Navier–Stokes Equation in a Cone with Cross-Sections in the Form of 3D Spheres, Depending on Time, and the Corresponding Basis

Abstract

The work establishes the unique solvability of a boundary value problem for a 3D linearized system of Navier–Stokes equations in a degenerate domain represented by a cone. The domain degenerates at the vertex of the cone at the initial moment of time, and, as a consequence of this fact, there are no initial conditions in the problem under consideration. First, the unique solvability of the initial-boundary value problem for the 3D linearized Navier–Stokes equations system in a truncated cone is established. Then, the original problem for the cone is approximated by a countable family of initial-boundary value problems in domains represented by truncated cones, which are constructed in a specially chosen manner. In the limit, the truncated cones will tend toward the original cone. The Faedo–Galerkin method is used to prove the unique solvability of initial-boundary value problems in each of the truncated cones. By carrying out the passage to the limit, we obtain the main result regarding the solvability of the boundary value problem in a cone.

1. Introduction

Navier–Stokes systems, which describe the movement of a fluid, have been the subject of research by many authors. Here, we only note the monographs [1,2,3], which are fundamental and have become classic works on this topic. In many applications, there is a need to study boundary value problems in domains with moving boundaries known as non-cylindrical domains [4,5,6,7,8,9,10,11,12,13,14].

For problems involving degenerate domains, earlier, in [15,16,17,18,19], we used the method of cutting off a family of neighborhoods of the degeneracy point of a domain. For the resulting family of boundary value problems, we applied a one-to-one transformation of them to problems in cylindrical domains and established their unique solvability in Sobolev classes. As a result, we obtained the unique solvability of a family of boundary value problems for truncated non-degenerate domains. This transformation introduced additional terms, causing the coefficients of the equations become dependent on the independent variables. In the work presented, we do things completely differently. Here, we use spectral decomposition and a priori estimates to solve the boundary value problems under study without the need for these preliminiary transformations. However, when establishing a priori estimates, additional difficulties arise due to the presence of normal derivatives on the boundary of the desired functions in integral identities that replace differential equations, boundary conditions, and initial conditions. In this work, we managed to overcome these and other emerging difficulties.

We also want to mention recent works [20,21] devoted to the study of the three-dimensional Navier–Stokes equation, the results of which may be of interest to readers.

First, in this paper, we study the solvability of the initial-boundary value problem for a 3D linearized system of Navier–Stokes equations in a truncated cone: $Q_{t_0}\equiv \{x=(x_1,x_2,x_3),t:\left|x\right|<t,t_0<t<T\}$ with a homogeneous Dirichlet boundary and initial conditions. We use the basis $\{w_k\left(x\right)=(w_k^{\left(1\right)}\left(x\right),w_k^{\left(2\right)}\left(x\right),$ $w_k^{\left(3\right)}\left(x\right)),k=1,2,3,\dots \}$, which is an independent system of functions in the solution space for a 3D linearized system of Stokes equations in the case of a unit ball. The existence of such a basis, for example, follows from the result of the work ([3], Chapter 1, Section 2.6). A time parameter t is introduced, subject to the following constraint: $t\in (t_0,T),0<t_0<T<\infty ,$ and the family of balls ${\mathsf{\Omega }}_t=\left\{\right|x|<t,t_0<t<T\},$ which forms the truncated cone $Q_{t_0}.$ By applying the aforementioned result on the basis of the unit ball to the problem with balls of variable radii, changing according to a linear law with respect to the time variable t, we obtain a time-dependent basis, i.e., $\{w_k(x,t)=(w_k^{\left(1\right)}(x,t),$ $w_k^{\left(2\right)}(x,t),w_k^{\left(3\right)}(x,t)),$ $k=1,2,3,\dots \}.$ It should be noted that the constructed basis will ensure the fulfillment of the incompressibility condition: $div{w}_k(x,t)=0,(x,t)\in Q_{t_0},k=1,2,3,\dots ,$ and, on the generatrix of the truncated cone of homogeneous Dirichlet boundary conditions, $w_k=0,(x,t)\in {\mathsf{\Sigma }}_{t_0}\equiv \{x=(x_1,x_2,x_3),t:\left|x\right|=t,t_0<t<T\}.$ Next, using the Faedo–Galerkin method, we establish, in Sobolev classes, the unique solvability of the initial-boundary value problem for a 3D time-dependent Stokes equations in a truncated cone. For this purpose, a priori estimates for Galerkin approximations are established and formulated in the form of a series of lemmas, and the passage to the limit is carried out using the methods of function theory and functional analysis. A priori estimates are found for the solution of the initial problem posed for a domain represented by a truncated cone. The latest a priori estimates allow us to obtain statements about the uniqueness and, additionally, about the differential properties of the desired solution. Next, the result regarding the unique solvability of the problem for a truncated cone is used to prove the main result of the work: the unique solvability of the boundary value problem for the 3D linearized system of Navier–Stokes equations in the degenerate domain represented by a cone. In conclusion, we indicate how the results obtained in the work can be developed for a degenerate curvilinear cone when the radii of the cone sections change according to the nonlinear law $\phi \left(t\right)$. We present a set of requirements that are sufficient to impose on the function $\phi \left(t\right)$.

2. Statement  of the  Initial-Boundary Value Problem and Boundary Value Problem for a Three-Dimensional Linearized Navier–Stokes System in Domains Represented by a Truncated Cone and a Cone, Respectively

Let $x=(x_1,x_2,x_3),Q_{t_0}={\{x,t:|x|}^2=x_1^2+x_2^2+x_3^2<t^2,0<t_0<t<T<\infty \}$ be a truncated cone, and ${\mathsf{\Omega }}_t=\{x:|x|<t\}$ be the section (representing a ball with radius equal to t) of the cone $Q_{t_0}$ for any fixed $t\in [t_0,T]$ with boundary $\partial {\mathsf{\Omega }}_t=\{x:|x|=t\}$; ${\mathsf{\Sigma }}_{t_0}=\{x,t:|x|=t,t_0<t<T\}$ is the generatrix of the truncated cone. Along with the truncated cone, we define the cone $Q=\{x,t:|x{|}^2=x_1^2+x_2^2+x_3^2<t^2,0<t<T<\infty \}$ with a vertex at the origin, $\mathsf{\Sigma }=\{x,t:|x|=t,0<t<T\}$ is the generatrix of the cone, and ${\mathsf{\Omega }}_t=\{x:|x|<t\}$ is the section of the cone Q for any fixed $t\in (0,T]$.

For any fixed $t\in [t_0,T]$, the boundary of the section ${\mathsf{\Omega }}_t$ of the cone $Q_{t_0}$ Figure 1, can be represented, for example, in the form

$$\partial {\mathsf{\Omega }}_t=\left\{x:\left|x\right|=t\right\}=\partial {\mathsf{\Omega }}_t^{+}\cup \partial {\mathsf{\Omega }}_t^{-},$$

(1)

where

$$\partial {\mathsf{\Omega }}_t^{+}=\left\{x:x_1=\sqrt{t^2-x_2^2-x_3^2},-\sqrt{t^2-x_3^2}<x_2<\sqrt{t^2-x_3^2};-t<x_3<t\right\},$$

(2)

$$\partial {\mathsf{\Omega }}_t^{-}=\left\{x:x_1=-\sqrt{t^2-x_2^2-x_3^2},-\sqrt{t^2-x_3^2}<x_2<\sqrt{t^2-x_3^2};-t<x_3<t\right\}.$$

(3)

Obviously, there are other representations different from (1). For definiteness, throughout the work, we will use the Representation (1) of a sphere, the surface of a ball of radius t.

In the truncated cone $Q_{t_0}$, we consider the initial-boundary value problem for the linear three-dimensional Navier–Stokes equation of determining the vector function $u(x,t)=\left\{u^{\left(1\right)}(x,t),u^{\left(2\right)}(x,t),\right.$ $\left.u^{\left(3\right)}(x,t)\right\}$ and scalar function $p(x,t)$:

$${\partial }_{t}u-\Delta u=f-\nabla p,(x,t)\in Q_{t_0},$$

(4)

$$divu=0,(x,t)\in Q_{t_0},$$

(5)

$$u=0,(x,t)\in {\mathsf{\Sigma }}_{t_0}\mathrm{is}\mathrm{the}\mathrm{lateral}\mathrm{surface}\mathrm{of}\mathrm{the}\mathrm{cone},$$

(6)

$$u=0,x\in {\mathsf{\Omega }}_{t_0}\mathrm{is}\mathrm{a}\mathrm{ball}\mathrm{with}\mathrm{radius}{t}_0,\mathrm{i}.\mathrm{e}.,\mathrm{the}\mathrm{lower}\mathrm{base}\mathrm{of}\mathrm{the}\mathrm{cone}.$$

(7)

Along with problem (4)–(7), we will also study the boundary value problem in the cone Q of determining the vector function $u(x,t)=\left\{u^{\left(1\right)}(x,t),u^{\left(2\right)}(x,t),u^{\left(3\right)}(x,t)\right\}$ and scalar function $p(x,t)$:

$${\partial }_{t}u-\Delta u=f-\nabla p,(x,t)\in Q,$$

(8)

$$divu=0,(x,t)\in Q,$$

(9)

$$u=0,(x,t)\in \mathsf{\Sigma }\mathrm{is}\mathrm{the}\mathrm{lateral}\mathrm{surface}\mathrm{of}\mathrm{the}\mathrm{cone}.$$

(10)

It should be noted that, in the boundary value problem (8)–(10), there is no initial condition (7), since the lower base of the cone degenerates into a point (the vertex of the cone).

