Variational formulations for the Euler and Navier–Stokes systems in fluid mechanics and related models

Fabio Silva Botelho

doi:10.1515/nleng-2025-0097

Article Open Access

Variational formulations for the Euler and Navier–Stokes systems in fluid mechanics and related models

Fabio Silva Botelho

Published/Copyright: March 12, 2025

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From the journal Nonlinear Engineering Volume 14 Issue 1

Abstract

In its first part, this article develops a variational formulation for the incompressible Euler system in fluid mechanics. The results are based on standard tools of calculus of variations and constrained optimization. In a second step, we present a variational formulation for a compressible Euler system in fluid mechanics assuming an approximately constant scalar field of temperature. We emphasize to obtain both the equations of Euler and Bernoulli as a single system of necessary conditions for a stationary point for the variational formulation in question. In the subsequent sections, we also present a variational formulation for a Navier–Stokes system in fluid mechanics. Finally, in the last section, we develop a duality principle applied to a Ginzburg–Landau-type equation.

Keywords: incompressible Euler system; compressible Euler system; variational formulations

MSC 2010: 76A02; 76N15; 49N15

1 Introduction

The first part of this article develops a variational formulation for the incompressible Euler system in fluid mechanics.

In such a variational formulation, the main functional is comprised of the kinetics energy subject to a mass conservation equation as a constraint. We highlight that the Lagrange multiplier corresponding to such a constraint is related to the fluid pressure field.

Such results are based on tools of calculus of variations and related Lagrange multipliers approach.

We highlight that the two main fields for such a system are a position one and a velocity one, which we specify below in the next lines.

As standard references in theoretical fluid mechanics, we would cite previous studies [1–5]. For related numerical methods, we would mention related works [6–8]. Concerning similar models, we would cite previous studies [9–14].

Finally, details on the Sobolev spaces involved may be found in the study by Adams and Fournier [15].

Let Ω ⊂ R 3 be an open, bounded, and connected set with a regular (Lipschitzian) boundary denoted by ∂ Ω .

Consider a fluid motion initially modeled as a particle system one, through the field of position r : Ω × [ 0 , T ] → R and field of velocities v , where v ( r ( x , t ) , t ) stands for the velocity at the position r ( x , t ) and time t ∈ [ 0 , T ] .

2 Variational formulation for the incompressible Euler equation

In this section, we develop in detail the concerning variational formulation for a incompressible case.

Assuming the Einstein convection of summing up repeated indices and denoting

r ( x , t ) = ( X 1 ( x , t ) , X 2 ( x , t ) , X 3 ( x , t ) ) ,

for such a motion, we also assume the mass conservation equation as a constraint, which stands for

∂ 2 X j ( x , t ) ∂ x j ∂ t = 0 , in Ω × [ 0 , T ] .

For a constant fluid density ρ > 0 , the system kinetics energy functional stands for

E c = 1 2 ∫ 0 T ∫ Ω ρ v ( r ( x , t ) , t ) ⋅ v ( r ( x , t ) , t ) d x d t .

Moreover, as usual, we generically denote

⟨ f , h ⟩ L 2 = ∫ 0 T ∫ Ω f h d x d t , ∀ f , h ∈ L 2 ( Ω × [ 0 , T ] ) ,

with a similar notation for a vectorial case.

For such a modeling, denoting by g = − g k the gravity field, we define the following variational formulation represented by the functional J : Y × V × Y 1 × Y 2 → R already including the concerning Lagrange multipliers for the constraints, where

(1) J ( v , r , λ , P ˆ ) = 1 2 ∫ 0 T ∫ Ω ρ v ( r ( x , t ) , t ) ⋅ v ( r ( x , t ) , t ) d x d t + λ , v ( r ( x , t ) , t ) − ∂ r ( x , t ) ∂ t L 2 − P ˆ , ∂ 2 X j ( x , t ) ∂ x j ∂ t L 2 − ∫ 0 T ∫ Ω ρ g ⋅ r ( x , t ) d x d t .

In order to simplify the analysis, we assume

V = C 0 2 ( Ω × [ 0 , T ] ) ,

Y 1 = C 0 1 ( Ω × [ 0 , T ] ) , Y 2 = C 0 2 ( Ω × [ 0 , T ] ) , and generically Y = C 2 .

We also denote

J ( v , r , λ , P ˆ ) = J 1 ( v ( r ( x , t ) , t ) , r ( x , t ) , λ , P ˆ ) .

Symbolically, the variation of J in v stands for

∂ J ( v , r , λ , P ˆ ) ∂ v = ∂ J 1 ( v ( r ( x , t ) , t ) , r ( x , t ) , λ , P ˆ ) ∂ v = 0 ,

so that

ρ v + λ = 0 .

Moreover, the variation of J in r stands for

(2) ∂ J ( v , r , λ , P ˆ ) ∂ r = ∂ J 1 ( v ( r ( x , t ) , t ) , r ( x , t ) , λ , P ˆ ) ∂ v ∂ v ∂ r + ∂ J 1 ( v ( r ( x , t ) , t ) , r ( x , t ) , λ , P ˆ ) ∂ r = ∂ J 1 ( v ( r ( x , t ) , t ) , r ( x , t ) , λ , P ˆ ) ∂ r = ∂ λ ∂ t − ∇ ∂ P ˆ ∂ t − ρ g .

The variation of J in λ stands for

v − ∂ r ∂ t = 0 .

Finally, the variation of J in P ˜ stands for

∂ 2 X j ( x , t ) ∂ x j ∂ t = 0 , in Ω × [ 0 , T ] .

Denoting v j = ∂ X j ( x , t ) ∂ t , this last equation stands for

∂ v j ∂ x j = 0 , in Ω × [ 0 , T ] .

In summary for the first two previous equations, we obtain

ρ v ( r ( x , t ) , t ) + λ = 0

and

∂ λ ∂ t − ∇ ∂ P ˆ ∂ t − ρ g = 0

so that

d ( ρ v ( r ( x , t ) , t ) ) d t + ∇ ∂ P ˆ ∂ t + ρ g = 0 ,

i.e.,

∂ ( ρ v ) ∂ t + ∂ ( ρ v ) ∂ X j ∂ X j ( x , t ) ∂ t + ∇ ∂ P ˆ ∂ t + ρ g = 0 ,

so that denoting the pressure field by

P = ∂ P ˆ ∂ t ,

we obtain

∂ ( ρ v ) ∂ t + ∂ ( ρ v ) ∂ X j v j + ∇ P + ρ g = 0 .

Specifically, in terms of coordinates, such a last incompressible Euler system stands for

∂ ( ρ v k ) ∂ t + ∂ ( ρ v k ) ∂ X j v j + ∂ P ∂ x k + ρ g k = 0 , in Ω × [ 0 , T ] ,

∀ k ∈ { 1 , 2 , 3 } , and

∂ v j ∂ x j = 0 ,

in Ω × [ 0 , T ] .

Remark 2.1

Here, we recall that generically for a scalar smooth function φ = φ ( r ( x , t ) , t ) , we have

∂ φ ∂ x j = ∂ φ ∂ X k ∂ X k ∂ x j .

