A Thermodynamically Consistent Master Equation Framework for the Two-Dimensional Incompressible Navier–Stokes Equations: Derivation, Convergence, and Global Regularity

Chapter 1 Introduction

The incompressible Navier–Stokes equations (hereafter referred to as the NS equations) constitute the fundamental governing equations for viscous fluid motion and play a central role across a wide range of disciplines in continuum mechanics and engineering [1,2,3,4]. Their mathematical analysis, particularly concerning the existence, uniqueness, and regularity of solutions, has been a major subject of investigation since the seminal introduction of weak solutions by Leray in the early twentieth century [5,6]. In particular, the global existence of smooth solutions in three dimensions remains one of the most prominent open problems in mathematical physics and has been designated as a Millennium Prize Problem by the Clay Mathematics Institute [7].

In contrast, the two-dimensional case exhibits a fundamentally different structure. In two dimensions, vorticity behaves as a scalar quantity, and the absence of the vortex stretching term enables control of the enstrophy. Owing to this structural property, not only global existence of weak solutions but also uniqueness and regularity are known to hold in the two-dimensional setting [8,9,10,11]. In this sense, the two-dimensional Navier–Stokes equations represent a completely resolved system from the standpoint of existence theory.

Despite this complete resolution, the classical proofs of existence rely predominantly on functional analytic techniques, including Galerkin approximations, energy inequalities, compactness arguments such as the Aubin–Lions lemma, and delicate treatments of the weak convergence of nonlinear terms [12,13,14,15]. The essential structure of these proofs is based on successive estimates and inequalities. Consequently, it cannot be said that the intrinsic physical structure of the equations—particularly conservation laws and dissipative mechanisms—has been explicitly incorporated as a central organizing principle in the construction of the existence theory.

From a physical standpoint, fluid equations are inherently governed by the interplay between conservation laws and irreversible processes. Specifically, the following principles simultaneously hold: the conservation of mass, momentum, and energy; the decay of energy due to viscous dissipation; and the second law of thermodynamics, expressed through entropy increase. In recent years, a unified framework capable of consistently describing these features has been proposed in the form of the GENERIC (General Equation for Non-Equilibrium Reversible–Irreversible Coupling) formalism. Within this framework, a wide class of physical systems can be represented through a decomposition into reversible (antisymmetric) and irreversible (symmetric positive semidefinite) structures [16,17,18]. Similarly, in the context of the Boltzmann equation and its hydrodynamic limits, entropy production plays a fundamental role in determining the macroscopic structure of the equations [19,20,21,22].

However, in the existence theory of the Navier–Stokes equations, such thermodynamic structures have not been systematically employed as a guiding principle for constructing proofs. In the classical approach, conservation laws and dissipative structures appear as properties that are verified a posteriori, rather than being explicitly embedded into the formulation from the outset. As a result, the logical structure of the theory remains largely analytical rather than structural.

The present work adopts a fundamentally different standpoint by incorporating conservation laws and the second law of thermodynamics directly at the level of equation construction. Specifically, a compressible master equation defined on a discrete network is introduced, where the state variables consist of density, momentum, and internal energy. The dynamics are decomposed into a reversible component represented by an antisymmetric operator, corresponding to energy-conserving interactions, and an irreversible component represented by a symmetric positive semidefinite operator, corresponding to entropy-producing dissipative processes [23,24,25,26,27]. Within this formulation, conservation laws are guaranteed by construction, energy dissipation emerges naturally from the properties of the operators, and the second law is encoded as the positivity of entropy production. These features are therefore not statements to be proved but direct consequences of the structural design of the system.

The central perspective of this work is to demonstrate that the incompressible Navier–Stokes equations arise naturally as a limiting case of this thermodynamically consistent master equation. Starting from the compressible formulation, the continuum limit and the low Mach number limit are considered. Under appropriate scaling, an incompressible system emerges in a manner analogous to the derivation of Euler and Navier–Stokes equations from the Boltzmann equation [19,20,28]. Although the limiting process is presented primarily at a formal level, the essential point is that the incompressible Navier–Stokes equations appear as a natural consequence of an underlying thermodynamically consistent parent system, rather than as an independently postulated model.

The objective of this paper is to establish, starting from such a thermodynamically consistent compressible master equation, a rigorous existence theory for the two-dimensional incompressible Navier–Stokes equations corresponding to its incompressible limit. More precisely, the following program is carried out. First, a compressible master equation satisfying both conservation laws and the second law of thermodynamics is formulated. Second, its continuum limit and incompressible limit are derived, leading to a reduced incompressible system. Third, a structure-preserving discrete approximation of this reduced system is constructed. Fourth, energy and enstrophy estimates are established at the discrete level. Fifth, the existence of weak solutions is obtained using compactness arguments of Aubin–Lions type. Finally, the specific structural properties of two-dimensional flows are exploited to upgrade weak solutions to strong solutions.

It is emphasized that the purpose of this work is not to establish a new existence result for the two-dimensional Navier–Stokes equations themselves. Rather, the essential significance lies in reconstructing the existence theory within a unified framework grounded in thermodynamic structure. From this perspective, the classical results are recovered not as isolated analytical achievements but as consequences of a more fundamental structural formulation.

The significance of the present approach can be summarized as follows. First, the existence theory is reformulated on the basis of structural principles, replacing the traditional reliance on analytical inequalities with a framework rooted in conservation laws and the second law of thermodynamics. Second, the incompressible Navier–Stokes equations are positioned as a limiting system of a compressible, thermodynamically consistent parent equation, thereby clarifying their physical origin. Third, in the two-dimensional setting, the full structure of the theory—from the existence of weak solutions to their promotion to strong solutions—is verified within a single coherent framework.

The remainder of this paper is organized as follows. Section 2 formulates the thermodynamically consistent compressible master equation and establishes its structural properties, including conservation laws and entropy production. Section 3 derives the continuum and incompressible limits, leading to the reduced incompressible system. Section 4 introduces a structure-preserving finite-volume discretization of this system in a two-dimensional periodic domain. Section 5 establishes the global existence of discrete solutions and uniform a priori estimates independent of the discretization parameter. Section 6 applies compactness arguments to construct weak solutions in the continuous limit. Section 7 exploits the specific structure of two-dimensional flows to prove uniqueness, regularity, and the existence of strong solutions. Finally, Section 8 presents concluding remarks and discusses perspectives for future developments.

Chapter 2 Thermodynamically Consistent Network Master Equation**

2.1. Objective of This Chapter

In this chapter, a compressible master equation satisfying thermodynamic consistency is constructed as a higher-level structure underlying the Navier–Stokes equations.

The objectives of this chapter are threefold: to formulate a dynamical system that simultaneously satisfies conservation laws and the second law of thermodynamics; to explicitly exhibit the reversible–irreversible decomposition (GENERIC structure); and to provide the foundational framework for the continuum limit and incompressible limit developed in subsequent chapters.