According to the monographs [1,2,3], we will give the designations of the spaces necessary in the work. For each $t\in (0,T]$, we introduce the notations of spaces ${\mathbf{V}}_t,$${\mathbf{H}}_t,$${\mathbf{L}}^2\left({\mathsf{\Omega }}_t\right),$ ${\mathbf{H}}_0^1\left({\mathsf{\Omega }}_t\right)$, and ${\mathbf{H}}^2\left({\mathsf{\Omega }}_t\right)$ used in studying the solvability of the initial-boundary value problem (4)–(7), and which we will use in the future:

$${\mathbf{V}}_t=\{v:v\in {\mathbf{H}}_0^1\left({\mathsf{\Omega }}_t\right)={\left(H_0^1\left({\mathsf{\Omega }}_t\right)\right)}^3,\mathrm{div}v=0\mathrm{on}{\mathsf{\Omega }}_t\},$$

$${\mathbf{H}}_t=\left\{v:v\in {\mathbf{L}}^2\left({\mathsf{\Omega }}_t\right),\mathrm{div}v=0\mathrm{on}{\mathsf{\Omega }}_t\right\},$$

$${\mathbf{L}}^2\left({\mathsf{\Omega }}_t\right)={\left(L^2\left({\mathsf{\Omega }}_t\right)\right)}^3,{\mathbf{H}}^2\left({\mathsf{\Omega }}_t\right)={\left(H^2\left({\mathsf{\Omega }}_t\right)\right)}^3.$$

The following dense embeddings take place:

$${\mathbf{V}}_t\subset {\mathbf{H}}_t\equiv {\mathbf{H}}_t^{\prime }\subset {\mathbf{V}}_t^{\prime },{\mathbf{H}}_0^1\left({\mathsf{\Omega }}_t\right)\subset {\mathbf{L}}^2\left({\mathsf{\Omega }}_t\right)\equiv {\left({\mathbf{L}}^2\left({\mathsf{\Omega }}_t\right)\right)}^{\prime }\subset {\mathbf{H}}^{-1}\left({\mathsf{\Omega }}_t\right),$$

and $\left(·,·\right),\left(\left(·,·\right)\right)$ are inner products in spaces ${\mathbf{H}}_t,{\mathbf{L}}^2\left({\mathsf{\Omega }}_t\right)$ and ${\mathbf{V}}_t,{\mathbf{H}}_0^1\left({\mathsf{\Omega }}_t\right),$ respectively. The Helmholtz decomposition of space ${\mathbf{L}}^2\left({\mathsf{\Omega }}_t\right)$: ${\mathbf{L}}^2\left({\mathsf{\Omega }}_t\right)={\mathbf{H}}_t\oplus {\mathbf{H}}_t^{\perp },$ where

$${\mathbf{H}}_t^{\perp }\mathrm{is}\mathrm{an}\mathrm{orthogonal}\mathrm{complement}\mathrm{to}{\mathbf{H}}_t\mathrm{in}\mathrm{the}\mathrm{space}{\mathbf{L}}^2\left({\mathsf{\Omega }}_t\right),$$

$${\mathbf{H}}_t^{\perp }=\{v:v\in {\mathbf{L}}^2\left({\mathsf{\Omega }}_t\right),v=\nabla u,u\in {\mathbf{H}}^1\left({\mathsf{\Omega }}_t\right)\},$$

$${\left({\mathbf{H}}_t\oplus {\mathbf{H}}_t^{\perp }\right)}^{\prime }\equiv {\left({\mathbf{L}}^2\left({\mathsf{\Omega }}_t\right)\right)}^{\prime }\equiv {\mathbf{L}}^2\left({\mathsf{\Omega }}_t\right)\equiv {\mathbf{H}}_t\oplus {\mathbf{H}}_t^{\perp },$$

and the “prime” symbol denotes a topologically dual space.

Using the introduced spaces, we will give meaningful formulations of the problems studied in the work.

Problem 1

(Truncated cone). Let $f\in L^2(t_0,T;{\mathbf{H}}_t).$ Establish the unique solvability of the initial-boundary value problem (4)–(7) in the space $u\in W_2^1(t_0,T;{\mathbf{H}}_t)\cap L^2(t_0,T;{\mathbf{H}}^2\left({\mathsf{\Omega }}_t\right)\cap {\mathbf{H}}_0^1\left({\mathsf{\Omega }}_t\right))$, $p\in L^2(t_0,T;{\mathbf{H}}^1\left({\mathsf{\Omega }}_t\right))$.

Problem 2

(Cone). Let in problem (8)–(10) $f\in L^2(0,T;{\mathbf{H}}_t).$ Establish the unique solvability of the boundary value problem (8)–(10) in the space $u\in W_2^1(0,T;{\mathbf{H}}_t)\cap L^2(0,T;{\mathbf{H}}^2\left({\mathsf{\Omega }}_t\right)\cap {\mathbf{H}}_0^1\left({\mathsf{\Omega }}_t\right)),p\in L^2(0,T;{\mathbf{H}}^1\left({\mathsf{\Omega }}_t\right)).$

In what follows, we first prove the unique solvability of Problem 1 for a truncated cone (Theorem 2). Section 3, Section 4 and Section 5 are devoted to the proof of Theorem 2. Using the statement of Theorem 2, we prove the unique solvability of Problem 2 (Theorem 3). The work ends with a brief conclusion.

3. Galerkin Approximations

Let, for each $t\in [t_0,T]$, the system of functions ${\left\{w_k(x,t),x\in {\mathsf{\Omega }}_t\right\}}_1^{\infty }$ be a basis in the space ${\mathbf{H}}^2\left({\mathsf{\Omega }}_t\right)\cap {\mathbf{H}}_0^1\left({\mathsf{\Omega }}_t\right)$, i.e., in particular, for each function $w_k(x,t)$ and for each $t\in [t_0,T]$, the following equalities take place:

$$div{w}_k(x,t)=0,x\in {\mathsf{\Omega }}_t;w_k(x,t)=0,x\in \partial {\mathsf{\Omega }}_t,$$

and, for each k, there exists a function $p_k(x,t)\in H^1\left({\mathsf{\Omega }}_t\right)$ such that the equations with the spectral parameter from (12) are satisfied. Thus, we assume that the basis is composed of the eigenfunctions of the spectral problem (12), which is orthonormal.

In the future, we will need the following proposition:

Let a variational formulation of the spectral problem for the Stokes operator be given:

$$w_k\in {\mathbf{V}}_t,\left(\left(w_k,u\right)\right)={\lambda }_k\left(w_k,u\right)\forall u\in {\mathbf{V}}_t,\forall {\mathsf{\Omega }}_t,t\in (t_0,T),$$

(11)

where $\left(·,·\right),\left(\left(·,·\right)\right)$ are inner products in the spaces ${\mathbf{H}}_t,{\mathbf{L}}^2\left({\mathsf{\Omega }}_t\right)$ and ${\mathbf{V}}_t,{\mathbf{H}}_0^1\left({\mathsf{\Omega }}_t\right),$ respectively. Obviously, the eigenfunctions $w_k(x,t)$ and eigenvalues ${\lambda }_k\left(t\right)$ will depend on the time variable t. As is known, problem (11) is interpreted as follows: for each k, there exists $p_k\in L^2\left({\mathsf{\Omega }}_t\right)$, such that

$$\left\{\begin{array}{ccc}-{\Delta }_x{w}_k+{\nabla }_x{p}_k={\lambda }_k{w}_k\hfill & \mathrm{in}\hfill & x\in {\mathsf{\Omega }}_t,\hfill \\ {div}_x{w}_k=0\hfill & \mathrm{in}\hfill & x\in {\mathsf{\Omega }}_t,\hfill \\ w_k=0\hfill & \mathrm{on}\hfill & x\in \partial {\mathsf{\Omega }}_t,\hfill \end{array}\right.t\in (t_0,T).$$

(12)

Let the section ${\mathsf{\Omega }}_t=\left\{\right|x|<t\}$ of the cone $Q_{t_0}$ be transformed into a ball of constant radius equal to one, $\mathsf{\Omega }=\left\{\right|y|<1\}$. We achieve this goal by the following transformation of independent variables:

$$y_j={\displaystyle \frac{x_j}{t}},j=1,2,3.$$

(13)

Thus, we have proved the following.

Proposition 1.

The elements of the basis $w_k(x,t)$ are differentiable with respect to t, i.e., there is a partial derivative

$${\partial }_t{w}_k(x,t)=-\sum _{j=1}^3{\displaystyle \frac{x_j}{t}}{\partial }_{x_j}{w}_k(x,t).$$

Then, by virtue of (13) and the replacement

$$v_k{\left(y\right)|}_{y=x/t}=v_k(x/t)=w_k(x,t),{\displaystyle \frac{q_k(x/t)}{t}}=p_k(x,t),$$

(14)

statements of the spectral problems (11) and (12) for the Stokes operator take the form, respectively,

$$v_k\in \mathbf{V},\left(\left(v_k,u\right)\right)={\tilde{\lambda }}_k\left(v_k,u\right)\forall u\in \mathbf{V},$$

(15)

$$\left\{\begin{array}{ccc}-{\Delta }_y{v}_k+{\nabla }_y{q}_k={\tilde{\lambda }}_k{v}_k\hfill & \mathrm{in}\hfill & y\in \mathsf{\Omega },\hfill \\ {div}_y{v}_k=0\hfill & \mathrm{in}\hfill & y\in \mathsf{\Omega },\hfill \\ v_k=0\hfill & \mathrm{on}\hfill & y\in \partial \mathsf{\Omega },\hfill \end{array}\right.$$

(16)

where $\left(·,·\right),\left(\left(·,·\right)\right)$ are inner products in the spaces $\mathbf{H},{\mathbf{L}}^2\left(\mathsf{\Omega }\right)$ and $\mathbf{V},{\mathbf{H}}_0^1\left(\mathsf{\Omega }\right),$ respectively, and the eigenvalues ${\tilde{\lambda }}_k=t^2{\lambda }_k\left(t\right)$ are constant, i.e., ${\lambda }_k\left(t\right)={\tilde{\lambda }}_k/t^2$. The designations of the spaces $\mathbf{H}$ and $\mathbf{V}$ correspond to the domain $\mathsf{\Omega },$ which are defined in the same way as the spaces ${\mathbf{H}}_t$ and ${\mathbf{V}}_t$ for the domain ${\mathsf{\Omega }}_t$.