Thus, the equation

∂ ( ρ v k ) ∂ t + ∂ ( ρ v k ) ∂ X j v j + ∂ P ∂ x k + ρ g k = 0

stands for

∂ ( ρ v k ) ∂ t + A j l ∂ ( ρ v k ) ∂ x l v j + ∂ P ∂ x k + ρ g k = 0 .

where

{ A k j } = ∂ X k ∂ x j − 1 .

Finally, with the Eulerian identification

X k ( x , t ) ≈ x k ,

we obtain

{ A k j } ≈ I d ,

where I d denotes the identity matrix, so that we obtain the final format for such an Euler equation:

∂ ( ρ v k ) ∂ t + ∂ ( ρ v k ) ∂ x j v j + ∂ P ∂ x k + ρ g k = 0 , in Ω × [ 0 , T ] ,

∀ k ∈ { 1 , 2 , 3 } .

The objective of this section is complete.

3 Variational formulation for the compressible Euler equation

In this section, we develop in detail a concerning variational formulation for the compressible case.

We highlight that the necessary conditions for an extremal point of such a functional comprise both the Euler and Bernoulli equations in the same system.

It is our understanding such a system (including the Euler and Bernoulli equations) as necessary conditions for a stationary point of a single functional is an important advance in the mathematical theory of fluid mechanics.

We start now to describe such a variational formulation.

Let Ω ⊂ R 3 be an open, bounded, and connected set with a regular (Lipschitzian) boundary denoted by ∂ Ω .

Assuming the Einstein convection of summing up repeated indices and denoting

r ( x , t ) = ( X 1 ( x , t ) , X 2 ( x , t ) , X 3 ( x , t ) ) ,

for such a compressible motion, we also assume the mass conservation equation as a constraint, which stands for

d ρ d t + ρ ∂ ∂ x j ∂ X j ( x , t ) ∂ t = 0 , in Ω × [ 0 , T ] .

Here, we have assumed an approximately constant temperature and a non-constant fluid density ρ = ρ ( r ( x , t ) , t ) > 0 so that the system kinetics energy functional stands for

E c = 1 2 ∫ 0 T ∫ Ω ρ v ( r ( x , t ) , t ) ⋅ v ( r ( x , t ) , t ) d x d t .

For such a modeling, denoting again by g = − g k the gravity field, we define the following variational formulation represented by the functional J : Y × V × Y 1 × Y 2 × Y 3 → R already including the concerning Lagrange multipliers for the constraints, where

(3) J ( v , r , λ , P ˆ , ρ ) = 1 2 ∫ 0 T ∫ Ω ρ v ( r ( x , t ) , t ) ⋅ v ( r ( x , t ) , t ) d x d t + λ , v ( r ( x , t ) , t ) − ∂ r ( x , t ) ∂ t L 2 − P ˆ , d ρ d t + ρ ∂ ∂ x j ∂ X j ( x , t ) ∂ t L 2 − ∫ 0 T ∫ Ω ρ g ⋅ r ( x , t ) d x d t .

In order to simplify the analysis, we assume

V = C 0 2 ( Ω × [ 0 , T ] ) ,

Y 1 = C 0 1 ( Ω × [ 0 , T ] ) , Y 2 = C 0 2 ( Ω × [ 0 , T ] ) , and generically, Y = Y 3 = C 2 .

We also denote

J ( v , r , λ , P ˆ , ρ ) = J 1 ( v ( r ( x , t ) , t ) , r ( x , t ) , λ , P ˆ , ρ ( r ( x , t ) , t ) ) .

Symbolically, the variation of J in v stands for

∂ J ( v , r , λ , P ˆ , ρ ( r ( x , t ) , t ) ) ∂ v = ∂ J 1 ( v ( r ( x , t ) , t ) , r ( x , t ) , λ , P ˆ , ρ ( r ( x , t ) , t ) ) ∂ v = 0 ,

so that

ρ v + λ = 0 .

Moreover, the variation of J in r stands for

(4) ∂ J ( v , r , λ , P ˆ , ρ ) ∂ r = ∂ J 1 ( v ( r ( x , t ) , t ) , r ( x , t ) , λ , P ˆ , ρ ) ∂ v ∂ v ∂ r + ∂ J 1 ( v ( r ( x , t ) , t ) , r ( x , t ) , λ , P ˆ , ρ ) ∂ ρ ∂ ρ ∂ r + ∂ J 1 ( v ( r ( x , t ) , t ) , r ( x , t ) , λ , P ˆ , ρ ) ∂ r = ∂ J 1 ( v ( r ( x , t ) , t ) , r ( x , t ) , λ , P ˆ , ρ ) ∂ r = ∂ λ ∂ t − ∂ ( ∇ ( ρ P ˆ ) ) ∂ t − ρ g ,

where to clarify the notation, we highlight that denoting

ρ P ˆ = ρ ( r ( x , t ) , t ) P ˆ ( x , t ) ≡ W ( x , t ) ,

we have

∂ ( ∇ ( ρ P ˆ ) ) ∂ t = ∂ ∇ ( W ( x , t ) ) ∂ t = ∇ ∂ W ( x , t ) ∂ t .

The variation of J in λ stands for

v − ∂ r ∂ t = 0 .

The variation of J in P ˆ stands for

d ρ d t + ρ ∂ ∂ x j ∂ X j ( x , t ) ∂ t = 0 , in Ω × [ 0 , T ] .

Denoting v j = ∂ X j ( x , t ) ∂ t , this last equation stands for

d ρ d t + ρ ∂ ( v j ) ∂ x j = 0 , in Ω × [ 0 , T ] .

Finally, the variation of J in ρ stands for

∂ J ( v , r , λ , P ˆ , ρ ( r ( x , t ) , t ) ) ∂ ρ = ∂ J 1 ( v ( r ( x , t ) , t ) , r ( x , t ) , λ , P ˆ , ρ ( r ( x , t ) , t ) ) ∂ ρ = 0 ,

so that,

1 2 v ⋅ v + ∂ P ˆ ∂ t − P ˆ ∂ v j ∂ x j − g ⋅ r = 0 , in Ω × [ 0 , T ] .

In summary, for the first two previous equations, we obtain

ρ v ( r ( x , t ) , t ) + λ = 0

and

∂ λ ∂ t − ∂ ( ∇ ( ρ P ˆ ) ) ∂ t − ρ g = 0 .

so that

d ( ρ v ( r ( x , t ) , t ) ) d t + ∂ ( ∇ ( ρ P ˆ ) ) ∂ t + ρ g = 0 ,

i.e.,

∂ ( ρ v ) ∂ t + ∂ ( ρ v ) ∂ X j ∂ X j ( x , t ) ∂ t + ∂ ( ∇ ( ρ P ˆ ) ) ∂ t + ρ g = 0 .