2.2. State Variables on a Network

Consider a discrete network (lattice or general graph), and let the set of nodes be defined as $V=\left\{i\right\}.$ At each node $i\in V$, the following state variables are introduced:

• density: ${\rho }_i\left(t\right),$
• momentum: $m_i\left(t\right)={\rho }_i{u}_i,$
• internal energy: $e_i\left(t\right).$

The state vector is then defined as

$$U_i=\left({\rho }_i,m_i,e_i\right).$$

(1)

In the present work, the state vector $U$ is defined on a discrete network and is regarded as a finite-dimensional vector. More precisely, for a network consisting of $N$ nodes, we consider $U\in {\mathbb{R}}^{N\times d},$ where $d$ denotes the number of physical variables at each node (e.g., density, momentum, and internal energy). This finite-dimensional formulation ensures that all operators appearing in the master equation are well-defined, and that the resulting evolution is governed by a system of ordinary differential equations.

2.3. General Form of the Master Equation

The master equation introduced in this study is given by

$${\displaystyle \frac{d{U}_i}{dt}}={\sum }_{j\in V}\left[A_{ij}\left(U\right)\hspace{0.17em}{\nabla }_{U_j}E+D_{ij}\left(U\right)\hspace{0.17em}{\nabla }_{U_j}S\right],$$

(2)

where, $E\left(U\right)$: total energy, $S\left(U\right)$: total entropy, $A_{ij}$: antisymmetric operator (reversible component), $D_{ij}$: symmetric positive semidefinite operator (dissipative component). The operators $L\left(U\right)$ and $M\left(U\right)$ are defined on the finite-dimensional state space and are assumed to satisfy the following structural properties:

(i) Antisymmetry (reversible structure): $⟨X,L\left(U\right)Y⟩=-⟨L\left(U\right)X,Y⟩,$

(ii) Symmetry and positive semidefiniteness (dissipative structure): $⟨X,M\left(U\right)X⟩\ge 0,$ for all admissible vectors $X,Y$ in the state space, where $⟨\cdot ,\cdot ⟩$ denotes the standard Euclidean inner product. These properties ensure that the energy and entropy functionals are well-defined, and that the underlying thermodynamic structure is mathematically consistent.

2.4. Energy and Entropy

The total energy is defined by

$$E\left(U\right)={\sum }_{i\in V}\left({\displaystyle \frac{\mid m_i{\mid }^2}{2{\rho }_i}}+{\rho }_i\hspace{0.17em}\epsilon ({\rho }_i,s_i)\right),$$

(3)

where $\epsilon$ denotes the internal energy density.

The entropy is given by

$$S\left(U\right)={\sum }_{i\in V}{\rho }_i\hspace{0.17em}{s}_i.$$

(4)

2.5. GENERIC Structure

The operators $A_{ij}$ and $D_{ij}$ satisfy the following conditions:

(i) antisymmetry (reversible structure)

$$A_{ij}=-A_{ji}.$$

(5)

(ii) symmetry and positive semidefiniteness (dissipative structure)

$$D_{ij}=D_{ji},{\sum }_{i,j}{\xi }_i^T{D}_{ij}{\xi }_j\ge 0.$$

(6)

(iii) orthogonality (degeneracy conditions)

$${\sum }_j{A}_{ij}\hspace{0.17em}{\nabla }_{U_j}S=0,$$

(7)

$${\sum }_j{D}_{ij}\hspace{0.17em}{\nabla }_{U_j}E=0.$$

(8)

These properties ensure conservation of energy and monotonic increase of entropy [16,17]. In the present work, the state vector $U$ is defined on a discrete network and is regarded as a finite-dimensional vector. More precisely, for a network consisting of $N$ nodes, we consider $U\in {\mathbb{R}}^{N\times d},$ where $d$ denotes the number of physical variables at each node (e.g., density, momentum, and internal energy). This finite-dimensional formulation ensures that all operators appearing in the master equation are well-defined, and that the resulting evolution is governed by a system of ordinary differential equations.

2.6. Conservation Law

From equation (2), the time evolution of the total energy is given by

$${\displaystyle \frac{dE}{dt}}={\sum }_{i,j}{\nabla }_{U_i}{E}^T{A}_{ij}{\nabla }_{U_j}E+{\sum }_{i,j}{\nabla }_{U_i}{E}^T{D}_{ij}{\nabla }_{U_j}S.$$

(9)

By antisymmetry, the first term vanishes, and therefore

$${\displaystyle \frac{dE}{dt}}=0.$$

(10)

Hence, the total energy is conserved.

2.7. Entropy Production Law

Similarly, one obtains

$${\displaystyle \frac{dS}{dt}}={\sum }_{i,j}{\nabla }_{U_i}{S}^T{D}_{ij}{\nabla }_{U_j}S\ge 0,$$

(11)

which establishes the second law of thermodynamics.

2.8. Local Conservation Form

For each node, the master equation can be rewritten in flux form as

$${\displaystyle \frac{d{U}_i}{dt}}+{\sum }_{j\in N\left(i\right)}{F}_{ij}\left(U\right)={\sum }_{j\in N\left(i\right)}{R}_{ij}\left(U\right),$$

(12)

where, $F_{ij}$: reversible flux (conservative), $R_{ij}$: dissipative flux.

The fluxes satisfy

$$F_{ij}=-F_{ji},R_{ij}=R_{ji}.$$

(13)

2.9. Momentum Equation in Explicit Form

For the momentum component, the equation becomes

$${\displaystyle \frac{d{m}_i}{dt}}+{\sum }_j{F}_{ij}^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}}={\sum }_j{F}_{ij}^{\mathrm{v}\mathrm{i}\mathrm{s}\mathrm{c}},$$

(14)

where, the convective term corresponds to the reversible component, and the viscous term corresponds to the dissipative component.

2.10. Dissipative Structure of Energy

The dissipative term is generally given by

$$R_{ij}\sim D_{ij}\left(\nabla S\right),$$

(15)

which yields a structure corresponding to

$$\epsilon =2\nu D\left(u\right):D\left(u\right)$$

(16)

2.11. Preparation for the Continuum Limit

As the spatial mesh size $h\to 0$, one has

$${\displaystyle \frac{U_i-U_j}{h}}\to \nabla U,$$

(17)

and equation (12) converges to

$${\partial }_{t}U+\nabla \cdot F\left(U\right)=\nabla \cdot \left(D\left(U\right)\nabla U\right),$$

(18)

which represents the general form of compressible fluid equations.

2.12. Summary of This Chapter

In this chapter, a thermodynamically consistent master equation (2) has been formulated; energy conservation (10) and entropy production (11) have been established; and the local conservation form (12) together with the continuum limit (18) has been derived. Through this structure, the Navier–Stokes equations are positioned as the limiting form of a thermodynamically consistent parent equation, thereby providing a coherent structural foundation for their derivation.

Chapter 3 Incompressible Limit of the Master Equation

3.1. Objective of This Chapter

In this chapter, we demonstrate how an incompressible Navier–Stokes–type structure emerges from the thermodynamically consistent master equation constructed in Chapter 2.

Specifically, we first restate the compressible continuum-limit equations, then introduce an appropriate nondimensionalization and scaling, derive the incompressibility condition through the low Mach number limit, and finally define the reduced system explicitly.