Now, we can show that the partial derivative of the eigenfunction $w_k(x,t)$ with respect to the variable t exists. Namely, according to (14), we have

$${\partial }_t{w}_k(x,t)=-\sum _{j=1}^3{\displaystyle \frac{x_j}{t^2}}{\partial }_{y_j}{v}_k{\left(y\right)|}_{y=x/t}=-\sum _{j=1}^3{\displaystyle \frac{x_j}{t}}{\partial }_{x_j}{w}_k(x,t),$$

(17)

since, according to (14), we have

$${\partial }_{y_j}{v}_k{\left(y\right)|}_{y=x/t}=t{\partial }_{x_j}{w}_k(x,t),j=1,2,3.$$

(18)

Let us move on to Galerkin approximations. It should be noted that we considered similar issues in [7,8,9,15,16,17,18,19]. Using this basis, to solve the initial-boundary value problem (4)–(7), we introduce Galerkin approximations:

$$u_N(x,t)=\sum _{k=1}^N{c}_{Nk}\left(t\right)w_k(x,t),$$

(19)

where the functions $c_{Nk}\left(t\right)$ are unknown and must be determined. Then, scalarly multiplying Equation (4) by $w_l(x,t)$ in the space $L^2\left({\mathsf{\Omega }}_t\right)$ and taking into account the incompressibility conditions (5) and boundary conditions (6), as well as the Galerkin approximations (19), we obtain a system of N ordinary differential equations for unknown functions $\{c_{Nk}\left(t\right)$, $k=\overline{1,N},t\in (t_0,T)\}$:

$$\begin{array}{c}{\displaystyle \sum _{k=1}^N}\left({\partial }_t\left[c_{Nk}\left(t\right)w_k(x,t)\right],w_l(x,t)\right)\\ +{\displaystyle \sum _{k=1}^N}\left(c_{Nk}\left(t\right)\nabla w_k(x,t),\nabla w_l(x,t)\right)=\left(f(x,t),w_l(x,t)\right),l=\overline{1,N},\end{array}$$

(20)

where

$${\partial }_t{u}_N(x,t)={\partial }_t{\displaystyle \sum _{k=1}^N}{c}_{Nk}\left(t\right)w_k(x,t)={\displaystyle \sum _{k=1}^N}\left[c_{Nk}^{\prime }\left(t\right)w_k(x,t)+c_{Nk}\left(t\right){\partial }_t{w}_k(x,t)\right],$$

(21)

and, according to (17), we obtain

$${\partial }_t{w}_k(x,t)=-{\displaystyle \sum _{j=1}^3}{\displaystyle \frac{x_j}{t}}{\partial }_{x_j}{w}_k(x,t),$$

(22)

or

$$\begin{array}{c}{\displaystyle \sum _{k=1}^N}\left\{\left(w_k(x,t),w_l(x,t)\right)c_{Nk}^{\prime }\left(t\right)+\left[\left({\partial }_t{w}_k(x,t),w_l(x,t)\right)+\left(\nabla w_k(x,t),\nabla w_l(x,t)\right)\right]c_{Nk}\left(t\right)\right\}\\ =f_{Nl}\left(t\right)\equiv \left(f(x,t),w_l(x,t)\right),l=\overline{1,N},\end{array}$$

(23)

where the following equality is used:

$${\left(w_k(x,t),\nabla p(x,t)\right)}_{L^2\left({\mathsf{\Omega }}_t\right)}=-{\left(div{w}_k(x,t),p(x,t)\right)}_{L^2\left({\mathsf{\Omega }}_t\right)}=0,t\in (t_0,T).$$

Let us write the system (23) in matrix form:

$$A_N\left(t\right)c_N^{\prime }\left(t\right)+B_N\left(t\right)c_N\left(t\right)=f_N\left(t\right),t\in (t_0,T),$$

(24)

where the matrices $A_N\left(t\right),$ $B_N\left(t\right)$ and vectors $c_N\left(t\right),$ $f_N\left(t\right)$ are given by the following formulas:

$$A_N\left(t\right)={\parallel \left(w_k(x,t),w_l(x,t)\right)\parallel }_{k,l=1}^N,$$

$$B_N\left(t\right)={\parallel \left({\partial }_t{w}_k(x,t),w_l(x,t)\right)+\left(\nabla w_k(x,t),\nabla w_l(x,t)\right)\parallel }_{k,l=1}^N,$$

$$c_N\left(t\right)=\left(c_{N1}\left(t\right),\dots ,c_{NN}\left(t\right)\right),f_N\left(t\right)=\left(f_{N1}\left(t\right),\dots ,f_{NN}\left(t\right)\right).$$

Further, since, for each fixed variable value of t from the interval $[t_0,T]$, the system of functions ${\left\{w_k(x,t)\right\}}_{k=1}^{\infty }$ is linearly independent, then, from the initial condition (7), we directly obtain

$$\sum _{k=1}^N{c}_{Nk}\left(t\right)w_k{(x,t)|}_{t=t_0}=0⟺c_N\left(t_0\right)=0.$$

(25)

Since, for each $t\in (t_0,T)$, the matrix $A_N\left(t\right)$ is a Gram matrix, it is invertible for each fixed $N=1,2,3,\dots ,$ and, from (24) and (25), we obtain the following Cauchy problem:

$$\left\{\begin{array}{c}{c}_N^{\prime }\left(t\right)+A_N^{-1}\left(t\right)B_N\left(t\right)c_N\left(t\right)=A_N^{-1}{f}_N\left(t\right),t\in (t_0,T),\hfill \\ c^N\left(t_0\right)=0.\hfill \end{array}\right.$$

(26)

According to the well-known Caratheodory theorem ([22], Chapter 1, paragraph 1) for the Cauchy problem (26) with measurable coefficients, the following lemma takes place.

Lemma 1.

The Cauchy problem (26) has a unique absolutely continuous solution $c_N\left(t\right)=\{c_{N1}\left(t\right),...,c_{NN}\left(t\right)\},$ $t\in (t_0,T).$

Using this solution, we find an explicit formula for the Galerkin approximation (19), for which we will establish a priori estimates in the next section.

4. A Priori Estimates for Galerkin Approximations

We multiply equation (23) by $c_{Nl}\left(t\right)$ and add the resulting N-equations by index l. Then, considering the equality (19) and the relations (21)–(22), we obtain

$$\left({\partial }_t{u}_N(x,t),u_N(x,t)\right)+{\parallel u_N(x,t)\parallel }_{{\mathbf{V}}_t}^2=\left(f(x,t),u_N(x,t)\right),t\in (t_0,T).$$

(27)

Next, we will use the validity of the following proposition.

Proposition 2.

If, for the time being, we assume that the boundary conditions (6) are inhomogeneous, then, taking into account the representation of the sphere $\partial {\mathsf{\Omega }}_t$ (1)–(3) for the ball ${\mathsf{\Omega }}_t$, the first term on the left side of equality (27) will be written as follows:

$${\left({\partial }_t{u}_N(x,t),u_N(x,t)\right)}_{L^2\left({\mathsf{\Omega }}_t\right)}={\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\underset{-t}{\overset{t}{\int }}\left[\underset{-\sqrt{t^2-x_3^2}}{\overset{\sqrt{t^2-x_3^2}}{\int }}\left(\underset{-\sqrt{t^2-x_2^2-x_3^2}}{\overset{\sqrt{t^2-x_2^2-x_3^2}}{\int }}{\left|u_N(x,t)\right|}^{2}d{x}_1\right)d{x}_2\right]d{x}_3-I\left(t\right),$$

(28)

where, using ([23], paragraph. 626, formula (5), p. 257), from (28), we obtain

$$I\left(t\right)={\displaystyle \frac{1}{2}}\underset{-t}{\overset{t}{\int }}\underset{-\sqrt{t^2-x_3^2}}{\overset{\sqrt{t^2-x_3^2}}{\int }}\left[{\left|u_N\left(\sqrt{t^2-x_2^2-x_3^2},x_2,x_3,t\right)\right|}^2\right.$$

$$\left.+{\left|u_N\left(-\sqrt{t^2-x_2^2-x_3^2},x_2,x_3,t\right)\right|}^2\right]{\displaystyle \frac{t}{\sqrt{t^2-x_2^2-x_3^2}}}d{x}_{2}d{x}_3,$$

where the differential of the surface $d\partial {\mathsf{\Omega }}_t$ on the sphere $\partial {\mathsf{\Omega }}_t$ is determined by the formula

$$d\partial {\mathsf{\Omega }}_t=d{S}_t={\displaystyle \frac{t}{\sqrt{t^2-x_2^2-x_3^2}}}d{x}_{2}d{x}_3.$$

(29)

If the boundary conditions (6) are satisfied, i.e., they are homogeneous, then, from (28), we obtain

$$\begin{array}{c}\left({\partial }_t{u}_N(x,t),u_N(x,t)\right)\\ ={\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\underset{-t}{\overset{t}{\int }}\left[\underset{-\sqrt{t^2-x_3^2}}{\overset{\sqrt{t^2-x_3^2}}{\int }}\left(\underset{-\sqrt{t^2-x_2^2-x_3^2}}{\overset{\sqrt{t^2-x_2^2-x_3^2}}{\int }}{\left|u_N(x,t)\right|}^{2}d{x}_1\right)d{x}_2\right]d{x}_3={\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{∥u_N(x,t)∥}_{{\mathbf{H}}_t}^2,\end{array}$$

(30)

where the term $I\left(t\right)$ in (28) is identically equal to zero.