Defining the pressure field by

P = ∂ ( ρ P ˆ ) ∂ t ,

we obtain

∂ ( ∇ ( ρ P ˆ ) ) ∂ t = ∇ ∂ ( ρ P ˆ ) ∂ t = ∇ P ,

so that specifically in terms of coordinates, such a last compressible Euler system stands for

∂ ( ρ v k ) ∂ t + ∂ ( ρ v k ) ∂ X j v j + ∂ P ∂ x k + ρ g k = 0 , in Ω × [ 0 , T ] ,

∀ k ∈ { 1 , 2 , 3 } ,

d ρ ∂ t + ρ ∂ v j ∂ x j = 0 ,

in Ω × [ 0 , T ] , and

1 2 v ⋅ v + ∂ P ˆ ∂ t − P ˆ ∂ v j ∂ x j + g X 3 ( x , t ) = 0 , in Ω × [ 0 , T ] .

Remark 3.1

Recalling the pressure field is expressed by

P = ∂ ( ρ P ˆ ) ∂ t ,

we obtain

(5) 1 2 v ⋅ v + P ρ − 1 ρ d ρ d t P ˆ − P ˆ ∂ v j ∂ x j + g X 3 ( x , t ) = 1 2 v ⋅ v + P ρ + g X 3 ( x , t ) = 0 , in Ω × [ 0 , T ] .

In summary, we have obtain the following Bernoulli equation:

1 2 v ⋅ v + P ρ + g X 3 = 0 .

Remark 3.2

Here, we recall again that generically, for a scalar smooth function φ = φ ( r ( x , t ) , t ) , we have

∂ φ ∂ x j = ∂ φ ∂ X k ∂ X k ∂ x j .

Thus, the equation

∂ ( ρ v k ) ∂ t + ∂ ( ρ v k ) ∂ X j v j + ∂ P ∂ x k + ρ g k = 0

stands for

∂ ( ρ v k ) ∂ t + A j l ∂ ( ρ v k ) ∂ x l v j + ∂ P ∂ x k + ρ g k = 0 ,

where

{ A k j } = ∂ X k ∂ x j − 1 .

Finally, with the Eulerian identification

X k ( x , t ) ≈ x k ,

and denoting x 3 = z + c 1 , we obtain

{ A k j } ≈ I d ,

where I d denotes the identity matrix, so that we obtain the final format for such Euler equations:

∂ ( ρ v k ) ∂ t + ∂ ( ρ v k ) ∂ x j v j + ∂ P ∂ x k + ρ g k = 0 , in Ω × [ 0 , T ] ,

∀ k ∈ { 1 , 2 , 3 } ,

D ρ D t + ρ ∂ v j ∂ x j = 0 , in Ω × [ 0 , T ]

and

1 2 v ⋅ v + P ρ + g z = c , in Ω × [ 0 , T ] ,

where c = − c 1 g .

In particular, these last two equations are the well-known continuity and Bernoulli ones, respectively.

Finally, we clarify that with the identification X k ( x , t ) ≈ x k

d ρ ( r ( x , t ) , t ) d t = ∂ ρ ∂ t + ∂ ρ ∂ X j ∂ X j ∂ t

will stand for

D ρ ( x , t ) D t = ∂ ρ ( x , t ) ∂ t + ∂ ρ ( x , t ) ∂ x j v j .

The objective of this section is complete.

4 Note on the Navier–Stokes system in fluid mechanics

Consider a positive definite symmetric constant fourth-order tensor { A i j k l } and a related viscosity part functional J 5 , where

J 5 ( v , r , ( r ∘ ε ) ) = 1 2 ∫ 0 T ∫ Ω A i j k l v k , l ( r ( x , t ) , t ) X i , j ( ε x , t ) d x d t ,

where generically we denote

( r ∘ ε ) ( x , t ) = r ( ε x , t ) , v = { v j } ,

and

v j , k = ∂ v j ∂ x k , ∀ j , k ∈ { 1 , 2 , 3 } .

Considering also the notation and results of the previous sections, we define a functional J 3 corresponding to an initially approximate variational formulation for the incompressible Navier–Stokes system, where

(6) J 3 ( v , r , ( r ∘ ε ) , λ , P ˆ ) = − J 5 ( v , r , ( r ∘ ε ) ) + 1 2 ∫ 0 T ∫ Ω ρ v ( r ( x , t ) , t ) ⋅ v ( r ( x , t ) , t ) d x d t + λ , v ( r ( x , t ) , t ) − ∂ r ( x , t ) ∂ t L 2 − P ˆ , ∂ 2 X j ( x , t ) ∂ x j ∂ t L 2 − ∫ 0 T ∫ Ω ρ g ⋅ r ( x , t ) d x d t .

In order to obtain the Navier–Stokes system, we define the following specific variation:

(7) δ ˆ r J 3 ( v , r , ( r ∘ ε ) , λ , P ˆ ; φ ) = lim h → 0 [ J 3 ( v ( r ( x , t ) + h φ ( x , t ) ) , r ( x , t ) + h φ ( x , t ) , r ( ε x , t ) + h φ ( x , t ) , λ , P ˆ ) − J 3 ( v ( r ( x , t ) ) , r ( x , t ) , r ( ε x , t ) , λ , P ˆ ) ] ⁄ h ,

so that the extremal equation

δ ˆ r J 3 ( v , r , ( r ∘ ε ) , λ , P ˆ ; φ ) = 0 , ∀ φ ∈ C c ∞ ( Ω × [ 0 , T ] ; R 3 )

stands for

∂ J 3 ( v ( r ( x , t ) , t ) , r , ( r ∘ ε ) , λ , P ˆ ) ∂ v ∂ v ∂ r + [ A i j k l v k , l j ] + ∂ λ ∂ t − ∂ ∇ P ˆ ∂ t − ρ g = 0 ,

i.e.,

[ A i j k l v k , l j ] + ∂ λ ∂ t − ∂ ∇ P ˆ ∂ t − ρ g = 0

in Ω × [ 0 , T ] .

Remark 4.1

We highlight that the solution of this last equation is computable thorough the Newton’s method, for example, considering in each iteration a standard quadratic approximation of J 3 in φ through a Taylor series.

Moreover, the variation of J 3 in v stands for

∂ J 3 ( v ( r ( x , t ) , t ) , r , ( r ∘ ε ) , λ , P ˆ ) ∂ v = 0

so that

ρ v + λ + ε [ A i j k l X i , y j y l ( ε x , t ) ] = 0 ,

where y = ε x .

The variation of J 3 in λ stands for

v − ∂ r ( x , t ) ∂ t = 0 .

Finally, the variation of J 3 in P ˆ stands for

∂ 2 X j ( x , t ) ∂ x j ∂ t = 0 , in Ω × [ 0 , T ] .

Denoting v j = ∂ X j ( x , t ) ∂ t , this last equation stands for

∂ v j ∂ x j = 0 , in Ω × [ 0 , T ] .