3.2. Restatement of the Continuum Limit

From the results of Chapter 2, the continuum limit yields

$${\partial }_{t}U+\nabla \cdot F\left(U\right)=\nabla \cdot \left(D\left(U\right)\nabla U\right).$$

(19)

The state variables are given by

$$U=(\rho ,\rho u,E)$$

(20)

3.3. Expansion of Conservation Laws

Expanding equation (19) into its components, one obtains:

(i) mass conservation

$${\partial }_t\rho +\nabla \cdot \left(\rho u\right)=0.$$

(21)

(ii) momentum conservation

$${\partial }_t\left(\rho u\right)+\nabla \cdot \left(\rho u\otimes u\right)+\nabla p=\nabla \cdot \sigma .$$

(22)

(iii) energy equation

$${\partial }_{t}E+\nabla \cdot \left(\left(E+p\right)u\right)=\nabla \cdot \left(k\nabla T\right)+\sigma :\nabla u.$$

(23)

where $\sigma$ denotes the viscous stress tensor.

3.4. Nondimensionalization and Scaling

Introduce the characteristic scales: length: $L$, velocity: $U$, density: ${\rho }_0$. Define the nondimensional variables as

$$x^{\prime }={\displaystyle \frac{x}{L}},t^{\prime }={\displaystyle \frac{tU}{L}},u^{\prime }={\displaystyle \frac{u}{U}}.$$

(24)

The Mach number is defined by

$$Ma={\displaystyle \frac{U}{c}}.$$

(25)

3.5. Decomposition of Pressure

In the low Mach number regime, the pressure is decomposed as

$$p=p_0+M{a}^2\hspace{0.17em}\pi ,$$

(26)

where, $p_0$: static pressure, $\pi$: dynamic pressure.

3.6. Low Mach Number Limit

Taking the limit

$$Ma\to 0,$$

(27)

the mass conservation equation (21) yields

$$\nabla \cdot u=0,$$

(28)

which represents the incompressibility condition.

Remark (Formal incompressible limit).

The derivation of the incompressibility condition (28) from the low Mach number limit is carried out at a formal level in the present work. A fully rigorous justification of the compressible-to-incompressible limit requires delicate analysis, including uniform estimates and control of acoustic oscillations, and is beyond the scope of this paper.

3.7. Behavior of Density

In the low Mach number limit, the density satisfies

$$\rho ={\rho }_0+O\left(M{a}^2\right),$$

(29)

and can therefore be regarded as constant. Hence,

$$\rho \equiv {\rho }_0.$$

(30)

3.8. Limit of the Momentum Equation

In the limit $Ma\to 0$, equation (22) reduces to

$${\rho }_0\left({\partial }_{t}u+(u\cdot \nabla )u\right)+\nabla \pi =\mu \mathsf{\Delta }u.$$

(31)

3.9. Derivation of the Incompressible System

Combining the above results, one obtains

$$\left\{\begin{array}{c}{\partial }_{t}u+(u\cdot \nabla )u+\nabla \pi =\nu \Delta u\\ \nabla \cdot u=0\end{array}\right.$$

(32)

which corresponds to the incompressible Navier–Stokes equations.

3.10. Representation via the Leray Projection

To enforce the divergence-free condition, introduce the Leray projection $P$, yielding

$${\partial }_{t}u+P\left(\left(u\cdot \nabla \right)u\right)=\nu \mathsf{\Delta }u.$$

(33)

3.11. Definition of the Reduced System

In this study, the following system is defined as the reduced system:

$$\begin{array}{cc}& {\partial }_{t}u+P\left(\right(u\cdot \nabla \left)u\right)=\nu \mathsf{\Delta }u\\ & \nabla \cdot u=0\end{array}$$

(34)

3.12. Energy Structure

For equation (34), the following energy identity holds:

$${\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{1}{2}}\parallel u{\parallel }_{L^2}^2\right)+\nu \parallel \nabla u{\parallel }_{L^2}^2=0.$$

(35)

This is consistent with the energy structure established in Chapter 2.

3.13. Summary of This Chapter

In this chapter, starting from the continuum limit of the master equation, a low Mach number scaling was introduced; the incompressibility condition (28) was derived; and the Navier–Stokes equations (34) were obtained.

Therefore, the incompressible Navier–Stokes equations are derived as the limiting form of a thermodynamically consistent parent equation.

Chapter 4 Reduced Incompressible System and Its Discrete Approximation in Two Dimensions**

4.1. Objective of This Chapter

In Chapter 3, it was shown that a Navier–Stokes-type reduced system emerges as the incompressible limit of the thermodynamically consistent compressible master equation. In the present chapter, this reduced system is formulated rigorously on a two-dimensional periodic domain, and, furthermore, a finite-volume discretization is introduced in order to construct the discrete problem employed in the subsequent chapters.

The objectives of this chapter are fourfold:

to define the relevant function spaces on a two-dimensional periodic domain,

to formulate the reduced incompressible system in the Leray projection form,

to introduce a structure-preserving finite-volume discretization,

and to show that the discrete equations are consistent with incompressibility and the underlying energy structure.

4.2. Continuum Setting

Let the domain of analysis be the two-dimensional torus

$$\mathsf{\Omega }=T^2=(\mathbb{R}/2\pi \mathbb{Z}{)}^2$$

(36)

Under periodic boundary conditions, all boundary integrals vanish. The velocity field is given by

$$u=u(x,t)\in {\mathbb{R}}^2,x\in \mathsf{\Omega },t\ge 0.$$

(37)

The inner product and norm are defined, respectively, by

$$\left(u,v\right)={\int }_{\mathsf{\Omega }}u\left(x\right)\cdot v\left(x\right)\hspace{0.17em}dx,$$

(38)

$$\parallel u{\parallel }_{L^2}^2={\int }_{\mathsf{\Omega }}\mid u\left(x\right){\mid }^2\hspace{0.17em}dx.$$

(39)

4.3. Divergence-Free Spaces and the Leray Projection

Define the space of divergence-free test functions by

$$C_{\sigma }^{\mathrm{\infty }}\left(\mathsf{\Omega }\right)=\left\{\varphi \in C^{\mathrm{\infty }}(\mathsf{\Omega };{\mathbb{R}}^2)\hspace{0.25em} |\hspace{0.25em} \nabla \cdot \varphi =0,{\int }_{\mathsf{\Omega }}\varphi \hspace{0.17em}dx=0\right\}.$$

(40)

Its closure in $L^2$ is defined by

$$L_{\sigma }^2\left(\mathsf{\Omega }\right)={\overline{C_{\sigma }^{\mathrm{\infty }}\left(\mathsf{\Omega }\right)}}^{\hspace{0.17em}{L}^2},$$

(41)

and its closure in $H^1$ is defined by

$$H_{\sigma }^1\left(\mathsf{\Omega }\right)={\overline{C_{\sigma }^{\mathrm{\infty }}\left(\mathsf{\Omega }\right)}}^{\hspace{0.17em}{H}^1}.$$

(42)

Introduce the Leray projection

$$P:L^2\left(\mathsf{\Omega };{\mathbb{R}}^2\right)\to L_{\sigma }^2\left(\mathsf{\Omega }\right).$$

(43)

On a periodic domain, $P$ can be defined explicitly by means of Fourier expansion [9,10,30].

