solutions for the Cauchy problem of the rotating shallow water equations with physical viscosity are obtained. In [17], local-time well-posedness and breakdown for solutions of regularized Saint-Venant equations are established. In [18], global solutions to Saint-Venant equations are investigated by a method of an additional argument. In [19], under regular initial data with small energy but possibly large oscillations, the global well-posedness of classical solution for Cauchy problem of two-dimensional chemotaxis-shallow water system is studied. In the note [20], a mathematical analysis, based on the theory of semigroups of operators on Hilbert space, of a linearized problem involving the Saint-Venant equations is given.

In this paper, we are especially interested in question of what conditions the initial data uo, vo and the topography of the bottom f must verify in order that Problem (1.1) has classical solutions. Here, by a classical solution to the Saint-Venant equations we mean a solution which is along with its derivatives that appear in the equations of class C([0, ro) x R). In other words, (u, v) belongs to the space C:([0, ro) x R) x C1 ([0, ro) x R) of continuously differentiable functions on [0, ro) x R. The main assumptions on the functions u0, v0 and f are

(H1) u0,v0 € C 1(R), 0 < u0 ^ B, 0 ^ v0 ^ B on R for some positive constant B.

(H2) f € C([0, ro), C 1(R)), 0 < |dxf | < B on [0, ro) x R.

Our approach is based on the use of the fixed point theory for the sum of operators in Banach spaces.

The paper is organized as follows. In the next section, we describe a new topological approach which uses fixed point abstract theory of the sum of two operators. In Section 3 we give some properties of solutions of Problem (1.1). These properties concern integral representation of the solutions and some energy estimates to be made more precise later on. In Section 4 we prove existence and multiplicity of solutions for the Saint-Venant system (1.1). Finally, in Section 5 an example illustrating our main results will be given.

2. On Fixed Points for the Sum of Two Operators

In this section, some definitions and results related to fixed point theory will be given. We will recall two approaches for the existence of fixed points for the sum of two operators.

2.1. First approach.

Theorem 2.1. Let E be a Banach space and

Ei = {x € E : ||x|| < R},

with R > 0. Consider two operators T and S, where

Tx = —ex, x € Ei,

with e > 0 and S : E1 ^ E be continuous and such that

(i) (/ — S)(E1) resides in a compact subset of E and

(ii) {x £ E : x = \(I - S)x, \\x\\ = R} = 0, for any A € (0, . Then there exists x* € Ex such that

Tx + Sx — x .

Theorem 2.1 will be used to prove Theorem 4.1 and its proof can be found in [21]. Let us recall the proof of Theorem 2.1 for convenience. < Define

/ 1 \ _ i~\x, if Hxll ^ Re, \ W=lft> if ||x|| > Re.

Then r (—— S)) : E\ —> E\ is continuous and compact. Hence and the Schauder fixed point theorem, it follows that there exists x * € E1 so that

r -S)x*^ =x*.

Assume that -\{I - S)x* (£ Ex. Then

R 1

and

R1

and hence, ||x*|| = R. This contradicts the condition (ii). Therefore —\{I — S)x* € E\ and

x* = r - S)x*^j = - S)x*

or

* | C1 * *

e^c I tS ^^ — x , Tx + Sx — x .

This completes the proof. >

2.2. Second approach. Let E be a real Banach space. Definition 2.1. A closed, convex set P in E is said to be cone if

1) ax € P for any a ^ 0 and for any x € P,

2) x, —x € P implies x — 0.

Definition 2.2. A mapping K : E ^ E is said to be completely continuous if it is continuous and maps bounded sets into relatively compact sets.

Definition 2.3. Let X and Y be real Banach spaces. A mapping K : X ^ Y is said to be expansive if there exists a constant h > 1 such that

||Kx — Ky||y ^ h||x — y||x

for any x, y € X.

or

The following result (its proof relies on [22, Proposition 2.16]) will be used to prove Theorem 4.2.