In summary, for the first two previous equations, we obtain

ρ v ( r ( x , t ) , t ) + λ = O ( ε )

and

[ A i j k l v k , l j ] + ∂ λ ∂ t − ∇ ∂ P ˆ ∂ t − ρ g = 0

so that

− [ A i j k l v k , l j ] + d ( ρ v ( r ( x , t ) , t ) ) d t + ∇ ∂ P ˆ ∂ t + ρ g = O ( ε ) ,

i.e.,

− [ A i j k l v k , l j ] + ∂ ( ρ v ) ∂ t + ∂ ( ρ v ) ∂ X j ∂ X j ( x , t ) ∂ t + ∇ ∂ P ˆ ∂ t + ρ g = O ( ε ) ,

so that denoting the pressure field by

P = ∂ P ˆ ∂ t ,

we obtain

− [ A i j k l v k , l j ] + ∂ ( ρ v ) ∂ t + ∂ ( ρ v ) ∂ X j v j + ∇ P + ρ g = O ( ε ) .

Letting ε → 0 , specifically in terms of coordinates, such a last incompressible Navier–Stokes system stands for

− [ A k j r l v r , j l ] + ∂ ( ρ v k ) ∂ t + ∂ ( ρ v k ) ∂ X j v j + ∂ P ∂ x k + ρ g k = 0 , in Ω × [ 0 , T ] ,

∀ k ∈ { 1 , 2 , 3 } , and

∂ v j ∂ x j = 0 ,

in Ω × [ 0 , T ] .

Remark 4.2

Here, we recall that generically for a scalar smooth function φ = φ ( r ( x , t ) , t ) , we have

∂ φ ∂ x j = ∂ φ ∂ X k ∂ X k ∂ x j .

Thus, the equation

− [ A k j r l v r , l j ] + ∂ ( ρ v k ) ∂ t + ∂ ( ρ v k ) ∂ X j v j + ∂ P ∂ x k + ρ g k = 0

stands for

− [ A k j r l v r , l j ] + ∂ ( ρ v k ) ∂ t + C j l ∂ ( ρ v k ) ∂ x l v j + ∂ P ∂ x k + ρ g k = 0 ,

where

{ C k j } = ∂ X k ∂ x j − 1 .

With the Eulerian identification

X k ( x , t ) ≈ x k ,

we obtain

{ C k j } ≈ I d ,

where I d denotes the identity matrix, so that we obtain the final format for such an Navier–Stokes system:

− ∂ 2 ( A k j r l v r ) ∂ x l ∂ x j + ∂ ( ρ v k ) ∂ t + ∂ ( ρ v k ) ∂ x j v j + ∂ P ∂ x k + ρ g k = 0 , in Ω × [ 0 , T ] ,

∀ k ∈ { 1 , 2 , 3 } , and

∂ v j ∂ x j = 0 , in Ω × [ 0 , T ] .

Finally, similarly as in the previous sections, we may obtain the following variational formulation for the compressible Navier–Stokes system for an approximately constant scalar field of temperature:

(8) J 8 ( v , r , ( r ∘ ε ) , λ , P ˆ , ρ ) = − 1 2 ∫ 0 T ∫ Ω A i j k l v k , l ( r ( x , t ) , t ) X i , j ( ε x , t ) d x d t × 1 2 ∫ 0 T ∫ Ω ρ v ( r ( x , t ) , t ) ⋅ v ( r ( x , t ) , t ) d x d t + λ , v ( r ( x , t ) , t ) − ∂ r ( x , t ) ∂ t L 2 − P ˆ , d ρ ∂ t + ρ ∂ ∂ x j ∂ X j ( x , t ) ∂ t L 2 − ∫ 0 T ∫ Ω ρ g ⋅ r ( x , t ) d x d t ,

obtaining also the following corresponding Navier–Stokes system:

− ∂ 2 ( A k j r l v r ) ∂ x l ∂ x j + ∂ ( ρ v k ) ∂ t + ∂ ( ρ v k ) ∂ x j v j + ∂ P ∂ x k + ρ g k = 0 , in Ω × [ 0 , T ] ,

∀ k ∈ { 1 , 2 , 3 } ,

D ρ D t + ρ ∂ v j ∂ x j = 0 , in Ω × [ 0 , T ] ,

and

1 2 v ⋅ v + P ρ + g z = c , in Ω × [ 0 , T ] ,

where c = − c 1 g .

We highlight that the specific viscosity tensor { A i j k l } depends on the fluid physical nature.

The objective of this section is complete.

5 Duality principle for a related model in superconductivity

This section develops a duality principle applicable to a large class of models in the calculus of variations. We present applications to a Ginzburg–Landau-type equation through a D.C. approach.

A similar result may be found in the preprint [16].

It is worth mentioning that the results on duality theory here addressed and developed are inspired mainly in the approaches of Telega et al. presented in the articles [17–20]. Other main reference is the D.C. approach found in the article by Toland [21].

Moreover, details on the Sobolev spaces involved may be found in the study by Adams and Fournier [15], and basic theoretical results in superconductivity may be found in previous studies [22,23].

Similar results and models are addressed in previous studies [9,11,13,14,24–26].

Basic results on convex analysis are addressed in previous studies [27,28]. Finally, other related results may be found in previous studies [29,30].

Now we start to describe the primal variational formulation for the Ginzburg–Landau model in superconductivity in question.

Let Ω ⊂ R 3 be an open, bounded, and connected set with a regular (Lipschitzian) boundary by ∂ Ω .

Define the Ginzburg–Landau-type functional J : V → R , by

(9) J ( u ) = γ 2 ∫ Ω ∇ u ⋅ ∇ u d x + α 2 ∫ Ω ( u 2 − β ) 2 d x − ⟨ u , f ⟩ L 2 ,

where

V = W 0 1,2 ( Ω ) ,

γ > 0 , α > 0 , β > 0 , and f ∈ L 2 ( Ω ) . We also denote

Y = Y * = L 2 ( Ω )

and

Y 1 = Y 1 * = L 2 ( Ω ; R 3 ) .

6 Main duality principle and related dual variational formulation

In this section, we develop in detail the main duality principle and respective convex dual variational formulation for the model in question. We highlight that some similar results have been obtained in the preprint [31].

Define the functionals F 1 : Y 1 → R , F 2 : V × Y * → R and F 3 : V → R by

(10) F 1 ( ∇ u ) = γ 2 ∫ Ω ∇ u ⋅ ∇ u d x ,

(11) F 2 ( u , v 0 * ) = ⟨ u 2 , v 0 * ⟩ L 2 − ⟨ u , f ⟩ L 2 + K 2 ∫ Ω u 2 d x − 1 2 α ∫ Ω ( v 0 * ) 2 d x − β ∫ Ω v 0 * d x

and

F 3 ( u ) = K 2 ∫ Ω u 2 d x ,

where K > 0 is a real constant to be specified.

Also, we define the polar functionals F 1 * : Y 1 * → R , F 2 * : Y 1 * × [ Y * ] 2 → R , and F 3 * : Y * → R , by

(12) F 1 * ( v 1 * ) = sup w ∈ Y { ⟨ w , v 1 * ⟩ L 2 − F 1 ( w ) } = 1 2 γ ∫ Ω ∣ v 1 * ∣ 2 d x ,

(13) F 2 * ( v 1 * , v 0 * , z * ) = sup u ∈ V { − ⟨ ∇ u , v 1 * ⟩ L 2 + ⟨ u , + z * ⟩ L 2 − F 2 ( u , v 0 * ) } = 1 2 ∫ Ω ( div v 1 * + z * + f ) 2 ( 2 v 0 * + K ) d x + 1 2 α ∫ Ω ( v 0 * ) 2 d x + β ∫ Ω v 0 * d x ,

if ‖ v 0 * ‖ ∞ ≤ K ⁄ 4 , and

(14) F 3 * ( z * ) = sup w ∈ Y { ⟨ w , z * ⟩ L 2 − F 3 ( w ) } = 1 2 K ∫ Ω ( z * ) 2 d x .