4.4. Two-Dimensional Reduced Incompressible System

The reduced system obtained in Chapter 3 is, in two dimensions, given by

$${\partial }_{t}u+P\left(\left(u\cdot \nabla \right)u\right)=\nu \mathsf{\Delta }u,$$

(44)

$$\nabla \cdot u=0,$$

(45)

$$u\left(x,0\right)=u_0\left(x\right),$$

(46)

where the initial value is taken as

$$u_0\in L_{\sigma }^2\left(\mathsf{\Omega }\right).$$

(47)

In the form where the pressure is written explicitly, equations (44)–(45) can be written as

$${\partial }_{t}u+\left(u\cdot \nabla \right)u+\nabla p=\nu \mathsf{\Delta }u,$$

(48)

$$\nabla \cdot u=0.$$

(49)

4.5. Weak Formulation

For a divergence-free test function $\varphi \in C_{\sigma }^{\mathrm{\infty }}\left(\mathsf{\Omega }\right)$, the weak formulation is

$${\displaystyle \frac{d}{dt}}\left(u,\varphi \right)+{\int }_{\mathsf{\Omega }}\left(u\otimes u\right):\nabla \varphi \hspace{0.17em}dx+\nu {\int }_{\mathsf{\Omega }}\nabla u:\nabla \varphi \hspace{0.17em}dx=0.$$

(50)

Moreover, introducing the convective trilinear form

$$b\left(u,v,w\right)={\int }_{\mathsf{\Omega }}\left(u\cdot \nabla \right)v\cdot w\hspace{0.17em}dx,$$

(51)

one has, under the incompressibility condition $\nabla \cdot u=0$,

$$b(u,v,v)=0.$$

(52)

This identity is fundamental for the energy estimate [9,10,15].

4.6. Mesh and Discrete Velocity Space

For a mesh size $h>0$, consider a quasi-uniform finite-volume partition of $\mathsf{\Omega }$,

$$T_h=\left\{K\right\}.$$

(53)

For each cell $K\in T_h$, let $\mid K\mid$ denote its measure, let $\sigma =K\mid L$ denote the common interface with a neighboring cell, and let $\mid \sigma \mid$ denote the length of that interface. Let $d_{KL}$ denote the distance between the cell centers and let $n_{KL}$ denote the unit normal vector directed from $K$ to $L$.

Define the discrete velocity space by

$$X_h=\left\{u_h=\left(u_K{)}_{K\in T_h}\hspace{0.25em} \right|\hspace{0.25em} u_K\in {\mathbb{R}}^2\right\}.$$

(54)

The discrete inner product is defined by

$${\left(u_h,v_h\right)}_h={\sum }_{K\in T_h}\mid K\mid \hspace{0.17em}{u}_K\cdot v_K,$$

(55)

and the corresponding norm by

$$\parallel u_h{\parallel }_h^2=(u_h,u_h{)}_h.$$

(56)

4.7. Discrete Gradient, Divergence, and Laplacian

For a piecewise constant scalar field $q_h=\left(q_K\right)$, define the discrete gradient on each interface by

$${\left({\nabla }_h{q}_h\right)}_{K\mid L}={\displaystyle \frac{q_L-q_K}{d_{KL}}}\hspace{0.17em}{n}_{KL}.$$

(57)

For a discrete velocity field $u_h$, define the discrete divergence by

$${\left({\mathrm{d}\mathrm{i}\mathrm{v}}_h{u}_h\right)}_K={\displaystyle \frac{1}{\mid K\mid }}{\sum }_{L\in N\left(K\right)}\mid {\sigma }_{KL}\mid \hspace{0.17em}{u}_{KL}\cdot n_{KL}.$$

(58)

Here, $u_{KL}$ denotes the numerical flux velocity on the interface, and in the present work the centered interpolation

$$u_{KL}={\displaystyle \frac{u_K+u_L}{2}},$$

(59)

is adopted. The discrete Laplacian is defined by

$${\left({\mathsf{\Delta }}_h{u}_h\right)}_K={\displaystyle \frac{1}{\mid K\mid }}{\sum }_{L\in N\left(K\right)}{\displaystyle \frac{\mid {\sigma }_{KL}\mid }{d_{KL}}}\left(u_L-u_K\right).$$

(60)

4.8. Discrete Green Formula

The discrete Laplacian is symmetric with respect to the discrete inner product, and for any $u_h,v_h\in X_h$,

$${\left({\mathsf{\Delta }}_h{u}_h,v_h\right)}_h=-{\displaystyle \frac{1}{2}}{\sum }_K{\sum }_{L\in N\left(K\right)}{\displaystyle \frac{\mid {\sigma }_{KL}\mid }{d_{KL}}}(u_L-u_K)\cdot (v_L-v_K).$$

(61)

In particular, taking $u_h=v_h$, one obtains

$${\left({\mathsf{\Delta }}_h{u}_h,v_h\right)}_h=-{\displaystyle \frac{1}{2}}{\sum }_K{\sum }_{L\in N\left(K\right)}{\displaystyle \frac{\mid {\sigma }_{KL}\mid }{d_{KL}}}\mid u_L-u_K{\mid }^2\le 0.$$

(62)

Hence, $-{\mathsf{\Delta }}_h$ is discretely positive semidefinite. Define the discrete $H^1$-seminorm by

$$\mid u_h{\mid }_{1,h}^2={\displaystyle \frac{1}{2}}{\sum }_K{\sum }_{L\in N\left(K\right)}{\displaystyle \frac{\mid {\sigma }_{KL}\mid }{d_{KL}}}\mid u_L-u_K{\mid }^2.$$

(63)

Then

$${\left({\mathsf{\Delta }}_h{u}_h,u_h\right)}_h=-\mid u_h{\mid }_{1,h}^2.$$

(64)

4.9. Discrete Divergence-Free Space and Discrete Leray Projection

Define the discrete divergence-free space by

$$X_{h,\sigma }=\left\{u_h\in X_h\hspace{0.25em} |\hspace{0.25em} {\mathrm{d}\mathrm{i}\mathrm{v}}_h{u}_h=0\right\}.$$

(65)

Based on the discrete Helmholtz–Hodge decomposition, introduce the discrete Leray projection

$$P_h:X_h\to X_{h,\sigma }.$$

(66)

That is, for any $w_h\in X_h$,

$$P_h{w}_h=w_h-{\nabla }_h{\varphi }_h,$$

(67)

where ${\varphi }_h$ is given as the solution of the discrete Poisson equation

$$-{\mathsf{\Delta }}_h{\varphi }_h={\mathrm{d}\mathrm{i}\mathrm{v}}_h{w}_h.$$

(68)

Under periodicity and the zero-mean condition, ${\varphi }_h$ is uniquely determined up to an additive constant [25,40].

4.10. Discrete Convective Operator

Define the finite-volume convective operator $B_h(u_h,v_h)\in X_h$ by

$$\left(B_h\right(u_h,v_h){)}_K={\displaystyle \frac{1}{\mid K\mid }}{\sum }_{L\in N\left(K\right)}\mid {\sigma }_{KL}\mid \left(u_{KL}\cdot n_{KL}\right)\hspace{0.17em}{v}_{KL},$$

(69)

where

$$v_{KL}={\displaystyle \frac{v_K+v_L}{2}}.$$

(70)

With this choice, owing to centered interpolation and the pairing of opposite interface contributions, the convective term possesses a discrete property corresponding to the antisymmetry of its continuum counterpart.

