On duality principles and related convex dual formulations suitable for local and global non-convex variational optimization

Fabio Silva Botelho

doi:10.1515/nleng-2022-0343

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On duality principles and related convex dual formulations suitable for local and global non-convex variational optimization

Published/Copyright: October 31, 2023

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From the journal Nonlinear Engineering Volume 12 Issue 1

Abstract

This article develops duality principles, a related convex dual formulation and primal dual formulations suitable for the local and global optimization of non convex primal formulations for a large class of models in physics and engineering. The results are based on standard tools of functional analysis, calculus of variations and duality theory. In particular, we develop applications to a Ginzburg–Landau type equation. Other applications include primal dual variational formulations for a Burger’s type equation and a Navier–Stokes system. We emphasize the novelty here is that the first dual variational formulation developed is convex for a primal formulation which is originally non-convex. Finally, we also highlight the primal dual variational formulations presented have a large region of convexity around any of their critical points.

Keywords: convex dual variational formulations; duality principles for non-convex primal local and global optimization; Ginzburg–Landau type equation

MSC 2010: 49N15

1 Introduction

In the first part of this article, we establish a duality principle and a related convex dual formulation suitable for the local optimization of a primal formulation for a large class of models in non-convex optimization. We highlight that the first dual variational formulation presented is convex, and such a feature may be very useful for a large class of similar models, in particularly for large systems in a three or higher dimensional context.

For such a large system, the convexity obtained is relevant for an easier numerical computation, since in such a case of strict convexity, the standard Newton, Newton-type, and other similar methods are always convergent.

We also emphasize that the main duality principle is applied to the Ginzburg–Landau system in superconductivity in the absence of a magnetic field.

Other applications include primal dual formulations for a Burger’s type equation and a Navier–Stokes system.

For the duality principles, the results are based on the works of Telega and Bielski [1–4] and on a DC optimization approach developed in the study by Toland [5].

About the other references, details on the Sobolev spaces involved are found in the study by Adams and Fournier [6]. Related and more recent results on convex analysis and duality theory are addressed in the previous studies [7–11]. In particular, the results in the present work are extensions, and improvements of those results found in the recent book [12] and the recent article [13], which by the way, are also based on the previous articles [1–4]. Similar models on the superconductivity physics may be found in the studies by Annet and Landau [14,15].

Finally, in the last section, we develop a duality principle for the quasi-convex relation of a general model in the vectorial calculus of variations.

Remark 1.1

We may generically denote

∫ Ω [ ( − γ ∇ 2 + K I d ) − 1 v * ] v * d x

simply by

∫ Ω ( v * ) 2 − γ ∇ 2 + K d x ,

where I d denotes a concerning identity operator. Here, we will denote − γ ∇ 2 + K I d simply as − γ ∇ 2 + K .

Other similar notations may be used along this text as their indicated meaning are sufficiently clear.

Finally, ∇ 2 denotes the Laplace operator, and for real constants K 2 > 0 and K 1 > 0 , the notation K 2 ≫ K 1 means that K 2 > 0 is much larger than K 1 > 0 .

Now we present some basic definitions and statements.

Definition 1.2

Let V be a Banach space. We define the topological dual space of V , denoted by V ′ , as the set of all continuous and linear functionals defined on V .

We assume V ′ may be represented through another Banach space denoted by V * and a bilinear form

⟨ ⋅ , ⋅ ⟩ V : V × V * → R .

More specifically, for each f ∈ V ′ , we suppose there exists a unique u * ∈ V * such that

f ( u ) = ⟨ u , u * ⟩ V , ∀ u ∈ V .

Moreover, we define the norm of f , which is denoted by

‖ f ‖ V *

‖ f ‖ V * = sup { ∣ ⟨ u , u * ⟩ V : u ∈ V and ‖ u ‖ V ≤ 1 } ≡ ‖ u * ‖ V * .

For an open, bounded, and connected set Ω ⊂ R N and Y = Y * = L 2 ( Ω ) , we recall that

⟨ u , u * ⟩ L 2 = ∫ Ω u u * d x .

More specifically, for each continuous and linear functional f : Y → R , there exists a unique u * ∈ Y * = L 2 ( Ω ) such that

f ( u ) = ∫ Ω u u * d x , ∀ u ∈ Y = L 2 ( Ω ) .

Definition 1.3

(The first variation) Let V be a Banach space. Let F : V → R be a functional.

We define the first variation of F at u ∈ V on the direction φ ∈ V , denoted by δ F ( u ; φ ) as follows:

δ F ( u ; φ ) = lim ε → 0 F ( u + ε φ ) − F ( u ) ε ,

if such a limit exists.

If there exists u * ∈ V * such that

δ F ( u ; φ ) = ⟨ φ , u * ⟩ V , ∀ φ ∈ V ,

we say that F is Gâteaux differentiable at u . Moreover, in such a case, u * ∈ V * is said to be the Gâteaux derivative of F at u .

We may also denote

u * = δ F ( u ) or u * = ∂ F ( u ) ∂ u .

Definition 1.4

(The second variation) Let V be a Banach space. Let F : V → R be a functional.

We define the second variation of F at u ∈ V on the directions φ ∈ V and φ 1 ∈ V , denoted by δ 2 F ( u ; φ , φ 1 ) , as

δ 2 F ( u ; φ , φ 1 ) = lim ε → 0 δ F ( u + ε φ 1 ; φ ) − δ F ( u ; φ ) ε ,

if such a limit exists.

Definition 1.5

(Polar functional) Let V be a Banach space, and let F : V → R be a functional.