For K 3 = 3 , define now,

A + = { u ∈ V : u f ≥ 0 , in Ω } , V 1 = { u ∈ A + : ‖ u ‖ ∞ ≤ K 3 } , B * = { v 0 * ∈ Y * : ‖ v 0 * ‖ ∞ ≤ K ⁄ 4 } , D * = { v 1 * ∈ Y 1 * : ‖ div v 1 * ‖ ∞ ≤ ‖ f ‖ ∞ + K K 3 ⁄ 2 } ,

and

E * = { z * ∈ Y * : ‖ z * ‖ ∞ ≤ K K 3 and z * f ≥ 0 , in Ω } .

Define also, J * : D * × B * × E * → R by

J * ( v 1 * , v 0 * , z * ) = − F 1 * ( v 1 * ) − F 2 * ( v 1 * , v 0 * , z * ) + F 3 * ( z * ) .

Let ( v ˆ 1 * , v ˆ 0 * , z ˆ * ) ∈ D * × B * × E * be such that

δ J * ( v ˆ 1 * , v ˆ 0 * , z ˆ * ) = 0 .

From the variation of J * in v 1 * , we obtain

∇ div v ˆ 1 * + z ˆ * + f 2 v ˆ 0 * + K − v ˆ 1 * γ = 0 .

Let u 0 ∈ V 1 be such that

u 0 = div v ˆ 1 * + z ˆ * + f 2 v ˆ 0 * + K .

Hence,

v ˆ 1 * = γ ∇ u 0 .

From the variation of J * in z * , we obtain

− v ˆ 1 * + z ˆ * + f ( 2 v ˆ 0 * + K ) + z ˆ * K = 0 ,

so that

− u 0 + z ˆ * K = 0 ,

i.e.,

z ˆ * = K u 0 .

From such results, we may infer that

( 2 v ˆ 0 * + K ) u 0 = div v ˆ 1 * + z ˆ * + f = γ ∇ 2 u 0 + K u 0 + f .

Consequently, we obtain

− γ ∇ 2 u 0 + 2 v ˆ 0 * u 0 − f = 0 .

Moreover, from the variation of J * in v 0 * , we obtain

− v ˆ 0 * α + div v ˆ 1 * + z ˆ * + f 2 v ˆ 0 * + K 2 − β = 0 ,

so that

v ˆ 0 * = α ( u 0 2 − β ) .

Combining the pieces, we obtain

− γ ∇ 2 u 0 + 2 α ( u 0 2 − β ) u 0 − f = 0 , in Ω ,

so that

δ J ( u 0 ) = 0 .

Let α 1 ∈ R be such that

α 1 = inf u ∈ V 1 J ( u ) .

Observe that

(15) α 1 ≤ J ( u ) = F 1 ( ∇ u ) + α 2 ∫ Ω ( u 2 − β ) 2 d x − ⟨ u , f ⟩ L 2 + F 3 ( u ) − ⟨ u , z * ⟩ L 2 + ⟨ u , z * ⟩ L 2 − F 3 ( u ) ≤ F 1 ( ∇ u ) + α 2 ∫ Ω ( u 2 − β ) 2 d x − ⟨ u , f ⟩ L 2 + F 3 ( u ) − ⟨ u , z * ⟩ L 2 + sup w ∈ Y { ⟨ w , z * ⟩ L 2 − F 3 ( w ) } = F 1 ( ∇ u ) + α 2 ∫ Ω ( u 2 − β ) 2 d x − ⟨ u , f ⟩ L 2 + F 3 ( u ) − ⟨ u , z * ⟩ L 2 + F 3 * ( z * ) ,

∀ u ∈ V 1 , z * ∈ E * .

Define now F 5 : V → R , by

F 5 ( u ) = F 1 ( ∇ u ) + α 2 ∫ Ω ( u 2 − β ) 2 d x − ⟨ u , f ⟩ L 2 + F 3 ( u ) .

Define also the polar functional F 5 * : Y * → R by

F 5 * ( z * ) = sup u ∈ V 1 { ⟨ u , z * ⟩ L 2 − F 5 ( u ) } ,

so that from this and the previous results,

(16) α 1 ≤ J ( u ) = inf u ∈ V 1 { F 5 ( u ) − ⟨ u , z * ⟩ L 2 } + F 3 * ( z * ) = − F 5 * ( z * ) + F 3 * ( z * ) ,

∀ u ∈ V 1 , z * ∈ E * .

Here, we assume K > 0 is such that

K ≫ max { K 3 , α , β , γ , 1 ⁄ α , 1 ⁄ γ } ,

Observe that

− F 5 * ( z * ) = inf u ∈ V 1 { − ⟨ u , z * ⟩ L 2 + F 5 ( u ) } .

Define

H ( u , z * ) = − ⟨ u , z * ⟩ L 2 + F 5 ( u ) .

Hence,

(17) H ( u , z * ) = F 1 ( ∇ u ) − ⟨ ∇ u , v 1 * ⟩ L 2 − ⟨ u 2 − β , v 0 * ⟩ L 2 + α 2 ∫ Ω ( u 2 − β ) 2 d x + ⟨ ∇ u , v 1 * ⟩ L 2 + ⟨ u 2 − β , v 0 * ⟩ L 2 + K 2 ∫ Ω u 2 d x − ⟨ u , f ⟩ L 2 − ⟨ u , z * ⟩ L 2 . ≥ inf w ∈ Y { − ⟨ w , v 1 * ⟩ L 2 + F 1 ( w ) } + inf w 1 ∈ Y − ⟨ w 1 , v 0 * ⟩ L 2 + α 2 ∫ Ω w 1 2 d x + inf u ∈ V { − ⟨ u , div v 1 * ⟩ L 2 + ⟨ u 2 − β , v 0 * ⟩ L 2 + K 2 ∫ Ω u 2 d x − ⟨ u , f ⟩ L 2 − ⟨ u , z * ⟩ L 2 . = − F 1 * ( v 1 * ) − F 2 * ( v 1 * , v 0 * , z * ) ,

∀ v 1 * ∈ D * , v 0 * ∈ B * .

In summary, we obtain

− F 5 * ( z * ) = inf u ∈ V H ( u , z * ) ≥ sup ( v 1 * , v 0 * ) ∈ D * × B * { − F 1 * ( v 1 * ) − F 2 * ( v 1 * , v 0 * , z * ) } .

For the appropriate choice of constant K > 0 previously specified, from the general results on convex analysis, we have

− F 5 * ( z * ) = sup ( v 1 * , v 0 * ) ∈ D * × B * { − F 1 * ( v 1 * ) − F 2 * ( v 1 * , v 0 * , z * ) } .