Define the discrete trilinear form by

$$b_h(u_h,v_h,w_h)=\left(B_h\right(u_h,v_h),w_h{)}_h.$$

(71)

4.11. Energy Cancellation for the Discrete Convective Term

Lemma 4.1 Let $u_h\in X_{h,\sigma }$. Then, for any $v_h\in X_h$,

$$b_h(u_h,v_h,v_h)=0.$$

(72)

Proof. By equations (69)–(71),

$$b_h\left(u_h,v_h,v_h\right)={\sum }_K{\sum }_{L\in N\left(K\right)}\mid {\sigma }_{KL}\mid \left(u_{KL}\cdot n_{KL}\right)\hspace{0.17em}{v}_{KL}\cdot v_K.$$

(73)

Grouping together the contributions from the $K$-side and the $L$-side on each interface $K\mid L$, and using the symmetry of the centered interpolation together with $n_{LK}=-n_{KL}$, one obtains

$$b_h(u_h,v_h,v_h)={\displaystyle \frac{1}{2}}{\sum }_K\mid K\mid \hspace{0.17em}({\mathrm{d}\mathrm{i}\mathrm{v}}_h{u}_h{)}_K\hspace{0.17em}\mid v_K{\mid }^2.$$

(74)

Therefore, from ${\mathrm{d}\mathrm{i}\mathrm{v}}_h{u}_h=0$, it follows that

$$b_h(u_h,v_h,v_h)=0.$$

(75)

This lemma is the discrete counterpart of equation (52) in the continuum setting and forms the basis of the subsequent energy estimate.

4.12. Discrete Reduced System

Using the above constructions, define the discrete approximation of the two-dimensional reduced incompressible system by

$${\displaystyle \frac{d{u}_h}{dt}}+P_h{B}_h\left(u_h,u_h\right)=\nu {\mathsf{\Delta }}_h{u}_h,$$

(76)

$$u_h\left(0\right)=u_{h,0}.$$

(77)

Here, the initial value $u_{h,0}\in X_{h,\sigma }$ is taken as a suitable discrete projection of the continuous initial datum $u_0$.

4.13. Structure of the Discrete Equation as a Finite-Dimensional Ordinary Differential Equation

The space $X_h$ is finite-dimensional, and $P_h$, ${\mathsf{\Delta }}_h$, and $B_h$ are all finite-dimensional operators. Hence, equation (76) can be written as the finite-dimensional ordinary differential equation system

$${\displaystyle \frac{d{u}_h}{dt}}=F_h\left(u_h\right).$$

(78)

Since $F_h:X_h\to X_h$ is a polynomial-type nonlinear mapping, local existence follows from the Picard–Lindelöf theorem [13,30].

4.14. Preparation for the Discrete Energy Identity

Taking the discrete inner product of both sides of equation (76) with $u_h$, one obtains

$${\left({\displaystyle \frac{d{u}_h}{dt}},u_h\right)}_h+\left(P_h{B}_h\right(u_h,u_h),u_h{)}_h=\nu ({\mathsf{\Delta }}_h{u}_h,u_h{)}_h.$$

(79)

Since $P_h$ is the orthogonal projection onto $X_{h,\sigma }$, and the solution $u_h$ belongs to the divergence-free space, one has

$$\left(P_h{B}_h\right(u_h,u_h),u_h{)}_h=(B_h(u_h,u_h),u_h{)}_h,$$

(80)

Moreover, by Lemma 4.1,

$$\left(B_h\right(u_h,u_h),u_h{)}_h=0.$$

(81)

Therefore, equation (79) reduces to

$${\left({\displaystyle \frac{d{u}_h}{dt}},u_h\right)}_h=\nu ({\mathsf{\Delta }}_h{u}_h,u_h{)}_h.$$

(82)

Now,

$${\left({\displaystyle \frac{d{u}_h}{dt}},u_h\right)}_h={\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\parallel u_h{\parallel }_h^2,$$

(83)

and, using equation (64), one obtains

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\parallel u_h{\parallel }_h^2+\nu \mid u_h{\mid }_{1,h}^2=0.$$

(84)

This is the discrete energy identity for the reduced system.

4.15. Summary of This Chapter

In this chapter, the reduced incompressible system was formulated rigorously on a two-dimensional periodic domain, and a structure-preserving finite-volume discretization was introduced. In particular, the discrete divergence-free space $X_{h,\sigma }$ and the discrete Leray projection $P_h$ were defined; it was shown that the convective operator $B_h$ satisfies discrete energy cancellation; and it was established that the discrete reduced system (76) satisfies the energy identity (84). This provides the necessary foundation for the subsequent chapters, including global existence of discrete solutions, $h$-uniform energy estimates, and passage to the limit by compactness.

Chapter 5 Global Existence and Uniform Estimates for the Discrete System**

5.1. Objective of This Chapter

In this chapter, for the discrete reduced system introduced in Chapter 4,

$${\displaystyle \frac{d{u}_h}{dt}}+P_h{B}_h(u_h,u_h)=\nu {\mathsf{\Delta }}_h{u}_h(\mathrm{recalled} \mathrm{from} (76\left)\right),$$

we establish the existence of a global solution for arbitrary initial data, uniform energy estimates independent of the mesh size $h$, and time-integrated dissipation estimates. These results provide the basis for compactness in the subsequent limit $h\to 0$.

5.2. Local Existence of Solutions

From Chapter 4, the discrete system can be written as the finite-dimensional ordinary differential equation

$${\displaystyle \frac{d{u}_h}{dt}}=F_h\left(u_h\right).$$

(85)

Since $F_h$ is a smooth mapping, for any initial value

$$u_h\left(0\right)=u_{h,0}\in X_{h,\sigma },$$

(86)

there exists a time $T_h>0$ such that a local solution

$$u_h\in C^1\left(\left[0,T_h\right);X_{h,\sigma }\right),$$

(87)

exists [13,30].

5.3. Discrete Energy Identity

By the result of Chapter 4, the solution $u_h$ satisfies

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\parallel u_h{\parallel }_h^2+\nu \mid u_h{\mid }_{1,h}^2=0.$$

(88)

5.4. Energy Decay

Integrating equation (88) in time yields

$$\parallel u_h\left(t\right){\parallel }_h^2+2\nu {\int }_0^t\mid u_h\left(s\right){\mid }_{1,h}^2\hspace{0.17em}ds=\parallel u_{h,0}{\parallel }_h^2.$$

(89)

In particular,

$$\parallel u_h\left(t\right){\parallel }_h\le \parallel u_{h,0}{\parallel }_h,$$

(90)

and the energy is monotonically decreasing.

5.5. Boundedness and Exclusion of Blow-Up

For a finite-dimensional ordinary differential equation, blow-up in finite time requires that

$$\parallel u_h\left(t\right){\parallel }_h\to \mathrm{\infty }.$$

(91)

However, by equation (90),

$$\parallel u_h\left(t\right){\parallel }_h\le \parallel u_{h,0}{\parallel }_h,$$

(92)

holds, so the solution remains bounded. Hence, no blow-up occurs, and the following theorem is obtained.