In our case, according to equality (30), identity (27) will be written as

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{∥u_N(x,t)∥}_{{\mathbf{H}}_t}^2+{∥u_N(x,t)∥}_{{\mathbf{V}}_t}^2=\left(f(x,t),u_N(x,t)\right).$$

(31)

By virtue of equality (31), applying Gronwall’s lemma, we establish the validity of the following lemma.

Lemma 2.

For the Galerkin approximation $u_N(x,t)$ (19) of the initial-boundary value Problem (4)–(7), the following a priori estimate is valid:

$$\parallel u_N{(x,t)\parallel }_{L^{\infty }(t_0,T;{\mathbf{H}}_t)}+\parallel u_N{(x,t)\parallel }_{L^2(t_0,T;{\mathbf{V}}_t)}\le K_1{\parallel f(x,t)\parallel }_{L^2(t_0,T;{\mathbf{H}}_t)},$$

(32)

where the constant value $K_1$ does not depend on N and t.

Lemma 2 allows us to establish the following theorem on the weak solvability of the initial-boundary value problems (4)–(7) ([3], chapter III, §1, Theorem 1.1).

Theorem 1.

Let $f\in L^2(t_0,T;{\mathbf{V}}_t^{\prime }).$ Then, the initial-boundary value problem (4)–(7) has a unique solution $\left\{u\right(x,t),\nabla p(x,t\left)\right\}$, such that

$$u\in W(t_0,T)\equiv \left\{v:v\in L^2(t_0,T;{\mathbf{V}}_t),{\partial }_{t}v\in L^2(t_0,T;{\mathbf{V}}_t^{\prime })\right\},\nabla p\in L^2(t_0,T;H^{-1}\left({\mathsf{\Omega }}_t\right)),$$

(33)

and the following a priori estimates hold:

$${\parallel u\parallel }_{W(t_0,T)}\le K_u{\parallel f\parallel }_{L^2(t_0,T;{\mathbf{V}}_t^{\prime })}{,\parallel \nabla p\parallel }_{L^2(t_0,T;H^{-1}\left({\mathsf{\Omega }}_t\right))}\le K_p{\parallel f\parallel }_{L^2(t_0,T;{\mathbf{V}}_t^{\prime })},$$

(34)

where the constants $K_u$ and $K_p$ depend only on the measure of the domain Ω, $t_0,$ and T.

From (33) and the estimates (34) of Theorem 1, the following corollary follows.

Corollary 1.

Let $f\in L^2(t_0,T;{\mathbf{H}}_t).$ Then, according to the Trace theorem, the following estimates hold:

$$\parallel {\partial }_{\overrightarrow{n}}{u}_N{(x,t)\parallel }_{L^2(t_0,T;{\mathbf{H}}^{-3/2}(\partial {\mathsf{\Omega }}_t))}\le K_{u1}{\parallel f\parallel }_{L^2(t_0,T;{\mathbf{V}}_t^{\prime })}\le K_{u3}{\parallel f\parallel }_{L^2(t_0,T;{\mathbf{H}}_t)},$$

(35)

$$\parallel {\partial }_{\overrightarrow{n}}{p\parallel }_{L^2(t_0,T;H^{-3/2}(\partial {\mathsf{\Omega }}_t))}\le K_{p1}{\parallel f\parallel }_{L^2(t_0,T;{\mathbf{V}}_t^{\prime })}\le K_{p2}{\parallel f\parallel }_{L^2(t_0,T;{\mathbf{H}}_t)},$$

(36)

$$\parallel {\partial }_t{u}_N{(x,t)\parallel }_{L^2(t_0,T;{\mathbf{H}}^{-3/2}(\partial {\mathsf{\Omega }}_t))}\le K_{u2}{\parallel f\parallel }_{L^2(t_0,T;{\mathbf{V}}_t^{\prime })}\le K_{u4}{\parallel f\parallel }_{L^2(t_0,T;{\mathbf{H}}_t)},$$

(37)

where $\overrightarrow{n}$ is the unit outward normal to the boundary $\partial \mathsf{\Omega }$.

Now, let us establish a stronger a priori estimate contained in the statement of the following lemma.

Lemma 3.

For the Galerkin approximation $u_N(x,t)$ (19) of the initial-boundary value problems (4)–(7), the following a priori estimate is valid:

$$\parallel u_N{(x,t)\parallel }_{L^{\infty }(t_0,T;{\mathbf{V}}_t)}+\parallel u_N{(x,t)\parallel }_{L^2(t_0,T;{\mathbf{V}}_t\cap {\mathbf{H}}^2\left({\mathsf{\Omega }}_t\right))}\le K_2{\parallel f(x,t)\parallel }_{L^2(t_0,T;{\mathbf{H}}_t)},$$

(38)

where the constant value $K_2$ does not depend on N and t.

First of all, let us make the following propositions, which will be taken into account in the proof of Lemma 3.

Proposition 3.

According to Theorem 1.5 from ([3], chapter I, §1), for each $t\in (t_0,T),$ the orthogonal complement ${\mathbf{H}}_t^{\perp }$ (to ${\mathbf{H}}_t$ in the space $L^2\left({\mathsf{\Omega }}_t\right)$) decomposes into the direct product ${\mathbf{H}}_t^{\perp }={\mathbf{H}}_{1t}\otimes {\mathbf{H}}_{2t}$, where

$${\mathbf{H}}_{1t}=\left\{u\in L^2\left({\mathsf{\Omega }}_t\right),u=\nabla p,p\in H^1\left({\mathsf{\Omega }}_t\right),\Delta p=0\right\},$$

$${\mathbf{H}}_{2t}=\left\{u\in L^2\left({\mathsf{\Omega }}_t\right),u=\nabla p,p\in H_0^1\left({\mathsf{\Omega }}_t\right)\right\}.$$

Then, the following equalities hold true:

$$\begin{array}{c}-{\left(\Delta u_N(x,t),\nabla p(x,t)\right)}_{L^2\left({\mathsf{\Omega }}_t\right)}\\ =\left\{\begin{array}{cc}-{\left(\Delta u_N,p\right)}_{L^2(\partial {\mathsf{\Omega }}_t)}+{\left(\Delta div{u}_N,p\right)}_{L^2\left({\mathsf{\Omega }}_t\right)}=0,\hfill & \nabla p\in L^2(t_0,T;{\mathbf{H}}_{2t}),\hfill \\ -{\left({\partial }_{\overrightarrow{n}}{u}_N,{\partial }_{\overrightarrow{n}}p\right)}_{H^{-3/2}(\partial {\mathsf{\Omega }}_t)},\hfill & \nabla p\in L^2(t_0,T;{\mathbf{H}}_{1t}),\hfill \end{array}\right\}\mathrm{for}\forall t\in (t_0,T),\end{array}$$

(39)

where $\overrightarrow{n}$ is the unit outward normal to the boundary $\partial {\mathsf{\Omega }}_t$, and, according to the statements of Lemma 2 and inequalities (35)–(36) from Corollary 1, we obtain the following estimate:

$$\left|{\left({\partial }_{\overrightarrow{n}}{u}_N,{\partial }_{\overrightarrow{n}}p\right)}_{H^{-3/2}(\partial {\mathsf{\Omega }}_t)}\right|\le K_{u3}{K}_{p2}{\parallel f\parallel }_{{\mathbf{H}}_t}^2\mathrm{for}\forall t\in (t_0,T),\mathrm{if}\nabla p\in L^2(t_0,T;{\mathbf{H}}_{1t}).$$

(40)

Further, considering the spectral problem (12), we have:

$${\left(\nabla p(x,t),-\Delta w_l(x,t)+\nabla p_l(x,t)\right)}_{L^2\left({\mathsf{\Omega }}_t\right)}$$

$$={\lambda }_l\left(t\right){\left(\nabla p(x,t),w_l(x,t)\right)}_{L^2\left({\mathsf{\Omega }}_t\right)}=0\mathrm{for}\forall t\in (t_0,T),l\in \overline{1,N},$$

or, multiplying the previous equalities by $c_{Nl}\left(t\right)$ and summing the resulting expressions over the index l, we obtain:

$$\begin{array}{c}{\left(\nabla p(x,t),-\Delta u_N(x,t)+\nabla {\displaystyle \sum _{l=1}^N}{c}_{Nl}\left(t\right)p_l(x,t)\right)}_{L^2\left({\mathsf{\Omega }}_t\right)}\\ ={\left(\nabla p(x,t),{\displaystyle \sum _{l=1}^N}{\lambda }_l\left(t\right)c_{Nl}\left(t\right)w_l(x,t)\right)}_{L^2\left({\mathsf{\Omega }}_t\right)}=0\mathrm{for}\forall t\in (t_0,T).\end{array}$$

(41)

Further, from (39)–(41), as well as (33) and the estimates (34) of Theorem 1, we conclude that the vector function

$$P_N(x,t)=\left\{P_{N1}(x,t),P_{N2}(x,t),P_{N3}(x,t)\right\}\equiv \nabla \left(\sum _{l=1}^N{c}_{Nl}\left(t\right)p_l(x,t)\right)$$

(42)

is bounded in the space ${\left(L^2\left({\mathsf{\Omega }}_t\right)\right)}^3$ for $\forall t\in (t_0,T),$ and its norm is bounded by the norm ${\parallel f\parallel }_{{\mathbf{H}}_t}$ for $\forall t\in (t_0,T),$ where $f(x,t)\in L^2(t_0,T;{\mathbf{H}}_t)$.