Moreover, also from the previous results, we obtain

(18) α 1 = inf u ∈ V 1 J ( u ) ≤ inf z * ∈ E * { − F 5 * ( z * ) + F 3 * ( z * ) } .

From the general results in Toland [21], we may infer that

(19) α 1 = inf u ∈ V 1 J ( u ) = inf z * ∈ E * { − F 5 * ( z * ) + F 3 * ( z * ) } ,

so that

(20) α 1 = inf u ∈ V 1 J ( u ) = inf z * ∈ E * { − F 5 * ( z * ) + F 3 * ( z * ) } = inf z * ∈ E * { sup ( v 1 * , v 0 * ) ∈ B * × D * { J * ( v 1 * , v 0 * , z * ) } } . ≥ sup ( v 1 * , v 0 * ) ∈ B * × D * { inf z * ∈ E * { J * ( v 1 * , v 0 * , z * ) } } .

Observe that J * is quadratic in z * .

Suppose

∂ 2 J * ( v ˆ 1 * , v ˆ 0 * , z * ) ∂ ( z * ) 2 > 0 .

Under such assumptions, we have

J * ( v ˆ 1 * , v ˆ 0 * , z ˆ * ) = inf z * ∈ E * J * ( v ˆ 1 * , v ˆ 0 * , z * ) .

Remark 6.1

Observe that the condition

∂ 2 J * ( v ˆ 1 * , v ˆ 0 * , z * ) ∂ ( z * ) 2 > 0

clearly is equivalent

− 1 v ˆ 0 * + K + 1 K > 0 ,

which is equivalent to

v ˆ 0 * > 0 , in Ω .

Such a condition is completely equivalent to the positive definiteness of the membrane tensor for the problem addressed by Telega and Bielski [17, 18], obtained initially in 1985. Indeed, such a condition

v ˆ 0 * > 0 , in Ω ,

here for a simpler case, is exactly the same as those obtained in 1985.

In such a sense, our work just complements such original results of Telega et al. [17], Telega and Bielski in [18].

The novelty here is that, for an appropriate sufficiently large of K > 0 , we have obtained a dual formulation expressed as a difference between two convex functionals, the so-called D.C. approach, which lead us to develop a very robust and efficient algorithm to compute a concerning critical point for the model in question.

We also highlight such a condition

v ˆ 0 * > 0 , in Ω ,

is not satisfied with the specified essential boundary conditions

u = 0 ,

on ∂ Ω .

However, this condition may be satisfied for a more general case with natural conditions

∇ u ⋅ n = 0 ,

on ∂ Ω .

Moreover, from an evident convexity of J * in ( v 1 * , v 0 * ) , we obtain

J * ( v ˆ 1 * , v ˆ 0 * , z ˆ * ) = sup ( v 1 * , v 0 * ) ∈ D * × B * J * ( v 1 * , v 0 * , z ˆ * ) .

Consequently, from such results and a standard Saddle point theorem, we may infer that

(21) J * ( v ˆ 1 * , v ˆ 0 * , z ˆ * ) = inf z * ∈ E * { sup ( v 1 * , v 0 * ) ∈ D * × B * J * ( v 1 * , v 0 * , z * ) } = sup ( v 1 * , v 0 * ) ∈ D * × B * { inf z * ∈ E * J * ( v 1 * , v 0 * , z * ) } .

Moreover, from the Legendre transform properties, we may obtain

J * ( v ˆ 1 * , v ˆ 0 * , z ˆ * ) = J ( u 0 ) .

Combining the pieces, we obtain

(22) J ( u 0 ) = inf u ∈ V 1 J ( u ) = J * ( v ˆ 1 * , v ˆ 0 * , z ˆ * ) = inf z * ∈ E * { sup ( v 1 * , v 0 * ) ∈ D * × B * J * ( v 1 * , v 0 * , z * ) } = sup ( v 1 * , v 0 * ) ∈ D * × B * { inf z * ∈ E * J * ( v 1 * , v 0 * , z * ) } .

The objective of this section is complete.

6.1 Numerical example

We have obtained a critical point for J * through the following algorithm (with some slight differences):

Set n = 1 , δ = 1 0 − 4 , K = 20 , and ( z * ) n ≡ 0.8 .
Calculate ( ( v 1 * ) n , ( v 0 * ) n ) ∈ D * × B * such that
∂ J * ( ( v 1 * ) n , ( v 0 * ) n , ( z * ) n ) ∂ v 1 * = 0
and
∂ J * ( ( v 1 * ) n , ( v 0 * ) n , ( z * ) n ) ∂ v 0 * = 0 .
Set
u n = div ( v 1 * ) n + ( z * ) n + f 2 ( v 0 * ) n + K
and
( z * ) n + 1 = K u n .
If ‖ ( z * ) n + 1 − ( z * ) n ‖ ∞ ⁄ K ≤ δ or n > 100 , then stop. Otherwise, n ≔ n + 1 and go item 2.

We have obtained a critical point ( v ˆ 1 * , v ˆ 0 * , z ˆ * ) ∈ D * × B * × E * for J * , for an one-dimensional case, with the following values:

γ = 0.1 , α = β = 1 , f ≡ 1 , on Ω = [ 0,1 ] and K = 20 .

For the corresponding solution

u 0 = div v ˆ 1 * + z ˆ * + f 2 v ˆ 0 * + K ,

please see Figure 1.

$Figure 1 Graph of function u 0 ( x ) {u}_{0}\left(x) on the interval [ 0,1 ] \left[\mathrm{0,1}] .$

Figure 1

Graph of function u 0 ( x ) on the interval [ 0,1 ] .

Here, we present the software through which we have obtained such numerical results.

******************************

clear all
global m8 d yo vo A1 B A u z K
m8=100;
d=1/m8;
K=20;
A=1;
B=1;
A1=0.1;
yo(:,1)=ones(m8,1);
z(:,1)=0.8*ones(m8,1);
xo=0.95*ones(2*m8,1);
x1=xo;
b14=1;
k1=1;
while ( b 14 > 1 0 − 4 ) && ( k 1 < 100 )

k1
k1=k1+1;
b12=1;
k=1;
while ( b 12 > 1 0 − 4 ) && ( k < 50 )
k
k=k+1;
X=fminunc(’funOct202492’,xo);
b12=max(abs(X-xo))
xo=X;
u(m8/2,1)
end;
b14=max(abs(x1-xo))
z=K*u;
x1=xo;
u5(k1,1)=u(m8/2,1);
end;
for i=1:m8
x3(i,1)=i*d;
end;
plot(x3,u)

****************************

With the auxiliary function “funOct202492”

************************

function S=funOct202492(x)
global m8 d yo vo A1 u A B K z
Id=eye(m8);
for i=1:m8
v1(i,1)=x(i,1);
vo(i,1)=x(i+m8,1);
end;
for i=1:m8-1
dv1(i,1)=(v1(i+1,1)-v1(i,1))/d;
end;
S=0;
for i=1:m8-1
S = S + v 1 ( i , 1 ) 2 ⁄ 2 ⁄ A 1 + ( d v 1 ( i , 1 ) + y o ( i , 1 ) + z ( i , 1 ) ) 2 / ( 2 * v o ( i , 1 ) + K ) ⁄ 2 + v o ( i , 1 ) 2 / A ⁄ 2 + B * v o ( i , 1 ) ;
S = S + z ( i , 1 ) 2 ⁄ 2 ⁄ K ;
end;
for i=1:m8-1
u(i,1)=(dv1(i,1)+yo(i,1)+z(i,1))/(2*vo(i,1)+K);
end;
u(m8,1)=0;

***********************************************

6.2 Duality principle for an exactly penalized primal variational formulation

In this section, we develop in detail the main duality principle and respective dual variational formulation for an exactly penalized primal formulation.

