Potential Theory for the Equation of Small Oscillations of a Rotating Fluid

Kapitonov, Boris V.

doi:10.1070/sm1980v037n04abeh002095

Math. USSR Sb.

1980

DOI: 10.1070/sm1980v037n04abeh002095

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Potential Theory for the Equation of Small Oscillations of a Rotating Fluid

Boris V. Kapitonov¹

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1996

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Cited by 23 publications

(16 citation statements)

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“…In particular, 2-D problems in multiply connected domains were studied in [15,16,17,18,19,20]. Initial boundary value problems in simply connected domains were studied for 3-D equation of inertial waves and for similar equations in [8,9,10,14,30], but 3-D problems were not treated in multiply connected domains before. The aim of the present paper is to consider initial-boundary value problem in 3-D multiply connected domain (interior and exterior) with Dirichlet boundary condition.…”

mentioning

confidence: 99%

“…This equation can be easily computed by discretization and inversion of a matrix. Initial-boundary value problems for some equations of composite type in a 3-D simply connected domain were reduced to integral equations in [8,10,14]. However, the method of reduction to integral equations suggested in these papers for simply connected do-…”

mentioning

confidence: 99%

“…This equation comes from ocean dynamics. Different initial-boundary value problems for analogous equations were treated in [8,9,10,14,15,16,17,18,19,20,30]. In particular, 2-D problems in multiply connected domains were studied in [15,16,17,18,19,20].…”

mentioning

confidence: 99%

“…However, these conditions were omitted in [14] together with analysis of uniqueness of a solution, and so exterior problems in [14] are not well-posed.…”

mentioning

confidence: 99%

“…The aim of the present paper is to consider initial-boundary value problem in 3-D multiply connected domain (interior and exterior) with Dirichlet boundary condition. To prove existence theorem for our problem we use dynamic potentials [14] and the method of boundary integral equations. Uniqueness of a solution is also studied.…”

mentioning

confidence: 99%

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Evolutionary equation of inertial waves in 3‐D multiply connected domain with Dirichlet boundary condition

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2001

International Journal of Mathematics and Mathematical Sciences

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Abstract. We study initial-boundary value problem for an equation of composite type in 3-D multiply connected domain. This equation governs nonsteady inertial waves in rotating fluids. The solution of the problem is obtained in the form of dynamic potentials, which density obeys the uniquely solvable integral equation. Thereby the existence theorem is proved. Besides, the uniqueness of the solution is studied. All results hold for interior domains and for exterior domains with appropriate conditions at infinity. Modern advances in the theory of waves are mostly concerned with nonlinear phenomena [4,3,5,6]. However, there exist some types of linear waves which are not well studied yet, for example, nonsteady inertial waves, that is, internal waves in rotating fluids. These waves are governed by PDEs of composite type and of high order. Such PDEs possess both elliptic and hyperbolic characteristics, and so they share properties of both elliptic and hyperbolic equations [13,23,24,25,26]. Equations of composite type are not well studied even in linear case and they are not covered by existing classifications of PDEs. In particular, they do not belong to equations of principal part, though elliptic, parabolic, and hyperbolic equations belong [7].Inertial wave equation (see (1.2)) is an evolutionary equation of fourth order and composite type. This equation comes from ocean dynamics. Different initial-boundary value problems for analogous equations were treated in [8,9,10,14,15,16,17,18,19,20,30]. In particular, 2-D problems in multiply connected domains were studied in [15,16,17,18,19,20]. Initial boundary value problems in simply connected domains were studied for 3-D equation of inertial waves and for similar equations in [8,9,10,14,30], but 3-D problems were not treated in multiply connected domains before. The aim of the present paper is to consider initial-boundary value problem in 3-D multiply connected domain (interior and exterior) with Dirichlet boundary condition. To prove existence theorem for our problem we use dynamic potentials [14] and the method of boundary integral equations. Uniqueness of a solution is also studied. It should be stressed that the integral equation obtained in the present paper is a uniquely solvable Fredholm equation of the second kind and index zero. This equation can be easily computed by discretization and inversion of a matrix. Initial-boundary value problems for some equations of composite type in a 3-D simply connected domain were reduced to integral equations in [8,10,14]. However, the method of reduction to integral equations suggested in these papers for simply connected do-