Proposition 4.

The following relation holds true:

$$\begin{array}{c}{\left({\partial }_t{u}_N(x,t),-\Delta u_N(x,t)+{\displaystyle \sum _{l=1}^N}{c}_{Nl}\left(t\right)\nabla p_l(x,t)\right)}_{L^2\left({\mathsf{\Omega }}_t\right)}=-{\left({\partial }_t{u}_N(x,t),{\partial }_{\overrightarrow{n}}{u}_N(x,t)\right)}_{L^2(\partial {\mathsf{\Omega }}_t)}\\ +{\displaystyle \frac{1}{2}}·{\displaystyle \frac{d}{dt}}{∥\nabla u_N(x,t)∥}_{L^2\left({\mathsf{\Omega }}_t\right)}^2+{\left({\partial }_t{u}_N(x,t),{\displaystyle \sum _{l=1}^N}{c}_{Nl}\left(t\right)\nabla p_l(x,t)\right)}_{L^2\left({\mathsf{\Omega }}_t\right)}.\end{array}$$

(43)

Further, from (34) and (41)–(43), respectively, we have:

$$\parallel {\partial }_t{u}_N{(x,t)\parallel }_{{\mathbf{L}}^2\left({\mathsf{\Omega }}_t\right)}\le K_u{\parallel f\parallel }_{{\mathbf{V}}_t^{\prime }},\parallel P_N{(x,t)\parallel }_{{\mathbf{L}}^2\left({\mathsf{\Omega }}_t\right)}\le K_P{\parallel f\parallel }_{{\mathbf{V}}_t^{\prime }}.$$

(44)

Proposition 5.

The first term in (57) can be written as

$$\begin{array}{c}-\left({\partial }_t{u}_N(x,t),\Delta u_N(x,t)\right)={\displaystyle \frac{1}{2}}\underset{\partial {\mathsf{\Omega }}_t^{+}}{\int }{\left|{\partial }_{x_1}{u}_N\left(\sqrt{t^2-x_2^2-x_3^2},x_2,x_3,t\right)\right|}^2{\displaystyle \frac{t}{\sqrt{t^2-x_2^2-x_3^2}}}d{x}_{2}d{x}_3\\ +{\displaystyle \frac{1}{2}}\underset{\partial {\mathsf{\Omega }}_t^{-}}{\int }{\left|{\partial }_{x_1}{u}_N\left(-\sqrt{t^2-x_2^2-x_3^2},x_2,x_3,t\right)\right|}^2{\displaystyle \frac{t}{\sqrt{t^2-x_2^2-x_3^2}}}d{x}_{2}d{x}_3+{\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{∥\nabla u_N(x,t)∥}_{L^2\left({\mathsf{\Omega }}_t\right)}^2.\end{array}$$

(45)

Proof of Proposition 5.

The equality (45) follows from the following relations: from the boundary condition (6),

$$u_N(x,t)=0\mathrm{on}{\mathsf{\Sigma }}_{xt}=\{x,t:|x|=t,t\in [t_0,T]\},$$

(46)

we obtain

$${\displaystyle \frac{d}{dt}}{u}_N(x,t){{|}_{\left|x\right|=t}={\partial }_t{u}_N(x,t){\mid }_{\left|x\right|=t}+\nabla u_N(x,t){\partial }_{t}x|}_{\left|x\right|=t}=0\mathrm{on}{\mathsf{\Sigma }}_{xt},$$

(47)

where

$$\begin{array}{c}\nabla u_N(x,t){\partial }_t{x|}_{\left|x\right|=t}\\ =\left({\partial }_{x_1}{u}_N(x,t){\mid }_{\left|x\right|=t}{\displaystyle \frac{t}{\sqrt{t^2-x_2^2-x_3^2}}},{\partial }_{x_2}{u}_N(x,t){\mid }_{\left|x\right|=t}·0,{\partial }_{x_3}{u}_N(x,t){\mid }_{\left|x\right|=t}·0\right),\end{array}$$

(48)

i.e., the vector $\nabla u_N(x,t){\partial }_t{x|}_{\left|x\right|=t}$ has the representation

$$\nabla u_N(x,t){\partial }_t{x|}_{\left|x\right|=t}=\left\{{\partial }_{x_1}{u}_N(x,t){\mid }_{\left|x\right|=t}{\displaystyle \frac{t}{\sqrt{t^2-x_2^2-x_3^2}}},0,0\right\},$$

(49)

$$\partial {\mathsf{\Omega }}_t^{\pm }\equiv \left\{\left|x\right|=t\right\}=\left\{x_1=\pm \sqrt{t^2-x_2^2-x_3^2};-\sqrt{t^2-x_3^2}<x_2<\sqrt{t^2-x_3^2};-t<x_3<t\right\},$$

(50)

$${\partial }_t{u}_N(x,t){\mid }_{\left|x\right|=t}=-\left({\partial }_{x_1}{u}_N(x,t){\mid }_{\left|x\right|=t}{\displaystyle \frac{t}{\sqrt{t^2-x_2^2-x_3^2}}},0,0\right).$$

(51)

Thus, from (46)–(51), we have the relation

$$\begin{array}{c}{\left(\nabla u_N(x,t){\partial }_t{x|}_{\left|x\right|=t},\nabla u_N(x,t){\mid }_{\left|x\right|=t}\right)}_{R^3}\\ ={\left|{\partial }_{x_1}{u}_N\left(\pm \sqrt{t^2-x_2^2-x_3^2},x_2,x_3,t\right)\right|}^2{\displaystyle \frac{t}{\sqrt{t^2-x_2^2-x_3^2}}},\end{array}$$

(52)

and $x\in \partial {\mathsf{\Omega }}_t^{\pm }$ (50), respectively, for + and −.

Finally, using the relation (52), we obtain the required equality (45). □

The details of further transformations are summarized in the following proposition.

Proposition 6.

If the boundary conditions (6) are satisfied, i.e., they are homogeneous, then, for the first term on the left-hand side of (57), we will have

$$\begin{array}{c}-{\left({\partial }_t{u}_N(x,t),\Delta u_N(x,t)\right)}_{L^2\left({\mathsf{\Omega }}_t\right)}=-{\left({\partial }_t{u}_N(x,t),\nabla u_N(x,t)\right)}_{L^2(\partial {\mathsf{\Omega }}_t)}\hfill \\ +{\left({\partial }_t\nabla u_N(x,t),\nabla u_N(x,t)\right)}_{L^2\left({\mathsf{\Omega }}_t\right)}={\displaystyle \frac{1}{2}}\underset{\partial {\mathsf{\Omega }}_t^{+}}{\int }{\left|{\partial }_{x_1}{u}_N\left(\sqrt{t^2-x_2^2-x_3^2},x_2,x_3,t\right)\right|}^2\hfill \\ {\displaystyle \frac{t}{\sqrt{t^2-x_2^2-x_3^2}}}d{x}_{2}d{x}_3·{\displaystyle \frac{t}{\sqrt{t^2-x_2^2-x_3^2}}}d{x}_{2}d{x}_3+{\displaystyle \frac{1}{2}}\underset{\partial {\mathsf{\Omega }}_t^{-}}{\int }{\left|{\partial }_{x_1}{u}_N\left(-\sqrt{t^2-x_2^2-x_3^2},x_2,x_3,t\right)\right|}^2\hfill \\ +{\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{∥\nabla u_N(x,t)∥}_{L^2\left({\mathsf{\Omega }}_t\right)}^2,\hfill \end{array}$$

(53)

where, using ([23], sec. 626, formula (5), p. 257), for the surface differential $d\partial {\mathsf{\Omega }}_t$ on the sphere $\partial {\mathsf{\Omega }}_t,$ we obtain

$$d\partial {\mathsf{\Omega }}_t=d{S}_t={\displaystyle \frac{t}{\sqrt{t^2-x_2^2-x_3^2}}}d{x}_{2}d{x}_3.$$

Next, from (53), we obtain

$$-\left({\partial }_t{u}_N(x,t),\Delta u_N(x,t)\right)={\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\underset{{\mathsf{\Omega }}_t}{\int }{\left|\nabla u_N(x,t)\right|}^{2}d{\mathsf{\Omega }}_t+{\displaystyle \frac{1}{2}}\underset{\partial {\mathsf{\Omega }}_t}{\int }{\left|{\partial }_{x_1}{u}_N(x,t)\right|}^{2}d\partial {\mathsf{\Omega }}_t.$$

(54)

Now, from (57) and (54), it directly follows that

$$\begin{array}{c}{\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{∥\nabla u_N(x,t)∥}_{L^2\left({\mathsf{\Omega }}_t\right)}^2+{∥\Delta u_N(x,t)∥}_{L^2\left({\mathsf{\Omega }}_t\right)}^2\hfill \\ +{\left({\partial }_t{u}_N(x,t)-\Delta u_N(x,t),{\displaystyle \sum _{l=1}^N}{c}_{Nl}\left(t\right)\nabla p_l(x,t)\right)}_{L^2\left({\mathsf{\Omega }}_t\right)}\hfill \\ =-{\displaystyle \frac{1}{2}}\underset{\partial {\mathsf{\Omega }}_t}{\int }{\left|{\partial }_{x_1}{u}_N(x,t)\right|}^{2}d\partial {\mathsf{\Omega }}_t-{\left(f(x,t),\Delta u_N(x,t)\right)}_{L^2\left({\mathsf{\Omega }}_t\right)}\hfill \\ +{\left(f(x,t),{\displaystyle \sum _{l=1}^N}{c}_{Nl}\left(t\right)\nabla p_l(x,t)\right)}_{L^2\left({\mathsf{\Omega }}_t\right)}.\hfill \end{array}$$

(55)

Remark 1.