Define the functionals F 1 : Y 1 → R , F 2 : V × Y * → R , F 3 : V × Y * → R , and F 4 : V → R by

(23) F 1 ( ∇ u ) = γ 2 ∫ Ω ∇ u ⋅ ∇ u d x ,

(24) F 2 ( u , v 0 * ) = ⟨ u 2 , v 0 * ⟩ L 2 − ⟨ u , f ⟩ L 2 + K 2 ∫ Ω u 2 d x + K 1 2 ∫ Ω u 2 d x − 1 2 α ∫ Ω ( v 0 * ) 2 d x − β ∫ Ω v 0 * d x

F 3 ( u , v 0 * ) = 1 4 α K 3 2 ∫ Ω ( − γ ∇ 2 u + 2 v 0 * u − f ) 2 d x − K 1 2 ∫ Ω u 2 d x ,

and

F 4 ( u ) = K 2 ∫ Ω u 2 d x ,

where K , K 1 > 0 are the real constants to be specified.

Also, we define the polar functionals F 1 * : Y 1 * → R , F 2 * : Y 1 * × [ Y * ] 3 → R , F 3 * : [ Y * ] 2 → R , and F 4 * : Y * → R by

(25) F 1 * ( v 1 * ) = sup w ∈ Y { ⟨ w , v 1 * ⟩ L 2 − F 1 ( w ) } = 1 2 γ ∫ Ω ∣ v 1 * ∣ 2 d x ,

(26) F 2 * ( v 1 * , v 2 * , v 0 * , z * ) = sup u ∈ V { − ⟨ ∇ u , v 1 * ⟩ L 2 + ⟨ u , − v 2 * + z * ⟩ L 2 − F 2 ( u , v 0 * ) } = 1 2 ∫ Ω ( div v 1 * − v 2 * + z * + f ) 2 ( 2 v 0 * + K + K 1 ) d x + 1 2 α ∫ Ω ( v 0 * ) 2 d x + β ∫ Ω v 0 * d x ,

if ‖ 2 v 0 * ‖ ∞ ≤ K ⁄ 4 ,

(27) F 3 * ( v 2 * ) = sup u ∈ Y { ⟨ u , v 2 * ⟩ L 2 − F 3 ( u , v 0 * ) }

and

(28) F 4 * ( z * ) = sup w ∈ Y { ⟨ w , z * ⟩ L 2 − F 3 ( w ) } = 1 2 K ∫ Ω ( z * ) 2 d x .

Here, we specify again K 3 = 3 and

A + = { u ∈ V : u f ≥ 0 , in Ω } , V 1 = { u ∈ A + : ‖ u ‖ ∞ ≤ K 3 } , B * = { v 0 * ∈ Y * : ‖ v 0 * ‖ ∞ ≤ K ⁄ 4 } , D 1 * = { v 1 * ∈ Y 1 * : ‖ div v 1 * ‖ ∞ ≤ ‖ f ‖ ∞ + K K 3 ⁄ 2 } , D 2 * = { v 2 * ∈ Y * : ‖ v 2 * ‖ ∞ ≤ ( 5 ⁄ 4 ) K 1 } ,

and

E * = { z * ∈ Y * : ‖ z * ‖ ∞ ≤ K K 3 and z * f ≥ 0 , in Ω } .

We assume K > 0 is such that

K ≫ max { K 1 , K 3 , α , β , γ , 1 ⁄ α , 1 ⁄ γ } ,

and K 1 > 0 is such that

∂ 2 F 3 ( u , v 0 * ) ∂ u 2 ≥ 0 , ∀ v 0 * ∈ B * .

Define also, J 1 * : D 1 * × D 2 * × B * × E * → R by

J 1 * ( v 1 * , v 2 * , v 0 * , z * ) = − F 1 * ( v 1 * ) − F 2 * ( v 1 * , v 2 * , v 0 * , z * ) − F 3 * ( v 2 * ) + F 4 * ( z * ) .

Let ( v ˆ 1 * , v ˆ 2 * , v ˆ 0 * , z ˆ * ) ∈ D 1 * × D 2 * × B * × E * be such that

δ J 1 * ( v ˆ 1 * , v ˆ 2 * , v ˆ 0 * , z ˆ * ) = 0 .

From the variation of J 1 * in v 1 * , we obtain

∇ div v ˆ 1 * − v ˆ 2 * + z ˆ * + f 2 v ˆ 0 * + K + K 1 − v ˆ 1 * γ = 0 .

Let u 0 ∈ V 1 be such that

u 0 = div v ˆ 1 * − v ˆ 2 * + z ˆ * + f 2 v ˆ 0 * + K + K 1 .

Hence,

v ˆ 1 * = γ ∇ u 0 .

From the variation of J 1 * in z * , we obtain

− div v ˆ 1 * − v ˆ 2 * + z ˆ * + f ( 2 v ˆ 0 * + K + K 1 ) + z ˆ * K = 0 ,

so that

− u 0 + z ˆ * K = 0 ,

i.e.,

z ˆ * = K u 0 .

From the variation of J 1 * in v 2 * , similarly as in the previous sections, we obtain

div v ˆ 1 * − v ˆ 2 * + z ˆ * + f ( 2 v ˆ 0 * + K + K 1 ) − u ˆ = 0 ,

where defining

H 3 ( u , v ˆ 2 * , v ˆ 0 * ) = ⟨ u , v ˆ 2 * ⟩ L 2 − F 3 ( u , v ˆ 0 * ) ,

we have that u ˆ is such that

∂ H 3 ( u ˆ , v ˆ 2 * , v ˆ 0 * ) ∂ u = 0 ,

i.e., denoting A 5 = − γ ∇ 2 u 0 + 2 v 0 * u 0 − f , we have

(29) v ˆ 2 * = 1 2 α K 3 2 ( − γ ∇ 2 + 2 v ˆ 0 * ) ( − γ ∇ 2 u ˆ + 2 v 0 * u ˆ − f ) − K 1 u ˆ = 1 2 α K 3 2 ( − γ ∇ 2 + 2 v ˆ 0 * ) A 5 − K 1 u 0 .

From such results, we may infer that

( 2 v ˆ 0 * + K + K 1 ) u 0 = div v ˆ 1 * − v ˆ 2 * + z ˆ * + f = γ ∇ 2 u 0 − 1 2 α K 3 2 ( − γ ∇ 2 + 2 v ˆ 0 * ) A 5 + K 1 u 0 + K u 0 + f .