Theorem 2.2. Let S^ be a cone of a Banach space E; Q a subset of and U\, U2 and Us three open bounded subsets of such that IJ\ C U2 C Us and 0 € U\. Assume that T : Q —>• is an expansive mapping, S : Us —>■ E is a completely continuous and S(U3) C (/ —T)(Q). Suppose that (U2 \ U1) n Q / 0, (Us \ U2) nO/0, and there exists w0 € P>\{0} such that the following conditions hold:

(i) Sx = (I - T)(x - Awo), for all A > 0 and x € dUi n (Q + Awo);

(ii) there exists e > 0 such that Sx = (I — T)(Ax), for all A ^ 1+ e, x € dU2 and Ax € Q;

(iii) Sx = (I — T)(x — Aw0), for all A > 0 and x € dU3 n (Q + Aw0). Then T + S has at least two non-zero fixed points x1, x2 € P such that

£1 € dU2 n Q and x2 € (Z73 \ U2) n Q

or

œi € (U2 \ U\) n Q and £2 € (f/3 \ U2) n Q.

3. Integral Representation and a Priori Estimates for Solutions of Problem (1.1)

Let X = X1 x Xwhere X1 = C 1([0, to) x R). For (u, v) € X, define the operators Sj, Sf and S1 as follows.

x t

S^(u, v)(t,x) = y'(u(i,x1) — uo(x1)) dx1 + |u(t1,x)v(t1 ,x) dt1, (t,x) € [0, to) x R,

Sf(-u, v)(t,x) = J (u(t, xi)v(i, xi) — uo(xi)vo(xi)) dxi 0

t

+ J (u(ti,x)(v(ti,x))2+ ^k(u(ti,x))2^J dti o

t x

+ k / / u(ti,xi) dxf (ti,xi) dxi dti, (t,x) € [0, to) x R,

x

oo

Si(u,v)(t,x) = (S1 (u,v)(t,x),S?(u,v)(t,x)) , (t,x) € [0, to) x R.

Lemma 3.1. Suppose that (H1) and (H2) are satisfied. If (u, v) € X satisfies the equation

Si(u,v)(t,x) =0, (t,x) € [0, to) x R, (3.1)

then (u, v) is a solution of the IVP (1.1).

< Let (u, v) € X be a solution to the equation (3.1). Then

Sl(u, v)(t, x) = 0, S2(u, v)(t, x) = 0, (t, x) € [0, to) x R. (3.2)

x

We differentiate the first equation of (3.2) with respect to t and x and we find

dtu(t, x) + dx(uv)(t, x) = 0, (t, x) € [0, to) x r. We put t = 0 in the first equation of (3.2) and we arrive at

x

0

which we differentiate with respect to x and we find

u(0,x) = u0(x), x € R. Now, we differentiate the second equation of (3.2) with respect to t and x and we find

dt(uv)(t, x) + dx (v,v2 + (t, x) + ku(t, x)dxf(t, x) = 0, (t, x) € [0, oo) x R.

We put t = 0 in the second equation of (3.2) and we get

x

/(««>,*,MMO-„^.M,(xl))= o, x€R,

0

which we differentiate with respect to x and we obtain

u(0,x)v(0,x) — u0(x)v0(x) = 0, x € R,

whereupon

v(0,x) = v0(x), x € R. Thus, (u, v) is a solution to the IVP (1.1). This completes the proof. >

Lemma 3.2. Suppose that (H1) and (H2) are satisfied. Let h € C([0, to) x R) be a positive function almost everywhere on [0, to) x R. If (u, v) € X satisfies the following integral equations:

t x

J J(t — ti)2(x — x,)2h(ti, x,)S,(u, v)(t,,x.) dx, dt, =0, (t,x) € [0, to) x R, 00

and

t x

J J(t — ti)2(x — x.)2h(ti,xi)S2(u,v)(ti,x.) dx, dt, = 0, (t,x) € [0, to) x R, 00

then (u, v) is a solution to the IVP (1.1).

< We differentiate three times with respect to t and three times with respect to x the integral equations of Lemma 3.2 and we find

h(t,x)Si(u,v)(t,x) = 0, (t, x) € [0, to) x R,

whereupon

Si(u,v)(t,x) =0, (t, x) € [0, to) x R.