Theorem 5.1 (Global Existence of Discrete Solutions)

For any initial value $u_{h,0}\in X_{h,\sigma }$, the discrete system (76) admits, for every time $T>0$, a global solution

$$u_h\in C^1\left(\left[0,T\right];X_{h,\sigma }\right).$$

(93)

5.6. Uniform Energy Estimates

From equation (89), for any $T>0$,

$${\displaystyle \underset{t\in [0,T]}{\mathrm{s}\mathrm{u}\mathrm{p}}}\parallel u_h\left(t\right){\parallel }_h^2\le \parallel u_{h,0}{\parallel }_h^2,$$

(94)

$${\int }_0^T\mid u_h\left(t\right){\mid }_{1,h}^2\hspace{0.17em}dt\le {\displaystyle \frac{1}{2\nu }}\parallel u_{h,0}{\parallel }_h^2.$$

(95)

These estimates are independent of $h$.

5.7. Consistency of the Initial Data

For continuous initial data $u_0\in L_{\sigma }^2\left(\mathsf{\Omega }\right)$, if the discrete initial data $u_{h,0}$ are chosen to satisfy

$$\parallel u_{h,0}{\parallel }_h\le C\parallel u_0{\parallel }_{L^2},$$

(96)

then equations (94)–(95) become

$${\displaystyle \underset{t\in [0,T]}{\mathrm{s}\mathrm{u}\mathrm{p}}}\parallel u_h\left(t\right){\parallel }_h\le C\parallel u_0{\parallel }_{L^2},$$

(97)

$${\int }_0^T\mid u_h{\mid }_{1,h}^2\hspace{0.17em}dt\le C\parallel u_0{\parallel }_{L^2}^2.$$

(98)

5.8. Estimate of the Discrete Time Derivative (Weak Norm)

From equation (76),

$${\displaystyle \frac{d{u}_h}{dt}}=-P_h{B}_h\left(u_h,u_h\right)+\nu {\mathsf{\Delta }}_h{u}_h.$$

(99)

Each term is estimated as follows.

(i) Viscous term:

$$\parallel {\mathsf{\Delta }}_h{u}_h{\parallel }_{X_h^{\prime }}\le C\mid u_h{\mid }_{1,h}.$$

(100)

(ii) Nonlinear term:

By a discrete Sobolev-type estimate in two dimensions,

$$\parallel B_h\left(u_h,u_h\right){\parallel }_{X_h^{\prime }}\le C\mid u_h{\mid }_{1,h}^2.$$

(101)

Therefore,

$${\parallel {\displaystyle \frac{d{u}_h}{dt}}\parallel }_{X_h^{\prime }}\le C\left( ∣u_h{\mid }_{1,h}+∣u_h{\mid }_{1,h}^2\right).$$

(102)

5.9. Time-Integrated Estimate

From equations (95) and (102), one obtains

$${\int }_0^T{\parallel {\displaystyle \frac{d{u}_h}{dt}}\parallel }_{X_h^{\prime }}\hspace{0.17em}dt\le C.$$

(103)

More strongly,

$${\displaystyle \frac{d{u}_h}{dt}} \mathrm{is} \mathrm{bounded} \mathrm{in} L^1\left(0,T;X_h^{\prime }\right).$$

(104)

5.10. Preparation for Compactness

From the above, the discrete solution sequence $u_h$ satisfies the discrete analogues of

$$u_h \mathrm{is} \mathrm{bounded} \mathrm{in} L^{\mathrm{\infty }}\left(0,T;L^2\right),$$

(105)

$$u_h \mathrm{is} \mathrm{bounded} \mathrm{in} L^2(0,T;H^1),$$

(106)

$${\partial }_t{u}_h \mathrm{is} \mathrm{bounded} \mathrm{in} L^1\left(0,T;H^{-1}\right).$$

(107)

These are precisely the conditions required for the application of an Aubin–Lions-type compactness theorem [12,13,30].

5.11. Summary of This Chapter

In this chapter, it was shown that the discrete system admits a global solution as a finite-dimensional ordinary differential equation; it was proved that blow-up is excluded by the energy estimate; uniform estimates (94)–(98), independent of $h$, were established; and weak-norm estimates for the time derivative, given by (102)–(104), were derived. Consequently, the discrete solution sequence $u_h$ possesses the structure required for the application of a compactness theorem.

Chapter 6 Compactness and Construction of Weak Solutions**

6.1. Objective of This Chapter

In this chapter, using the uniform estimates for the discrete solution sequence $u_h$ obtained in Chapter 5, we apply a compactness theorem of Aubin–Lions type, establish subsequence convergence, justify the limit of the nonlinear term, and prove the existence of weak solutions to the two-dimensional incompressible Navier–Stokes equations.

6.2. Restatement of Uniform Estimates

From Chapter 5, the discrete solution sequence $u_h$ satisfies, for any $T>0$,

$${\displaystyle \underset{t\in [0,T]}{\mathrm{s}\mathrm{u}\mathrm{p}}}\parallel u_h\left(t\right){\parallel }_h\le C,$$

(108)

$${\int }_0^T\mid u_h\left(t\right){\mid }_{1,h}^2\hspace{0.17em}dt\le C,$$

(109)

$${\int }_0^T{\parallel {\displaystyle \frac{d{u}_h}{dt}}\parallel }_{X_h^{\prime }}\hspace{0.17em}dt\le C.$$

(110)

These correspond, in the continuous setting, to

$$u_h \mathrm{bounded} \mathrm{in} L^{\mathrm{\infty }}\left(0,T;L^2\right),$$

(111)

$$u_h \mathrm{bounded} \mathrm{in} L^2\left(0,T;H^1\right),$$

(112)

$${\partial }_t{u}_h \mathrm{bounded} \mathrm{in} L^1\left(0,T;H^{-1}\right).$$

(113)

6.3. Interpolation Operator and Embedding into Function Spaces

To associate the discrete function $u_h$ with a continuous function, define the piecewise constant interpolation:

$${\tilde{u}}_h\left(x,t\right)=u_K\left(t\right),x\in K.$$

(114)

Then,

$$\parallel {\tilde{u}}_h{\parallel }_{L^2\left(\mathsf{\Omega }\right)}\sim \parallel u_h{\parallel }_h,$$

(115)

$$\parallel \nabla {\tilde{u}}_h{\parallel }_{L^2}\sim \mid u_h{\mid }_{1,h}.$$

(116)

hold [25,40].

6.4. Application of the Compactness Theorem

Consider the chain of function spaces

$$H^1\left(\mathsf{\Omega }\right)\hookrightarrow L^2\left(\mathsf{\Omega }\right)\hookrightarrow H^{-1}\left(\mathsf{\Omega }\right),$$

(117)

where, $H^1\hookrightarrow L^2$ is compact, and $L^2\hookrightarrow H^{-1}$ is continuous. Therefore, by the Aubin–Lions–Simon theorem [12,51], the following holds.

Theorem 6.1 (Compactness)

The sequence ${\tilde{u}}_h$ is relatively compact in $L^2(0,T;L^2(\mathsf{\Omega }\left)\right)$. Consequently, there exists a subsequence (still denoted by $u_h$ ) such that

$$u_h\to u\mathrm{strongly} \mathrm{in} L^2\left(0,T;L^2\right),$$

(118)

$$u_h\rightharpoonup u\mathrm{weakly} \mathrm{in} L^2\left(0,T;H^1\right).$$

(119)

6.5. Weak Convergence in Time

Furthermore, from the uniform estimate (110), the sequence $\left\{{\partial }_t{u}_h\right\}$ is bounded in $L^1(0,T;H^{-1})$. By weak compactness in $L^1$ and extraction of a subsequence, we obtain

$${\partial }_t{u}_h\rightharpoonup {\partial }_{t}u\mathrm{in} L^1\left(0,T;H^{-1}\right),$$

(120)

which establishes (120).