Consequently, we obtain

− γ ∇ 2 u 0 + 2 v ˆ 0 * u 0 − f + 1 2 α K 3 2 ( − γ ∇ 2 + 2 v ˆ 0 * ) A 5 = 0 ,

so that

A 5 + 1 2 α K 3 2 ( − γ ∇ 2 + 2 v ˆ 0 * ) A 5 = 0 .

Hence, from this and the essential boundary conditions:

A 5 = 0 , on ∂ Ω ,

we obtain

A 5 = 0 ,

so that

− γ ∇ 2 u 0 + 2 v ˆ 0 * u 0 − f = 0 .

Moreover, with a similar reason, from the variation of J 1 * in v 0 * , we obtain

− v ˆ 0 * α + div v ˆ 1 * + z ˆ * + f 2 v ˆ 0 * + K + K 1 2 − β = 0 ,

so that

v ˆ 0 * = α ( u 0 2 − β ) .

Combining the pieces, we obtain

− γ ∇ 2 u 0 + 2 α ( u 0 2 − β ) u 0 − f = 0 , in Ω ,

so that

δ J ( u 0 ) = 0 .

Moreover, from the Legendre transform properties, we may obtain

J 1 * ( v ˆ 1 * , v ˆ 2 * , v ˆ 0 * , z ˆ * ) = J ( u 0 ) .

Observe that J 1 * is quadratic in z * .

Suppose

∂ 2 J 1 * ( v ˆ 1 * , v ˆ 2 * , v ˆ 0 * , z * ) ∂ ( z * ) 2 > 0 .

Hence, under such assumptions, we have

J 1 * ( v ˆ 1 * , v ˆ 2 * , v ˆ 0 * , z ˆ * ) = inf z * ∈ E * J 1 * ( v ˆ 1 * , v ˆ 2 * , v ˆ 0 * , z * ) .

Moreover, from an evident concavity of J 1 * in ( v 1 * , v 2 * , v 0 * ) on D 1 * × D 2 * × B * , we have

J 1 * ( v ˆ 1 * , v ˆ 2 * , v ˆ 0 * , z ˆ * ) = sup ( v 1 * , v 0 * , v 2 * ) ∈ D 1 * × D 2 * × B * J 1 * ( v 1 * , v 2 * , v 0 * , z ˆ * ) .

From such results and a standard Saddle point theorem, we may infer that

(30) J 1 * ( v ˆ 1 * , v ˆ 2 * , v ˆ 0 * , z ˆ * ) = inf z * ∈ E * { sup ( v 1 * , v 2 * , v 0 * ) ∈ D 1 * × D 2 * × B * J 1 * ( v 1 * , v 2 * , v 0 * , z * ) } = sup ( v 1 * , v 2 * , v 0 * ) ∈ D 1 * × D 2 * × B * { inf z * ∈ E * J 1 * ( v 1 * , v 2 * , v 0 * , z * ) } .

Therefore,

(31) J ( u 0 ) = J 1 * ( v ˆ 1 * , v ˆ 2 * , v ˆ 0 * , z ˆ * ) ≤ J 1 * ( v ˆ 1 * , v ˆ 2 * , v ˆ 0 * , z * ) ≤ F 1 ( ∇ u ) + F 2 ( u , v ˆ 0 * ) + F 3 ( u , v ˆ 0 * ) − ⟨ u , z * ⟩ L 2 + F 4 * ( z * ) ,

∀ u ∈ V 1 , z * ∈ E * .

In particular, for z * = K u , we obtain

(32) J ( u 0 ) = J 1 * ( v ˆ 1 * , v ˆ 2 * , v ˆ 0 * , z ˆ * ) ≤ F 1 ( ∇ u ) + F 2 ( u , v ˆ 0 * ) + F 3 ( u , v ˆ 0 * ) − F 4 ( u ) = γ 2 ∫ Ω ∇ u ⋅ ∇ u d x + ⟨ u 2 , v ˆ 0 * ⟩ L 2 − ⟨ u , f ⟩ L 2 − 1 2 α ∫ Ω ( v ˆ 0 * ) 2 d x − β ∫ Ω v ˆ 0 * d x + 1 4 α K 3 2 ∫ Ω ( − γ ∇ 2 u + 2 v ˆ 0 * u − f ) 2 d x ≤ γ 2 ∫ Ω ∇ u ⋅ ∇ u d x − ⟨ u , f ⟩ L 2 + sup v 0 * ∈ Y * ⟨ u 2 , v 0 * ⟩ L 2 − 1 2 α ∫ Ω ( v 0 * ) 2 d x − β ∫ Ω v 0 * d x + 1 4 α K 3 2 ∫ Ω ( − γ ∇ 2 u + 2 v ˆ 0 * u − f ) 2 d x = γ 2 ∫ Ω ∇ u ⋅ ∇ u d x − ⟨ u , f ⟩ L 2 + α 2 ∫ Ω ( u 2 − β ) 2 d x + 1 4 α K 3 2 ∫ Ω ( − γ ∇ 2 u + 2 v ˆ 0 * u − f ) 2 d x = J ( u ) + 1 4 α K 3 2 ∫ Ω ( − γ ∇ 2 u + 2 v ˆ 0 * u − f ) 2 d x ,

∀ u ∈ V 1 .

Combining the pieces, we obtain

(33) J ( u 0 ) = J ( u 0 ) + 1 4 α K 3 2 ∫ Ω ( − γ ∇ 2 u 0 + 2 v ˆ 0 * u 0 − f ) 2 d x = inf u ∈ V 1 J ( u ) + 1 4 α K 3 2 ∫ Ω ( − γ ∇ 2 u + 2 v ˆ 0 * u − f ) 2 d x = J * ( v ˆ 1 * , v ˆ 2 * , v ˆ 0 * , z ˆ * ) = inf z * ∈ E * { sup ( v 1 * , v 2 * , v 0 * ) ∈ D 1 * × D 2 * × B * J * ( v 1 * , v 2 * , v 0 * , z * ) } = sup ( v 1 * , v 2 * , v 0 * ) ∈ D 1 * × D 2 * × B * { inf z * ∈ E * J * ( v 1 * , v 2 * , v 0 * , z * ) } .

The objective of this section is complete.

7 Conclusion

In this article, we have developed variational formulations for both the incompressible and compressible Euler systems in fluid mechanics. We emphasize again both the variational formulations comprise the fluid kinetics energy as their main part, subject to a mass conservation equation as a constraint. Moreover, the Lagrange multiplier corresponding to such a constraint is related to the fluid pressure field.

It is also worth mentioning that for the compressible case, we have obtained both the Euler and Bernoulli equations in a single system as necessary conditions for a stationary point for the functional in question.

Finally, in the last sections, we have developed duality principles applied to a Ginzburg–Landau-type equation.

We highlight that the final variational formulation for such a mentioned duality principle is suitable for a primal one, which is originally non-convex.

Funding information: The authors state no funding involved.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Received: 2024-07-26

Revised: 2024-11-30

Accepted: 2025-02-09

Published Online: 2025-03-12

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Keywords for this article

incompressible Euler system; compressible Euler system; variational formulations

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