Hence and Lemma 3.1, we conclude that (u, v) is a solution to the IVP (1.1). This completes the proof. >

Now, let us prove some a priori estimates related to solutions of Problem (1.1). In the sequel, X = Xi x Xi, where Xi = Ci([0, to) x R) will be endowed with the norm

||(u,v)|| = max{||u||xi, ||v||x1}, (u,v) € X,

with

||u||Xi = max ^ sup |u(t,x)|, sup |ut(t,x)|, sup |ux(t, x)|

(t,x)e[0,ro)xR (t,x)e[0,ro)xR (t,x)e[0,ro)xR

provided it exists. Let

Bi = max { 2B, B2 [ 1 + ^k) , kB2, 2B2

Lemma 3.3. Under hypothesis (H1) and (H2) and for (u, v) € X with ||(u,v)|| ^ B, the following estimates hold:

and

|Si(u,v)(t,x)| < Bi(1 + t)(1 + |x|), (t,x) € [0, to) x R,

|S 2(u, v)(t, x) | < B 1 (1 + t)(1 + |x|), (t,x) € [0, to) x R.

< Suppose that (H1) and (H2) are satisfied and let (u, v) € X with ||(u,v)|| ^ B. (i) Estimation of S(u, v)(t, x)|, (t,x) € [0, to) x R :

IS i(u,v)(i,x)| =

(u(t,x i) — u0(xi)) dx i + u(ti,x)v(t i,x) dt i

<

(|u(t, x i)| + u0(x i)) dx i

+ |u(t i,x)||v(ti,x)| dt i

< 2B|x| + B 2t < B i (t + |x|) < B i (1 + t)(1 + |x|), (ii) Estimation of |S2(u, v)(t, x)|, (t, x) € [0, to) x R :

IS 2 (u,v)(t,x)| =

(u(t, x i)v(t, x i) — u0(x i)v0(x i)) dx i

t x

1

2

+ / [ u(ti,x)(v(ti,x)) + -k(u(ti, x)) )dti+k / u(ti,xi)dxf(ti,xi)dxidti

00

<

(|u(t, x i)||v(t, x i)| + U0(x i)v0(xi)) dx i

+ / [\u(t1,x)\(v(t1,x))2 + -k(u(j;1,x))2)dt1

t x

+ k J J |u(ti,xi)||dx f (t i,xi)| dx i

00

1

dti ^ 2B2\x\ + ( B2 + -kB2 ) t + kB2t\x\

< B i (t + |x| + t|x|) < B i (1 + |x|)(1+1). This completes the proof. >

t

x

t

x

x

t

t

x

Suppose

(H3) there exist a positive constant A and a nonnegative function g € C([0, to) x;

such

that

8(1 + t) (1 + t + t2) (1 + |x|) (1 + |x| + x2^ Jg(ti, xi)

t i, xi) dx i

dti < A, (t,x) € [0, to) x r.

In the last section, we will give an example for a function g that satisfies (H3). For (u, v) € X, define the operators S., S2 and S2 as follows.

t x

Si(u, v)(t,x) = J |(t - ti)2(x - xi)2g(ti,xi)Si(u,v)(ti,xi) dxi dt i, (t,x) € [0, to) x R,

o o

t x

S|(u, v)(t,x) = J j(t - ti)2(x - xi)2g(ti,xi)S2(u, v)(ti,xi) dxi dt i, (t, x) € [0, to) x R, oo

and

S2(u,v)(t,x) = (Si(u,v)(t,x),S2(u,v)(t,x)) , (t,x) € [0, to) x R. (3.3)

Lemma 3.4. Under hypothesis (H1), (H2) and (H3) and for (u, v) € X, with ||(u, v)|| ^ B, the following estimate holds:

||S2(u,v)| < AB 1.