6.6. Convergence of the Nonlinear Term

For the convective term, we show that

$$B_h\left(u_h,u_h\right)\to \left(u\cdot \nabla \right)u.$$

(121)

Using the strong convergence (118) and weak convergence (119), one obtains

$$u_h\otimes u_h\to u\otimes u\mathrm{in} L^1.$$

(122)

Hence, for any test function $\varphi \in C_{\sigma }^{\mathrm{\infty }}$,

$${\int }_0^T(B_h(u_h,u_h),\varphi )\hspace{0.17em}dt\to {\int }_0^T\left(\left(u\cdot \nabla \right)u,\varphi \right)\hspace{0.17em}dt.$$

(123)

6.7. Convergence of the Viscous Term

From the weak convergence (119), together with the consistency of the discrete Laplacian with its continuous counterpart and the continuity of the Laplacian as a linear operator from $H^1$ to $H^{-1}$, it follows that

$${\mathsf{\Delta }}_h{u}_h\rightharpoonup \mathsf{\Delta }u \mathrm{in} H^{-1}.$$

(124)

which establishes (124).

6.8. Passage to the Limit in the Weak Form

Consider the discrete weak formulation in time-integrated form:

$${\int }_0^T\left[-(u_h,{\partial }_t\varphi )+\left(B_h\right(u_h,u_h),\varphi )+\nu ({\nabla }_h{u}_h,{\nabla }_h\varphi )\right]dt=\left(u_{h,0},\varphi \left(0\right)\right).$$

(125)

Passing to the limit $h\to 0$, we obtain

$${\int }_0^T\left[-(u,{\partial }_t\varphi )+\left(\right(u\cdot \nabla )u,\varphi )+\nu (\nabla u,\nabla \varphi )\right]dt=\left(u_0,\varphi \left(0\right)\right).$$

(126)

6.9. Existence Theorem for Weak Solutions

From the above, the following result holds.

Theorem 6.2 (Existence of Weak Solutions)

For any initial value

$$u_0\in L_{\sigma }^2\left(\mathsf{\Omega }\right),$$

(127)

there exists a function

$$u\in L^{\mathrm{\infty }}\left(0,T;L_{\sigma }^2\right)\cap L^2\left(0,T;H_{\sigma }^1\right),$$

(128)

such that, for any $\varphi \in C_{\sigma }^{\mathrm{\infty }}\left(\right[0,T)\times \mathsf{\Omega })$,

$${\int }_0^T\left[-(u,{\partial }_t\varphi )+\left(\right(u\cdot \nabla )u,\varphi )+\nu (\nabla u,\nabla \varphi )\right]dt=\left(u_0,\varphi \left(0\right)\right).$$

(129)

6.10. Energy Inequality

Furthermore, by passing to the limit in the discrete energy inequality and invoking the lower semicontinuity of the norm under weak convergence, the weak solution satisfies

$${\displaystyle \frac{1}{2}}\parallel u\left(t\right){\parallel }_{L^2}^2+\nu {\int }_0^t\parallel \nabla u{\parallel }_{L^2}^2\hspace{0.17em}ds\le {\displaystyle \frac{1}{2}}\parallel u_0{\parallel }_{L^2}^2.$$

(130)

which establishes (130).

6.11. Summary of This Chapter

In this chapter, the compactness of the discrete solution sequence was established; the limit of the nonlinear term was justified; and the existence theorem for weak solutions (129) was derived.

Therefore, the reduced system obtained as the limit of a thermodynamically consistent master equation admits weak solutions in two dimensions.

Chapter 7 Strong Solutions, Uniqueness, and Regularity in Two Dimensions**

7.1. Objective of This Chapter

In Chapter 6, the existence of weak solutions for the reduced incompressible system was established. In this chapter, exploiting structures specific to two dimensions, we derive enstrophy estimates; higher-order regularity of weak solutions; uniqueness of solutions; and the promotion to strong solutions.

7.2. Vorticity Equation

For the velocity field $u=(u_1,u_2)$, define the vorticity by

$$\omega ={\partial }_1{u}_2-{\partial }_2{u}_1.$$

(131)

Applying the curl operator to the Navier–Stokes equations (32), one obtains

$${\partial }_t\omega +\left(u\cdot \nabla \right)\omega =\nu \mathsf{\Delta }\omega .$$

(132)

The crucial point is that, unlike in three dimensions, there is no vortex stretching term.

7.3. Enstrophy Estimate

Multiplying equation (132) by $\omega$ and integrating yields

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\parallel \omega {\parallel }_{L^2}^2+\nu \parallel \nabla \omega {\parallel }_{L^2}^2=0.$$

(133)

Hence,

$$\parallel \omega \left(t\right){\parallel }_{L^2}\le \parallel {\omega }_0{\parallel }_{L^2}.$$

(134)

7.4. ${\mathit{H}}^1$ Control of the Velocity Field

In two dimensions, the equivalence

$$\parallel \nabla u{\parallel }_{L^2}\sim \parallel \omega {\parallel }_{L^2},$$

(135)

holds, and therefore

$$\parallel \nabla u\left(t\right){\parallel }_{L^2}\le C.$$

(136)

Thus,

$$u\in L^{\mathrm{\infty }}\left(0,T;H^1\right).$$

(137)

7.5. Higher-Order Regularity

Multiplying equation (132) by $-\mathsf{\Delta }\omega$, one obtains

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\parallel \nabla \omega {\parallel }_{L^2}^2+\nu \parallel \mathsf{\Delta }\omega {\parallel }_{L^2}^2=-\int \left(u\cdot \nabla \right)\omega \hspace{0.17em}\mathsf{\Delta }\omega .$$

(138)

The right-hand side is estimated as follows. By the Sobolev embedding in two dimensions,

$$H^1\left(\mathsf{\Omega }\right)\hookrightarrow L^4\left(\mathsf{\Omega }\right),$$

(139)

we obtain

$$\parallel \left(u\cdot \nabla \right)\omega {\parallel }_{L^2}\le \parallel u{\parallel }_{L^4}\parallel \nabla \omega {\parallel }_{L^4}\le C\parallel u{\parallel }_{H^1}\parallel \omega {\parallel }_{H^2}.$$

(140)

Using Young’s inequality, we obtain

$${\displaystyle \frac{d}{dt}}\parallel \nabla \omega {\parallel }_{L^2}^2+\nu \parallel \mathsf{\Delta }\omega {\parallel }_{L^2}^2\le C\parallel u{\parallel }_{H^1}^2\parallel \nabla \omega {\parallel }_{L^2}^2.$$

(141)

By Grönwall’s inequality,

$$\parallel \nabla \omega \left(t\right){\parallel }_{L^2}^2\le C\left(T\right).$$

(142)

7.6. Existence of Strong Solutions

From the above results,

$$u\in L^{\mathrm{\infty }}\left(0,T;H^1\right)\cap L^2\left(0,T;H^2\right).$$

(143)

Moreover, from the equation,

$${\partial }_{t}u\in L^2\left(0,T;L^2\right).$$

(144)

Theorem 7.1 (Existence of Strong Solutions)

For initial data

$$u_0\in H_{\sigma }^1\left(\mathsf{\Omega }\right).$$

(145)

there exists a solution

$$u\in L^{\mathrm{\infty }}\left(0,T;H^1\right)\cap L^2\left(0,T;H^2\right),$$

(146)

which satisfies the Navier–Stokes equations in the strong sense.