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mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

“…However, these conditions were omitted in [14] together with analysis of uniqueness of a solution, and so exterior problems in [14] are not well-posed.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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Evolutionary equation of inertial waves in 3‐D multiply connected domain with Dirichlet boundary condition

Крутицкий

2001

International Journal of Mathematics and Mathematical Sciences

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On critical exponents for weak solutions to the Cauchy problem for one nonlinear equation with gradient nonlinearity

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Matveeva

2022

Math Methods in App Sciences

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In this paper, we consider the Cauchy problem for one nonclassical, third-order, partial differential equation with gradient nonlinearity |∇u(x, t)| q . The solution to this problem is understood in a weak sense. We show that for 1 < q ⩽ 3∕2, there are no local-in-time weak solutions of this problem with initial data u 0 (x) from the class U, whereas for q > 3∕2, such a solution exists. For 3∕2 < q ⩽ 2, the Cauchy problem proved not to have global-in-time weak solutions. For 3∕2 < q ⩽ 2, we show finite-time blow-up of unique local-in-time weak solution of the Cauchy problem without dependence from the initial functions from the class U. As a technique, we obtain Schauder-type estimates for potentials. We use them to investigate smoothness of the weak solution to the Cauchy problem for q = 2 and q ⩾ 4.

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The second initial boundary value problem for the gravitation-gyroscopic wave equation in exterior domains

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1996

Math Notes

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ABSTRACT. We study the existence and uniqueness of the solution to the second initial boundary value problem for the gravitation-gyroscopic wave equation in an exterior multiply connected domain with various types of conditions at infinity.KEY WORDS: equations of composite type, existence and uniqueness theorem, conditions at infinity.The gravitation-gyroscopic wave equation is a fourth-order evolution equation of composite type with two space variables. It has the form ~u ~u ~u 2 02uwhere Wl, w2 > 0 are constants. The solvability of exterior initial boundary value problems for this equation was studied in [1-3] without taking into account the conditions at infinity and without analyzing the uniqueness of the solution. Problems in half-space for equations of composite type were studied in [4]. Various problems in the case of three space variables for equations of these types were studied in [5][6]. In [7][8][9] solutions of some initial boundary value problems for the gravitation-gyroscopic wave equation were obtained in the exterior multiply connected domains formed by systems of cuts on the plane. Such problems describe nonstationary interior waves excited by the motion of several bodies in a stratified rotating fluid.We study the existence and uniqueness of the solution to the second initial boundary value problem for the gravitation-gyroscopic wave equation in an exterior multiply connected domain with various types of conditions at infinity. The role of the conditions at infinity for the correct statement of this problem is clarified. The proof of the solvability makes use of a new function-theoretic method which has never been used earlier in such problems.w Statement of the problem and uniqueness theorem Consider an exterior connected domain D with boundary F E C 2'~ on a plane with Cartesian coordinates x = (Xl, z2). Let F consist of N simple closed curves rl, ... , FN having no common points. We specify the direction of the circuit as positive if the domain D is on its right. Suppose that the boundary r is parametrized, z(s) = (xl(s), zu(S)) is the arc length s measured from some fixed point on each closed part of r. Suppose further that s increases in the positive direction and that there is a one-to-one correspondence between the values of s and the points of F (except for the above-mentioned fixed points). By n we denote the outward normal on F with respect to D. Let us introduce some notation and definitions which we shall need in what follows. Let u(t, z) = u(t, za, z~) be a sufficiently smooth function defined for t > 0 and x E D. Assuming that n is the outward normal at a point x(s) E F, we define the differential operator 2q't~ at the points E D by setting

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Potential Theory for the Equation of Small Oscillations of a Rotating Fluid

Cited by 23 publications

References 1 publication

Evolutionary equation of inertial waves in 3‐D multiply connected domain with Dirichlet boundary condition

Evolutionary equation of inertial waves in 3‐D multiply connected domain with Dirichlet boundary condition

On critical exponents for weak solutions to the Cauchy problem for one nonlinear equation with gradient nonlinearity

The second initial boundary value problem for the gravitation-gyroscopic wave equation in exterior domains

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