< Suppose that (H1), (H2) and (H3) are satisfied. Let (u, v) € X, with ||(u, v)|| ^ B. (i) Estimation of |S.(u, v)(t, x)|, (t,x) € [0, to) x R :

IS2 (u, v) (t, x) I =

t x

<

y y (t - t 2)2(x - x i)2g(t 2,x 2)S2(u, v)(t i, x i ) dx i dt i oo

y (t - t i )2(x - x i )2g(t 2î x i )|S {(u, v)(t 2î x i )| dx i o

x

y y (t - ti )2(x - x i )2g(t i,x i )(1+1 i )(1 + |x i |) dx i

t x

< 4B i (1 + t)t2(1 + |x|)x^ y y g(t i, x i )

dx

dt

t x

< 8B i (1 + t)(1 + t + t2)(1 + | x | )(1 + |x| + x2)f y g(t i, x i )

t , x ) dx

dt i ^ AB i.

t

x

ii) Estimation of \§iS\(u, v)(t, (t,x) € [0, oo) x R :

d

—S%(u,v)(t,x)

=2

t x

J j(t — t ,)(x — x ,)2g(t ,, x ,)S.(u, v)(t ,, x ,) dx , dt , 00

^ 2

< 2B,

(t — t ,)(x — x,)2g(t ,, x ,) |S.(u, v)(t ,, x ,)| dx ,

0

t x

dt

J J(t — t 1 )(x — x 1 )2g(t i,x 1 )(1 + t 1 )(1 + |x 11) dx 1

dt

t

< 8B,(1+ t)t(1 + |x|)x2 y

g(t1, x1) dx1

dt1

t x

< 8B,(1+1)(1+1 +12)(1 + |x|)(1 + |x| + x2) J Jg(t,,x.) dx,

(iii) Estimation of | v)(t, x) \, (t,x) € [0, oo) x R :

dt, ^ AB

d

—S%(u,v)(t,x)

=2

t x

J J(t — t ,)2(x — x ,)g(t ,,x ,)S.(u, v)(t ,,x ,) dx , dt ,

00

^ 2

(t — t ,)2|x — x ,|g(t ,, x ,)|S.(u, v)(t ,, x ,)| dx ,

dt

< 2B,

t x

J J(t — ti)2|x — x 1 |g(t i,x 1 )(1+1 i)(1 + |x 11) dx 1

dt

< 4Bi(1+ t)t2(1 + |x|)|x| J

g( t , x ) dx

dt

t x

< 8B i(1+1)(1+1 +12)(1 + |x|)(1 + |x| + x2) y Jg(t i, x,) dx,

dt i ^ AB i.

As above,

|s2(u,v)(t,x)|

Therefore

d

—S%(u,v)(t,x)

d

—S$(u,v)(t,x)

^ AB i, (t, x) € [0, to) x R.

||S2(u,v)|| < AB i

This completes the proof. >

t

x

x

t

x

x

4. Existence and Multiplicity of Nonnegative Solutions

In the sequel, suppose that the constants B and A which appear in the conditions (H1) and (H3), respectively, satisfy the following inequality:

(H4) ABX < B, where Bx = max {2B, B2 (l + \k) , kB2,2B2 } . Our first main result for existence of classical solutions of the IVP (1.1) is as follows. Theorem 4.1. Under hypotheses (H1), (H2), (H3) and (H4), the IVP (1.1) has at least one nonnegative solution (u, v) € C 1([0, to) x R) x C 1([0, to) x R).

< Choose e € (0,1), such that eB i(1 + A) < B. For (u,v) € X = C^[0, to) x R) x C 1 ([0, to) x R), we will write

(u,v) ^ 0 if u(t, x) ^ 0 and v(t,x) ^ 0 for any (t,x) € [0, to) x R.

Let Y denotes the set of all equi-continuous families in X with respect to the norm || • ||, Y = Y be the closure of Y, Y = Y U |(uo, vo)} and

Y = {(u,v) € Y : (u, v) ^ 0, ||(u,v)|| < b} . Note that Y is a compact set in X. For (u, v) € X, define the operators T(u, v)(t, x) = —e(u, v)(t, x), (t, x) € [0, to) x R,

S (u, v)(t, x) = (u, v)(t, x) + e(u, v)(t, x) + eS2(u, v)(t, x), (t, x) € [0, to) x R. For (u, v) € Y and by using Lemma 3.4, it follows that

||(1—S)(u,v)|| = ||e(u,v)—eS2(u,v)|| < e||(u,v)||+e||S2(u,v)|| < eB 1 +eAB1 = eB1 (1+A) < B.