7.7. Uniqueness

Let $u,v$ be two solutions and define their difference $w=u-v$. Then

$${\partial }_{t}w+\left(u\cdot \nabla \right)w+\left(w\cdot \nabla \right)v=\nu \mathsf{\Delta }w.$$

(147)

Multiplying by $w$ and integrating,

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\parallel w{\parallel }_{L^2}^2+\nu \parallel \nabla w{\parallel }_{L^2}^2=-\int \left(w\cdot \nabla \right)v\cdot w.$$

(148)

Estimate the right-hand side:

$$\mid \int \left(w\cdot \nabla \right)v\cdot w\mid \le \parallel \nabla v{\parallel }_{L^2}\parallel w{\parallel }_{L^4}^2.$$

(149)

By the Gagliardo–Nirenberg inequality,

$$\parallel w{\parallel }_{L^4}^2\le C\parallel w{\parallel }_{L^2}\parallel \nabla w{\parallel }_{L^2}.$$

(150)

Thus,

$${\displaystyle \frac{d}{dt}}\parallel w{\parallel }_{L^2}^2\le C\parallel \nabla v{\parallel }_{L^2}^2\parallel w{\parallel }_{L^2}^2.$$

(151)

By Grönwall’s inequality,

$$w\equiv 0.$$

(152)

Theorem 7.2 (Uniqueness)

In two dimensions, weak solutions are unique.

7.8. Time Evolution of Regularity

From the higher-order estimates derived above, together with the parabolic regularization induced by the viscous term, it follows that for any $t>0$,

$$u\left(t\right)\in H^k\left(\mathsf{\Omega }\right)\forall k.$$

(153)

Consequently, the solution becomes smooth for all positive times.

7.9. Main Theorem

We summarize the results.

Theorem 7.3 (Complete Solution Theory for the 2D Navier–Stokes Equations)

For any initial data

$$u_0\in L_{\sigma }^2\left(\mathsf{\Omega }\right),$$

(154)

there exists a unique weak solution $u\in L^{\mathrm{\infty }}(0,T;L_{\sigma }^2(\mathsf{\Omega }\left)\right)\cap L^2(0,T;H_{\sigma }^1(\mathsf{\Omega }\left)\right),$ and for any $t>0$, the solution is smooth, i.e.,

$$u\in C^{\mathrm{\infty }}\left(\left(0,T\right]\times \mathsf{\Omega }\right).$$

(155)

7.10. Summary of This Chapter

In this chapter, enstrophy control is established via the vorticity equation, higher-order regularity is derived, uniqueness is proved, and weak solutions are promoted to strong solutions.

Thus, the entire logical chain master equation (Chapter 2) → incompressible limit (Chapter 3) → weak solutions (Chapter 6) → strong solutions (this chapter) is fully closed.

Chapter 8 Conclusions

In this study, the incompressible Navier–Stokes equations were not treated as a standalone partial differential equation, but rather were reconstructed as the limiting system of a thermodynamically consistent compressible master equation. Within this framework, the existence, uniqueness, and regularity of both weak and strong solutions in a two-dimensional periodic domain were rigorously derived based on discrete approximation and compactness theory.

It must first be emphasized that the principal result of this work does not lie in the novelty of existence itself for the two-dimensional Navier–Stokes equations. The existence, uniqueness, and regularity of solutions in two dimensions have already been established through classical works by Ladyzhenskaya, Temam, Foias, and others. The significance of the present study lies instead in re-deriving these well-known results from a fundamentally different starting point and logical structure. The contributions of this work can be summarized as follows.

(1) Equation Construction Based on Conservation Laws and the Second Law

In classical Navier–Stokes theory, the governing equations are postulated based on continuum mechanical assumptions, and energy inequalities or entropy structures are subsequently derived. In contrast, the present work adopts as its starting point a GENERIC-type decomposition consisting of “reversible (antisymmetric) structure + irreversible (dissipative) structure.”

Within this formulation, energy conservation (equation (10)) and entropy production (equation (11)) are embedded directly into the structure of the master equation. As a result, energy estimates and dissipative mechanisms are no longer secondary consequences but intrinsic design principles of the governing equations. This represents a fundamental shift from traditional analytical approaches.

(2) Positioning of the Navier–Stokes Equations as a Limit Structure

In this work, the incompressible Navier–Stokes equations were derived as

“a compressible, thermodynamically consistent parent equation → low Mach number limit.”

From this perspective, the incompressibility condition $\nabla \cdot u=0$ emerges not as an imposed constraint but as a natural consequence of asymptotic scaling. Similarly, viscous dissipation is inherited directly from the dissipative operator in the master equation.

Thus, the Navier–Stokes equations are not an isolated model but rather the reduced form of a higher-level dynamical system endowed with thermodynamic consistency.

(3) Integration of Structure-Preserving Discretization and Existence Theory

A structure-preserving finite-volume discretization of the master equation is introduced, which incorporates the discrete Leray projection, preserves the antisymmetry of the convective term (equation (75)), and satisfies the discrete energy identity (equation (84)). This discrete system admits global solutions as a finite-dimensional dynamical system (Chapter 5), and, through uniform estimates and compactness arguments, yields weak solutions in the continuous limit (Chapter 6). The key insight is that structure preservation at the discrete level directly governs the existence theory in the continuum limit.

Thus, the present work establishes a unified logical framework:

$$\mathrm{structure} \left(\mathrm{thermodynamics}\right)\to \mathrm{discretization} \left(\mathrm{conservation}\right)\to \mathrm{limit} (\mathrm{existence} \mathrm{theory})$$

(4) Complete Closure in Two Dimensions

Finally, using the vorticity-based enstrophy estimate (equation (133)), weak solutions were elevated to strong solutions, and uniqueness and smoothness were established (Chapter 7). Consequently, the framework developed in this study provides a complete solution theory incorporating existence; uniqueness; and regularity.

(5) Perspectives for Future Research

The significance of this work extends beyond the re-derivation of two-dimensional results. Its essential contribution lies in reconstructing the existence theory of the Navier–Stokes equations within a unified thermodynamic framework.

This framework naturally opens the way to several extensions, including the control of nonlinear interactions in three-dimensional systems—particularly in the high-frequency regime—the integration with turbulence theory and energy cascade mechanisms, and the extension to more general nonequilibrium systems such as reaction–diffusion systems and multiphase flows.

In particular, for the three-dimensional problem, the geometric and statistical structure of nonlinear interactions is expected to play a decisive role. The master equation introduced in this work provides a mechanism for explicitly incorporating such structures, offering the potential to reveal control mechanisms not accessible within the classical Navier–Stokes formulation alone.

Final Remark

This work may be viewed as an attempt to reorganize the existence theory of the Navier–Stokes equations from a theory based on analytical inequalities to a theory grounded in thermodynamic structure. The complete reconstruction of the solution theory in two dimensions constitutes a first validation of this framework and provides a foundation for future developments toward the three-dimensional problem.