We have already established the estimates for the terms in the second row and the last term of relation (55). These are given by expressions (33)–(36), (39)–(42), and (44).

Therefore, it is sufficient to only establish the estimates for the two boundary integrals on the right-hand side of the equality in relation (55). For this purpose, it will be convenient to introduce the following notations for the sets:

$$X_1=\left(0,\sqrt{t^2-x_2^2-x_3^2}\right),X_{23}=\left(-\sqrt{t^2-x_3^2},\sqrt{t^2-x_3^2}\right)\times (-t,t).$$

Let us establish the estimate for the first term on the right-hand side of the equality in (55). Using the interpolation inequality from ([24], Theorem 5.9, pp. 140–141), we have

$$\begin{array}{c}{\int }_{\partial {\mathsf{\Omega }}_t^{+}}{\left|{\partial }_{x_1}{u}_N\left(\sqrt{t^2-x_2^2-x_3^2},x_2,x_3,t\right)\right|}^{2}d\partial {\mathsf{\Omega }}_t\le {∥{\partial }_{x_1}{u}_N(x_1,x_2,x_3,t)∥}_{L^{\infty }\left(X_1;L^2\left(X_{23}\right)\right)}^2\\ \le {\displaystyle \frac{K}{2}}{∥{\partial }_{x_1}{u}_N(x,t)∥}_{W_2^1\left(X_1;L^2\left(X_{23}\right)\right)}{∥{\partial }_{x_1}{u}_N(x,t)∥}_{L^2\left(X_1;L^2\left(X_{23}\right)\right)}\\ \le {\displaystyle \frac{K}{2}}\left[{∥{\partial }_{x_1}^2{u}_N(x,t)∥}_{L^2\left(X_1;L^2\left(X_{23}\right)\right)}+{∥{\partial }_{x_1}{u}_N(x,t)∥}_{L^2\left(X_1;L^2\left(X_{23}\right)\right)}\right]{∥{\partial }_{x_1}{u}_N(x,t)∥}_{L^2\left(X_1;L^2\left(X_{23}\right)\right)}\\ \le {\displaystyle \frac{K}{2}}{∥\Delta u_N(x,t)∥}_{L^2\left({\mathsf{\Omega }}_t\right)}{∥\nabla u_N(x,t)∥}_{L^2\left({\mathsf{\Omega }}_t\right)}+{\displaystyle \frac{K}{2}}{∥\nabla u_N(x,t)∥}_{L^2\left({\mathsf{\Omega }}_t\right)}^2.\end{array}$$

(56)

It should be noted that we can establish a similar estimate to (56) for the second term on the right-hand side of the equality in (55) as well.

Proof of Lemma 3.

In (20), by replacing the factor $w_l(x,t)$ with ${\lambda }_l\left(t\right)w_l(x,t)=-\Delta w_l(x,t)+\nabla p_l(x,t),t\in (t_0,T)$, we obtain

$$\begin{array}{c}{\left({\partial }_t{u}_N(x,t),-\Delta w_l(x,t)+\nabla p_l(x,t)\right)}_{L^2\left({\mathsf{\Omega }}_t\right)}+{\left(-\Delta u_N(x,t),-\Delta w_l(x,t)+\nabla p_l(x,t)\right)}_{L^2\left({\mathsf{\Omega }}_t\right)}\\ ={\left(f(x,t),-\Delta w_l(x,t)+\nabla p_l(x,t)\right)}_{L^2\left({\mathsf{\Omega }}_t\right)},t\in (t_0,T).\end{array}$$

(57)

Let $\nabla p_l\in {\mathbf{H}}_{2t}$. Using the relations from Proposition 3, multiplying the equality (57) by $c_{Nl}\left(t\right)$ and summing the resulting N equations over the index l, we obtain

$$\begin{array}{c}{\left({\partial }_t{u}_N(x,t),-\Delta u_N(x,t)+{\displaystyle \sum _{l=1}^N}{c}_{Nl}\left(t\right)\nabla p_l(x,t)\right)}_{L^2\left({\mathsf{\Omega }}_t\right)}+{∥\Delta u_N(x,t)∥}_{L^2\left({\mathsf{\Omega }}_t\right)}^2\\ \le \left|{\left(f(x,t),-\Delta u_N(x,t)+{\displaystyle \sum _{l=1}^N}{c}_{Nl}\left(t\right)\nabla p_l(x,t)\right)}_{L^2\left({\mathsf{\Omega }}_t\right)}\right|\le \left|{\left(f(x,t),-\Delta u_N(x,t)\right)}_{L^2\left({\mathsf{\Omega }}_t\right)}\right|\\ +\left|{\left(f(x,t),{\displaystyle \sum _{l=1}^N}\left|c_{Nl}\left(t\right)\nabla p_l(x,t)\right|\right)}_{L^2\left({\mathsf{\Omega }}_t\right)}\right|,t\in (t_0,T).\end{array}$$

(58)

Now, using (41)–(44), from (58), we obtain

$$\begin{array}{c}{\displaystyle \frac{1}{2}}·{\displaystyle \frac{d}{dt}}{∥\nabla u_N(x,t)∥}_{L^2\left({\mathsf{\Omega }}_t\right)}^2+{∥\Delta u_N(x,t)∥}_{L^2\left({\mathsf{\Omega }}_t\right)}^2\\ ={\left({\partial }_t{u}_N(x,t),{\partial }_{\overrightarrow{n}}{u}_N(x,t)\right)}_{L^2(\partial {\mathsf{\Omega }}_t)}-{\left({\partial }_t{u}_N(x,t),{\displaystyle \sum _{l=1}^N}{c}_{Nl}\left(t\right)\nabla p_l(x,t)\right)}_{L^2\left({\mathsf{\Omega }}_t\right)}\\ \le \left|{\left(f(x,t),-\Delta u_N(x,t)\right)}_{L^2\left({\mathsf{\Omega }}_t\right)}\right|+\left|{\left(f(x,t),{\displaystyle \sum _{l=1}^N}{c}_{Nl}\left(t\right)\nabla p_l(x,t)\right)}_{L^2\left({\mathsf{\Omega }}_t\right)}\right|,t\in (t_0,T).\end{array}$$

(59)

Here, we used Propositions 5 and 6.

From (58), using the estimates (34), (35), and (37), as well as (41) and (42), we obtain the statement of Lemma 3 for the case when $\nabla p_l\in {\mathbf{H}}_{2t}$.

Now, let $\nabla p_l\in {\mathbf{H}}_{1t}$. In this case, we additionally need to estimate the following term (see Equation (39)):

$${\left({\partial }_{\overrightarrow{n}}{u}_N,{\partial }_{\overrightarrow{n}}p\right)}_{H^{-3/2}(\partial {\mathsf{\Omega }}_t)},t\in (t_0,T).$$

(60)

The necessary estimates for the factors in (60) follow from (35) and (36). Thus, we obtain the statement of Lemma 3 for the case when $\nabla p_l\in {\mathbf{H}}_{1t}$.

Thus, according to (56) and Remark 1, from (55), we obtain

$$\begin{array}{c}{\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{∥\nabla u_N(x,t)∥}_{L^2\left({\mathsf{\Omega }}_t\right)}^2+{∥\Delta u_N(x,t)∥}_{L^2\left({\mathsf{\Omega }}_t\right)}^2\le K{∥\Delta u_N(x,t)∥}_{L^2\left({\mathsf{\Omega }}_t\right)}{∥\nabla u_N(x,t)∥}_{L^2\left({\mathsf{\Omega }}_t\right)}\\ +K{∥\nabla u_N(x,t)∥}_{L^2\left({\mathsf{\Omega }}_t\right)}^2+\left|{\left(f(x,t),\Delta u_N(x,t)\right)}_{L^2\left({\mathsf{\Omega }}_t\right)}\right|\\ +\left[K_u{K}_p+K_{u1}{K}_{p1}+C{K}_{p1}\right]{∥f(x,t)∥}_{{\mathbf{H}}_t}^2.\end{array}$$

(61)

Next, in Equation (61), by applying the Cauchy $\epsilon$-inequality and Gronwall’s inequality, we establish the required a priori estimate (38). □

Lemma 4.

For the Galerkin approximation $u_N(x,t)$ (19) of the initial-boundary value problem (4)–(7), the following a priori estimate holds:

$$\parallel {\partial }_t{u}_N{(x,t)\parallel }_{L^2(t_0,T;{\mathbf{H}}_t)}\le K_3{\parallel f(x,t)\parallel }_{L^2(t_0,T;{\mathbf{H}}_t)},$$

(62)

where the constant $K_3$ is independent of N and t.

Proof of Lemma 4.