Thus, S : Y ^ X is continuous and (I — S)(Y) resides in a compact subset of X. Now, suppose that there is a (u, v) € X so that ||(u, v)|| = B and

(u, v) = A(1 — S )(u, v)

or 1

— (u, v) = (I — S)(u, v) = —e(u, v) — eS^f/u, v), A

or

^ + e ) (u, v) = -eS2(u, v) for some A € (0, Hence, \\S2(u, v) || < ABX < B,

e5< Q + Q + e) ||(u,i;)|| = e\\S2(u, v) || <eB,

which is a contradiction. Hence and Theorem 2.1, it follows that the operator T + S has a fixed point (u*,v*) € Y. Therefore

(u*, v*)(t, x) = T (u*, v*)(t, x) + S(u*, v*)(t, x) = — e(u*, v* )(t, x)

+ (u*,v*)(t,x) + e(u*,v*)(t,x) + eS2(u*,v*)(t,x), (t,x) € [0, to) x R,

whereupon

0 = S2(u*,v *)(t,x), (t,x) € [0, to) x R.

Lemma 3.2 yields that (u *,v *) is a solution to the IVP (1.1). This completes the proof. >

In the sequel, suppose that the constants B and A which appear in the conditions (H1) and (H3), respectively, satisfy the following inequality:

(H5) ABi < |, where Bx = max {25, (l + \k) ,kB2,2B2} and L is a positive constant that satisfies the following conditions:

r <L KR^B, Ri > (+ 1 ) L,

\5m J

with r and Ri are positive constants and m > 0 is large enough.

Our second main result for existence and multiplicity of classical solutions of the IVP (1.1) is as follows.

Theorem 4.2. Under Hypotheses (H1), (H2), (H3) and (H5), the IVP (1.1) has at least two nonnegative solutions (u1, v1), (u2, v2) € C 1([0, to) x R) x C 1([0, to) x R). < Set X = C 1([0, to) x R) x C 1([0, to) x R) and let

P = {(u, v) € X : (u, v) ^ 0 on [0, to) x R} .

With P we will denote the set of all equi-continuous families in P For (u, v) € X, define the operators T1 and S3 as follows:

Ti(u, v)(t, x) = (1 + me)(u, v)(t, x) - , (t, x) € [0, 00) x R,

Ss(u, v)(t, x) = —eS2(-u, v)(t, x) — me(u,v)(t, x) — ' x) e [0> x

where e is a positive constant, m > 0 is large enough and the operator S2 is given by formula (3.3). Note that any fixed point (u, v) € X of the operator T1 + S3 is a solution to the IVP (1.1). Now, let us define

U1 = Pr = {(u,v) € P : ||(u,v)|| < r},

U = Pl = {(u,v) € P : ||(u,v)|| <L}, U3 = Pr! = {(u,v) € P : ||(u,v)|| < R1},

Q = P% = {(u,v) eP: ||(u,i;)|| ^ R2}, R2 = Ri + ~B1 + ^.

mm

1) For (u1,v1), (u2,v2) € Q, we have

||T1(u1,v1) - T1(u2,v2)| = (1 + me)|(u1,v1) - (u2,v2)||,

whereupon T\ : Q —> X is an expansive operator with a constant h = 1 + me > 1.

2) Let (u,v) £ P_Ri, then Lemma 3.4 yields

||S3(u,v)|| < e\\S2(u,v)\\ +me\\(u,v)\\ + e^ < e (ABl + mRl + ^

Therefore S3(P^1) is uniformly bounded. Since S3 : Pr1 —> X is continuous, we have that S3) is equi-continuous. Consequently S3 : Pr1 —> X is completely continuous.

3) Let (ui,vi) € 3>Rl. Set

(u2,v2) = (ui,vi) + —S2(ui,vi) + (

m \5m 5m

Note that S\(ui,vi) + f ^ 0, + f ^ 0 on [0, oo) x R. We have u2,v2 ^ 0 on

[0, to) x R and

m 5m m 5m

Therefore (u2,v2) € Q and

—em(u2, v2) = -em(ui,vi) - eS2(ui,Vi) - e (j^, - e ^

or

(.I - Ti)(u2,v2) = -em(u2,v2) + e (j^, = S3(ui,vi).

Consequently, S3(Pr) C (I - Ti)(Q).

4) Assume that for any (w0,z0) € P* = P \ {0} there exist A ^ 0 and (u,v) € dPr n (Q + A(w0, z0)) or (u, v) € dPRl n (Q + A(w0, z0)) such that

S3(u,v) = (I - Ti)((u,v) - A(w0,z0)).

Then

—eS2(u, v) - me(u, v) - e ^ = -me((u, v) - A(w0, ¿o)) + e (j5> 75

or

—S2(u, v) = Xm(w0,z0) + ( —,—

Hence,

|S2(U,V)| =

Xm(w0,z0) + (

L

>5-

This is a contradiction.

5) Let ei = Assume that there exist (u\,vi) € <9Pl and Ai ^ 1 + £\ such that Xi(ui,vi) € P_Ri and

Ss(ubv1) = (/ — T1)(A1(u1,v1)). (4.1)

Since («i,t>i) € 9Pl and X\(ui,vi) € P_ri; it follows that

2

Moreover,

. +1 )L<X1L = X1\\(u1,v1)\\^R1. 5m

-eS2(ui,Vi) -me(ui,vi) -e (y^yj^ = ~Xime(ui,Vi) + e

or

S2{ui,vi) + i -,-J = (Ai - l)m(«i,vi).

From here,

L / L L\

= (Ai — l)m||(«i,t>i)|| = (Al -1 )mL

L

2- ^

5

S2(ui,vi) + ( ^

and

2

7- + 1 ^ Ai, 5m

which is a contradiction.

Therefore all conditions of Theorem 4.2 hold. Hence, the IVP (1.1) has at least two solutions (ui,vi) and (u2,v2) so that

||(ui,vi)|| = L < II(U2,V2)y < Ri

or

r < ||(ui,vi)| < L < ||(U2,V2)|| < Ri. >

5. An Example

Below, we will illustrate our main results. Let

l + s11v/2 + s22 snV2

h(s) = log--—=-—, l(s) = arctan --_, s € M, s / ±1.

1 - siV2 + s22 1 - s22

Then

22\/2s10(l — s22) , ll\/2s10(l + s22) „ ,

h'(s) =----'—=-, l(s) = —--^--, se M, s^±l.

Therefore

-to < lim (1 + s + s2 + s3 + s4 + s5 + s6) h(s) < to

-to < lim (1 + s + s2 + s3 + s4 + s5 + s6) l(s) < to.

Hence, there exists a positive constant Ci so that

s22 1 sii

(1 + s + s2 + s3 + s4 + s5 + s6) ( —= log 1 + + ° + —arctan ° I ^ Ci,

v y44\/2 l-sllV2 + s22 22y/2 1 - s22 J

s € M. Note that lim^(s) = ^ and by [23, p. 707, Integral 79], we have

f dz 1 , l + zV2 + z2 1 zV2

H--= arctan ■

J 1 + z4 4^2 l-zV2 + z2 2yf2 1-z2'

Let

si0

g(s)=(l + ^)(l + S + S2)2' SGR'

and

gi(t,x) = Q(t)Q(x), t e [0, to), x e R.

Then there exists a constant C2 > 0 such that

t x

8(1+t) (1 + t + t2) (1+|x|) (1 + |x| + |x|2) / y gi(ti,xi)

dxi

0 0

dti < C2, (t,x) € [0, m)xR.

Let

Then

A

g(t,x) = — gi(t,x), (t, x) e [0, 00) x R. C2

t x

8(1 +1) (1+1 + t2) (1 + |x|) (1 + |x| + |x|2)/ Jg(ti,xi) dxi

0 0

dti < A, (t,x) € [0, to) x R,

i.e., (H3) holds. Now, consider the initial value problem

dtu + dx(uv) =0, t > 0, x € R, 4xu

+ (to2 + u2) + 1 + =0, i > 0, x € R,

3

«(0, x) = v(0, x) =-tt:, x € R,

V > V > 1 + xl6'

so that (HI) and (H2) hold, with B = 10, for example. Take B = 10 and A = Then

B1 = max {20, 2 ■ 102} =2 ■ 102

and

AB1 = ±- 2 • 102 < B.