Taking into account the validity of equality (21), we replace the factor $w_l(x,t)$ in (27) with ${\partial }_t{u}_N(x,t)$. Since, for the cross-section of the cone ${\mathsf{\Omega }}_t=\left\{\right|x|<t\}$, the inequality ${\displaystyle \sum _{j=1}^3}{\left|{\displaystyle \frac{x_j}{t}}\right|}^2<1$ holds, we obtain

$$\begin{array}{c}{∥{\partial }_t{u}_N(x,t)∥}_{L^2\left({\mathsf{\Omega }}_t\right)}^2-\left(\Delta u_N(x,t),{\partial }_t{u}_N(x,t)\right)\\ \le \left|\left(f(x,t),{\partial }_t{u}_N(x,t)\right)\right|+{∥\nabla p(x,t)∥}_{{\mathbf{V}}_t^{\prime }}{∥\nabla u_N(x,t)∥}_{{\mathbf{V}}_t},t\in (t_0,T).\end{array}$$

(63)

Here, the following relation is used:

$$\left|\sum _{l=1}^N{c}_{Nl}\left(t\right){\partial }_t{w}_l(x,t)\right|=\left|-\sum _{l=1}^N{c}_{Nl}\left(t\right)\sum _{j=1}^3{\displaystyle \frac{x_j}{t}}{\partial }_{x_j}{w}_l(x,t)\right|$$

$$=\left|-\sum _{j=1}^3{\displaystyle \frac{x_j}{t}}\sum _{l=1}^N{c}_{Nl}\left(t\right){\partial }_{x_j}{w}_l(x,t)\right|\le \left|\sum _{l=1}^N{c}_{Nl}\left(t\right){\partial }_{x_j}{w}_l(x,t)\right|\le \left|\nabla u_N(x,t)\right|.$$

Furthermore, according to the assertion of Theorem 1, for the last term in (63), we have the estimates

$${∥\nabla p(x,t)∥}_{{\mathbf{V}}_t^{\prime }}\le C_{p1}{∥\nabla p(x,t)∥}_{H^{-1}\left({\mathsf{\Omega }}_t\right)}\le C_p{\parallel f(x,t)\parallel }_{{\mathbf{H}}_t},t\in (t_0,T),$$

(64)

and, accordingly, by Lemma 2, we obtain

$${∥\nabla u_N(x,t)∥}_{{\mathbf{V}}_t}\le K_1{\parallel f(x,t)\parallel }_{{\mathbf{H}}_t},t\in (t_0,T).$$

(65)

Using the estimates (64)–(65), as well as relation (45), from (63), we obtain

$$\begin{array}{c}{∥{\partial }_t{u}_N(x,t)∥}_{L^2\left({\mathsf{\Omega }}_t\right)}^2+{\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{∥\nabla u_N(x,t)∥}_{L^2\left({\mathsf{\Omega }}_t\right)}^2\\ =-{\displaystyle \frac{1}{2}}\underset{\partial {\mathsf{\Omega }}_t^{+}}{\int }{\left|{\partial }_{x_1}{u}_N\left(\sqrt{t^2-x_2^2-x_3^2},x_2,x_3,t\right)\right|}^{2}d\partial {\mathsf{\Omega }}_t^{+}\\ -{\displaystyle \frac{1}{2}}\underset{\partial {\mathsf{\Omega }}_t^{-}}{\int }{\left|{\partial }_{x_1}{u}_N\left(-\sqrt{t^2-x_2^2-x_3^2},x_2,x_3,t\right)\right|}^{2}d\partial {\mathsf{\Omega }}_t^{-}\\ +\left(f(x,t),{\partial }_t{u}_N(x,t)\right)+{∥\nabla p(x,t)∥}_{{\mathbf{V}}_t^{\prime }}{∥\nabla u_N(x,t)∥}_{{\mathbf{V}}_t}.\end{array}$$

(66)

Next, based on the estimate (56), from (66), it follows

$$\begin{array}{c}{∥{\partial }_t{u}_N(x,t)∥}_{L^2\left({\mathsf{\Omega }}_t\right)}^2+{\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{∥\nabla u_N(x,t)∥}_{L^2\left({\mathsf{\Omega }}_t\right)}^2\le K{∥\Delta u_N(x,t)∥}_{L^2\left({\mathsf{\Omega }}_t\right)}{∥\nabla u_N(x,t)∥}_{L^2\left({\mathsf{\Omega }}_t\right)}\\ +K{∥\nabla u_N(x,t)∥}_{L^2\left({\mathsf{\Omega }}_t\right)}^2+\left(f(x,t),{\partial }_t{u}_N(x,t)\right)+{∥\nabla p(x,t)∥}_{{\mathbf{V}}_t^{\prime }}{∥\nabla u_N(x,t)∥}_{{\mathbf{V}}_t}.\end{array}$$

(67)

Using the assertions of Lemmas 2 and 3 to estimate the terms ${∥\nabla u_N(x,t)∥}_{L^2\left(Q_{t_0}\right)}$ and ${∥\Delta u_N(x,t)∥}_{L^2\left(Q_{t_0}\right)}$ in the right-hand side of (67), respectively, as well as Gronwall’s lemma, from (67), we establish the required estimate of Lemma 4. □

5. Unique Solvability of the Problem 1 (4)–(7)

The following theorem is true.

Theorem 2.

Let $f\in L^2(t_0,T;{\mathbf{H}}_t).$ Then, the initial-boundary value problem (4)–(7) has a unique solution $\left\{u(x,t)\in W_2^1(t_0,T;{\mathbf{H}}_t)\cap L^2(t_0,T;{\mathbf{H}}^2\left({\mathsf{\Omega }}_t\right)\cap {\mathbf{H}}_0^1\left({\mathsf{\Omega }}_t\right))\right.$, $\left.\nabla p\in L^2(t_0,T;{\mathbf{H}}_t)\right\},$ for which the following a priori estimates are met:

$${\parallel u(x,t)\parallel }_{W_2^1(t_0,T;{\mathbf{H}}_t)\cap L^2(t_0,T;{\mathbf{H}}^2\left({\mathsf{\Omega }}_t\right)\cap {\mathbf{H}}_0^1\left({\mathsf{\Omega }}_t\right))}\le C_1{\parallel f(x,t)\parallel }_{L^2(t_0,T;{\mathbf{H}}_t)},$$

$${\parallel \nabla p(x,t)\parallel }_{L^2(t_0,T;{\mathbf{H}}_t)}\le C_2{\parallel f(x,t)\parallel }_{L^2(t_0,T;{\mathbf{H}}_t)},$$

where the constant values $C_1$ and $C_2$ do not depend on $T.$

Proof of Theorem 2.

From the a priori estimates established in the previous section, we obtain the following bounded sequences:

$${\left\{u_N(x,t)\right\}}_{N=1}^{\infty }\subset L^{\infty }(t_0,T;{\mathbf{H}}_t)\cap L^2(t_0,T;{\mathbf{V}}_t),$$

(68)

$${\left\{u_N(x,t)\right\}}_{N=1}^{\infty }\subset L^{\infty }(t_0,T;{\mathbf{V}}_t)\cap L^2(t_0,T;{\mathbf{V}}_t\cap {\mathbf{H}}^2\left({\mathsf{\Omega }}_t\right)),$$

(69)

$${\left\{{\partial }_t{u}_N(x,t)\right\}}_{N=1}^{\infty }\subset L^2(t_0,T;{\mathbf{H}}_t).$$

(70)

From (68)–(70), it follows that there exist the following weakly convergent subsequences such that

$$u_{\mu }(x,t)\rightharpoonup z_1(x,t)*–\mathrm{weakly}\mathrm{in}{L}^{\infty }(t_0,T;{\mathbf{H}}_t)\cap L^2(t_0,T;{\mathbf{V}}_t),$$

(71)

$$u_{\mu }(x,t)\rightharpoonup z_2(x,t)*–\mathrm{weakly}\mathrm{in}{L}^{\infty }(t_0,T;{\mathbf{V}}_t)\cap L^2(t_0,T;{\mathbf{V}}_t\cap {\mathbf{H}}^2\left({\mathsf{\Omega }}_t\right)),$$

(72)

$${\partial }_t{u}_{\mu }(x,t)\rightharpoonup {\partial }_t{z}_3(x,t)\mathrm{weakly}\mathrm{in}{L}^2(t_0,T;{\mathbf{H}}_t),$$

(73)

moreover, the functions $z_1(x,t),$$z_2(x,t)$, and $z_3(x,t),$ as is known for (weakly) convergent sequences, will coincide, i.e., $z(x,t)\equiv z_1(x,t)\equiv z_2(x,t)\equiv z_3(x,t)$ on $Q_{t_0}$, possibly excluding a set of measure zero.

By analogy with relation (27), we write the initial-boundary value problem (4)–(7) in variational form. We have

$$\left({\partial }_t{u}_{\mu }(x,t),w(x,t)\right)-\left(\Delta u_{\mu }(x,t),w(x,t)\right)=\left(f(x,t),w(x,t)\right),\forall w(x,t)\in {\mathbf{H}}_t,t\in (t_0,T),$$

(74)

$$u_{\mu }(x,t_0)=0\forall \mu .$$

(75)

In (74)–(75), we can pass to the limit as $\mu \to \infty$; as a result, we obtain

$$\left({\partial }_{t}z(x,t),w(x,t)\right)-\left(\Delta z(x,t),w(x,t)\right)=\left(f(x,t),w(x,t)\right),\forall w(x,t)\in {\mathbf{H}}_t,t\in (t_0,T),$$

(76)

$$z(x,t_0)=0.$$

(77)

On the other hand, proofs of a priori estimates that we established for the Galerkin approximations (Lemmas 2–4) are carried over word for word to the function $z(x,t)$. From a priori estimates, we obtain the uniqueness of the solution to the initial-boundary value problem (4)–(7), i.e., the weak limit $z(x,t)$ coincides with the solution $u(x,t)$ of problem (4)–(7). As for the gradient components of the unknown pressure function $\nabla p(x,t)$, they are found from Equation (4) for the found fluid velocity $u(x,t)=\{u^{\left(1\right)}(x,t),u^{\left(2\right)}(x,t),u^{\left(3\right)}(x,t)\}$. An a priori estimate for the pressure gradient $\nabla p(x,t)$ is also established using Equation (4) and a priori estimates for the fluid velocity (Lemmas 2–4). Theorem 2 is completely proven. □

6. Unique Solvability of the Problem 2 (8)–(10)

The following theorem is true.