So, Condition (H4) is fulfilled. Thus, the conditions (H1), (H2), (H3) and (H4) are satisfied. Hence, by Theorem 4.1, it follows that Problem 6 has at least one nonnegative solution (u, v) € C 1([0, to) x R) x C 1([0, to) x R). In the sequel, take

Ri = B = 10, L = 5, r = 4, m = 1050, A = e =

Clearly,

1

To1'

r < L < R\ ^ B, e > 0, Ei>(-^ + l)L, AB\ <

5m 5

i.e., (H5) holds. Thus, the conditions (H1), (H2), (H3) and (H5) are satisfied. Hence, by Theorem 4.2, it follows that the initial value Problem 6 has at least two nonnegative solutions (ui,vi), (U2,V2) € Ci([0, to) x R) x Ci([0, to) x R).

Acknowledgements: The authors R. Azib, A. Kheloufi and K. Mebarki acknowledge support of "Direction Generale de la Recherche Scientifique et du Developpement Technologique (DGRSDT)", MESRS, Algeria.

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Received September 14, 2021 Riyadh Azib

Bejaia University, Laboratory of Applied Mathematics, Faculty of Exact Sciences, Bejaia 06000, Algeria,

Ph.D student of the Department of Mathematics

E-mail: [email protected], [email protected]

https://orcid.org/0000-0001-8162-8315

Svetlin Georgiev

Sofia University "St. Kliment Ohridski",

Faculty of Mathematics and Informatics,

15 Tzar Osvoboditel Blvd., Sofia 1504, Bulgaria,

Professor of the Department of Differential Equations

E-mail: svetlingeorgiev1@gmail. com

https://orcid.org/0000-0001-8015-4226

Arezki Kheloufi

Bejaia University, Laboratory of Applied Mathematics, Faculty of Exact Sciences, Bejaia 06000, Algeria,

Professor of the Department of Mathematics

E-mail: [email protected], [email protected]

https://orcid.org/0000-0001-5584-1454

Karima Mebarki

Bejaia University, Laboratory of Applied Mathematics, Faculty of Exact Sciences, Bejaia 06000, Algeria,

Professor of the Department of Mathematics

E-mail: [email protected], [email protected]

https://orcid.org/0000-0002-6679-5059

СУЩЕСТВОВАНИЕ ГЛОБАЛЬНЫХ КЛАССИЧЕСКИХ РЕШЕНИЙ ДЛЯ УРАВНЕНИЙ СЕН-ВЕНАНА

Азиб Р.1, Георгиев С.2, Хелоуфи А.1, Мебарки К.1

1 Университет Беджая, Алжир, 06000, Беджая; 2 Софийский университет имени святого Климента Охридского, Болгария, 1504, София, Бульвар Царя-Освободителя, 15

E-mail: [email protected], [email protected], [email protected], arezki [email protected], [email protected], [email protected], mebarqi_karima@hotmail. fr

Аннотация. В настоящее время исследования существования глобальных классических решений нелинейных эволюционных уравнений являются предметом активных математических исследований. В этой статье нас интересует классическая система уравнений мелкой воды, описывающая длинные поверхностные волны в жидкости переменной глубины. Эта система была предложена в 1871 г. Адмаром Жан-Клодом Барром де Сен-Венаном. А именно, мы исследуем начальную задачу для одномерных уравнений Сен-Венана. Нас особенно интересует вопрос, при каких достаточных

условиях должны верифицироваться начальные данные и топография дна, чтобы рассматриваемая система имела глобальные классические решения. Для доказательства наших основных результатов мы используем новый топологический подход, основанный на абстрактной теории суммы двух операторов в банаховых пространствах с фиксированной точкой. Эта основная и новая идея приводит к глобальным теоремам существования для многих интересных уравнений математической физики.

Ключевые слова: уравнения Сен-Венана, классическое решение, неподвижная точка, начальная задача.

AMS Subject Classification: 35Q35, 35A09, 35E15.

Образец цитирования: Azib, R., Georgiev, S., Kheloufi, A. and Mebarki, K. Existence of Global Classical Solutions for the Saint-Venant Equations // Владикавк. мат. журн.—2022.—Т. 24, № 3.—C. 21-36 (in English). DOI: 10.46698/x4972-4013-9236-n.