Theorem 3

(Main result). Let $f\in L^2(0,T;{\mathbf{H}}_t).$ Then, boundary value problem (8)–(10) has a unique solution

$$\left\{u(x,t)\in W_2^1(0,T;{\mathbf{H}}_t)\cap L^2(0,T;{\mathbf{H}}^2\left({\mathsf{\Omega }}_t\right)\cap {\mathbf{H}}_0^1\left({\mathsf{\Omega }}_t\right)),\nabla p\in L^2(0,T;{\mathbf{H}}_t)\right\},$$

for which the following a priori estimates are met:

$${\parallel u(x,t)\parallel }_{W_2^1(0,T;{\mathbf{H}}_t)\cap L^2(0,T;{\mathbf{H}}^2\left({\mathsf{\Omega }}_t\right)\cap {\mathbf{H}}_0^1\left({\mathsf{\Omega }}_t\right))}\le C_1{\parallel f(x,t)\parallel }_{L^2(0,T;{\mathbf{H}}_t)},$$

$${\parallel \nabla p(x,t)\parallel }_{L^2(0,T;{\mathbf{H}}_t)}\le C_2{\parallel f(x,t)\parallel }_{L^2(0,T;{\mathbf{H}}_t)},$$

where the constants $C_1$ and $C_2$ do not depend on $T.$

Problem 3

(Family of m-problems). In truncated cones $\{Q_{t_m}:|x|<t,t\in (t_m,T)\},$ where $\{t_m=1/m,m=m_1,m_1+1,m_1+2,...,\},$$1/m_1<T\}$, consider initial-boundary value problems for the linear 3D Navier–Stokes system of determining the vector function $u^m(x,t)=\left\{u^{\left(1m\right)}(x,t),u^{\left(2m\right)}(x,t),u^{\left(3m\right)}(x,t)\right\}$ and the scalar function $p^m(x,t)$:

$${\partial }_t{u}^m-\Delta u^m=f^m-\nabla p^m,(x,t)\in Q_{t_m},$$

(78)

$$div{u}^m=0,(x,t)\in Q_{t_m},$$

(79)

$$u^m=0,(x,t)\in {\mathsf{\Sigma }}_{t_m},$$

(80)

$$u^m=0,x\in {\mathsf{\Omega }}_{t_m}\mathit{is}a\mathit{ball}\mathit{of}\mathit{radius}{t}_m,i.e.,\mathit{the}\mathit{lower}\mathit{base}\mathit{of}\mathit{the}\mathit{cone},$$

(81)

where ${\mathsf{\Sigma }}_{t_m}=\{x,t_m:\left|x\right|=t,t\in (t_m,T)\}\mathit{is}\mathit{the}\mathit{lateral}\mathit{surface}\mathit{of}\mathit{the}\mathit{cone}\mathit{and}$ $f^m=\{$restriction of a given function $f\mathit{on}{Q}_{t_m}.\}$

For each of the family of initial-boundary value problems (78)–(81), the statement of Theorem 2 is true.

Theorem 4.

Let $f^m\in L^2(t_m,T;{\mathbf{H}}_t).$ Then, the initial-boundary value problem (78)–(81) has a unique solution

$$\left\{u^m(x,t)\in W_2^1(t_m,T;{\mathbf{H}}_t)\cap L^2(t_m,T;{\mathbf{H}}^2\left({\mathsf{\Omega }}_t\right)\cap {\mathbf{H}}_0^1\left({\mathsf{\Omega }}_t\right)),\nabla p\in L^2(t_m,T;{\mathbf{H}}_t)\right\},$$

for which the following a priori estimates hold:

$$\parallel u^m{(x,t)\parallel }_{W_2^1(t_m,T;{\mathbf{H}}_t)\cap L^2(t_m,T;{\mathbf{H}}^2\left({\mathsf{\Omega }}_t\right)\cap {\mathbf{H}}_0^1\left({\mathsf{\Omega }}_t\right))}\le C_1\parallel f^m{(x,t)\parallel }_{L^2(t_m,T;{\mathbf{H}}_t)}\le C_1{\parallel f(x,t)\parallel }_{L^2(0,T;{\mathbf{H}}_t)},$$

(82)

$$\parallel \nabla p^m{(x,t)\parallel }_{L^2(t_m,T;{\mathbf{H}}_t)}\le C_2\parallel f^m{(x,t)\parallel }_{L^2(t_m,T;{\mathbf{H}}_t)}\le C_2{\parallel f(x,t)\parallel }_{L^2(0,T;{\mathbf{H}}_t)},$$

(83)

where the constants $C_1$ and $C_2$ do not depend on m and $t.$

Proof of Theorem 3.

We extend each of the functions $\left\{u^m(x,t),\nabla p^m(x,t),f^m(x,t),\right.$$m=m_1,m_1+1,m_1$$\left.+2,\dots \right\}$ by zero onto the cone, denoting them by $\left\{\widetilde{u^m(x,t)},\widetilde{\nabla p^m(x,t)},\right.$ $\widetilde{f^m(x,t)},m=m_1,$ $m_1+1,$ $\left.m_1+2,...\right\}.$ These extended functions will satisfy the initial-boundary value problem (4)–(7) in variational form:

$$\left({\partial }_t\widetilde{u^m(x,t)}-\Delta \widetilde{u^m(x,t)},w(x,t)\right)=\left(\widetilde{f^m(x,t)},w(x,t)\right),\forall w(x,t)\in {\mathbf{H}}_t,t\in (t_0,T),$$

(84)

$$\widetilde{u^m(x,t_m)}=0\forall m,$$

(85)

and a priori estimates (82)–(83). Thus, we obtain bounded sequences $\left\{\widetilde{u^m(x,t)},\right.$ $\widetilde{\nabla p^m(x,t)}$, $m=m_1,m_1+1,$ $\left.m_1+2,\dots \right\}$ in the space

$$W_2^1(0,T;{\mathbf{H}}_t)\cap L^2(0,T;{\mathbf{H}}^2\left({\mathsf{\Omega }}_t\right)\cap {\mathbf{H}}_0^1\left({\mathsf{\Omega }}_t\right))\times L^2(0,T;{\mathbf{H}}_t),$$

from which we can extract weakly convergent subsequences

$$\left\{\widetilde{u^{\mu }(x,t)},\widetilde{\nabla p^{\mu }(x,t)},\mu =1,2,3,...\right\},$$

i.e., the following take place:

$$\widetilde{u^{\mu }(x,t)}\rightharpoonup U(x,t)\mathrm{weakly}\mathrm{in}{W}_2^1(0,T;{\mathbf{H}}_t)\cap L^2(0,T;{\mathbf{H}}^2\left({\mathsf{\Omega }}_t\right)\cap {\mathbf{H}}_0^1\left({\mathsf{\Omega }}_t\right)),$$

(86)

$$\widetilde{u^{\mu }(x,t)}\to U(x,t)\mathrm{strongly}\mathrm{in}{L}^2(0,T;{\mathbf{H}}_t)\cap L^2(0,T;{\mathbf{H}}_0^1\left({\mathsf{\Omega }}_t\right)),$$

(87)

$$\widetilde{\nabla p^{\mu }(x,t)}\rightharpoonup S(x,t)\equiv \nabla P(x,t)\mathrm{weakly}\mathrm{in}{L}^2(0,T;{\mathbf{H}}_t).$$

(88)

In Equations (84) and (85), replacing m with $\mu$, we can pass to the limit as $\mu \to \infty$, and, as a result, we obtain

$$\left({\partial }_{t}U(x,t),w(x,t)\right)-\left(\Delta U(x,t),w(x,t)\right)=\left(f(x,t),w(x,t)\right),\forall w(x,t)\in {\mathbf{H}}_t,t\in (t_0,T),$$

(89)

$$U(0,0)=0.$$

(90)

The components of the pressure gradient $S(x,t)$ are found in Equation (8). Thus, the existence of a solution to the boundary value problem (8)–(10) is proven. It remains to show the uniqueness of the solution, which follows from a priori estimates by contradiction. Assuming that the boundary value problem (8)–(10) has two different solutions, we obtain from a priori estimates that their difference is identically equal to zero. This implies the uniqueness statement in Theorem 3.

Theorem 3 is completely proven. □

7. Conclusions

In this work, the unique solvability of the initial-boundary value problem for the 3D time-dependent Stokes system in the truncated cone $Q_{t_0}$ is investigated. The unique solvability of the boundary value problem in the case of a cone, when the domain Q of independent variables $(x,t)$ degenerates at the initial moment of time at the top of the cone, is separately studied. The results of the work can also be developed for the case when the lengths of the radii of sections ${\mathsf{\Omega }}_t$ of the cone Q change according to the nonlinear law $\phi \left(t\right),t\in [0,T]$. In this case, it is enough to impose the following conditions on the function $\phi \left(t\right)$:

$$1^0.\phi \left(t\right)\in C^1\left([0,T]\right),\phi \left(0\right)=0,{\phi }^{\prime }\left(0\right)\ge C_0=\mathrm{const}>0,$$

$$2^0.\phi \left(t\right)\mathrm{is}\mathrm{a}\mathrm{non}\text{-}\mathrm{decreasing}\mathrm{function}\mathrm{on}\mathrm{the}\mathrm{interval}[0,T],$$

and the differential of the surface $d\partial {\mathsf{\Omega }}_t$ (29), in this case, on the sphere $\partial {\mathsf{\Omega }}_t$, will be determined by the formula

$$d\partial {\mathsf{\Omega }}_t=d{S}_t={\displaystyle \frac{\phi \left(t\right){\phi }^{\prime }\left(t\right)}{\sqrt{{\left[\phi \left(t\right)\right]}^2-x_2^2-x_3^2}}}d{x}_{2}d{x}_3.$$

It should be noted that, in the case we considered in this work, $\phi \left(t\right)=t.$ Obviously, conditions $1^0$ and $2^0$ are met.