We define the polar functional of F , denoted by F * : V * → R , by

F * ( u * ) = sup u ∈ V { ⟨ u , u * ⟩ V − F ( u ) } , ∀ u * ∈ V * .

Another important definition refers to the Legendre transform one and respective relevant propriety, which are summarized in the next theorem.

Theorem 1.6

(Legendre transform theorem) Let V be a Banach space, and let F : V → R be a twice continuously Fréchet differentiable functional.

Let u * ∈ V * . Assume there exists a unique u ˆ ∈ V such that

u * = ∂ F ( u ˆ ) ∂ u .

Suppose also

det ∂ 2 F ( u ) ∂ u 2 ≠ 0 ,

in a neighborhood of u ˆ .

Under such hypotheses, defining the Legendre transform of F at u * by F L * ( u * ) , where

F L * ( u * ) = ⟨ u ˆ , u * ⟩ V − F ( u ˆ ) ,

we have that

u ˆ = ∂ F L * ( u * ) ∂ u * .

Remark 1.7

Concerning such a last definition, observe that if F is convex on V , then the extremal condition

u * = ∂ F ( u ˆ ) ∂ u ,

corresponds to globally maximize

H ( u ) = ⟨ u , u * ⟩ V − F ( u )

on V , so that, in such a case,

F * ( u * ) = H ( u ˆ ) = ⟨ u ˆ , u * ⟩ V − F ( u ˆ ) = F L * ( u * ) .

Summarizing, if F is convex, under the hypotheses of the last theorem, the polar functional F * ( u * ) coincides with the Legendre transform of F on V * already denoted by F L * , that is,

F * ( u ) = F L * ( u * ) , ∀ u * ∈ V * .

2 The primal variational formulation and the dual functional definitions

At this point, we start to describe the primal and dual variational formulations.

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

Consider a functional J : V → R , where

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

Here, γ > 0 , α > 0 , β > 0 , and f ∈ L 2 ( Ω ) ∩ L ∞ ( Ω ) .

Moreover, V = W 0 1 , 2 ( Ω ) and we denote Y = Y * = L 2 ( Ω ) .

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

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

and

F 2 ( u ) = K 2 2 ∫ Ω u 2 d x .

At this point, we assume a finite dimensional version for this concerning model. For example, we may define a new domain for the primal functional considering the projection of V on the space spanned by the first N (in general N = 10 , is enough) eigen-vectors of the Laplace operator, corresponding to the first N eigen-values. On this new not relabeled finite dimensional space V , since v 0 * corresponds to a diagonal matrix, there exists c 0 > 0 such that

( − γ ∇ 2 + 2 v 0 * ) 2 ≥ c 0 I d ,

∀ v 0 * ∈ B * , where

B * = { v 0 * ∈ Y * : ‖ 2 v 0 * ‖ ∞ < K ∕ 2 } ,

for an appropriate real constant K > 0 .

We define also J 1 : V × Y → R by

J 1 ( u , v 0 * ) = F 1 ( u , v 0 * ) − F 2 ( u ) ,

and F 1 * : Y * × B * → R , F 2 * : Y * → R by

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

and,

(4) F 2 * ( v 2 * ) = sup u ∈ V { ⟨ u , v 2 * ⟩ L 2 − F 2 ( u ) } = 1 2 K 2 ∫ Ω ( v 2 * ) 2 d x ,

respectively.

Furthermore, we define

D * = { v 2 * ∈ Y * : ‖ v 2 * ‖ ∞ ≤ 5 K 2 ∕ 4 }

and J 1 * : D * × B * → R , by

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

Assuming 0 < α ≪ 1 (through a re-scaling, if necessary) and

K 2 ≫ K 1 ≫ K ≫ max { ‖ f ‖ ∞ , α , β , γ , 1 }

by directly computing δ 2 J 1 * ( v 2 * , v 0 * ) we may easily obtain that for such specified real constants, J 1 * in convex in ( v 2 * , v 0 * ) on D * × B * .

3 The main duality principle and a concerning convex dual formulation

Considering the statements and definitions presented in the previous section, we may prove the following theorem.

Theorem 3.1

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

δ J 1 * ( v ˆ 2 * , v ˆ 0 * ) = 0

and u 0 ∈ V be such that

u 0 = ∂ F 2 * ( v ˆ 2 * ) ∂ v 2 * .

Under such hypotheses, we have

δ J ( u 0 ) = 0

and

(5) − J ( u 0 ) = J 1 ( u 0 , v ˆ 0 * ) = inf u ∈ V J 1 ( u , v ˆ 0 * ) = inf ( v 2 * , v 0 * ) ∈ D * × B * J 1 * ( v 2 * , v 0 * ) = J 1 * ( v ˆ 2 * , v ˆ 0 * ) .

Proof

Observe that δ J 1 * ( v ˆ 2 * , v ˆ 0 * ) = 0 so that, since J 1 * is convex in ( v 2 * , v 0 * ) on D * × B * , we obtain

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

Now we will show that

δ J ( u 0 ) = 0 .

From

∂ J 1 * ( v ˆ 2 * , v ˆ 0 * ) ∂ v 2 * = 0

and

∂ F 2 * ( v ˆ 2 * ) ∂ v 2 * = u 0 ,

we have

− ∂ F 1 * ( v ˆ 2 * , v 0 * ) ∂ v 2 * + u 0 = 0

and

v ˆ 2 * − K 2 u 0 = 0 .

Observe now that denoting

H ( v 2 * , v 0 * , u ) = ⟨ u , v 2 * ⟩ L 2 − F 1 ( u , v 0 * ) ,

there exists u ˆ ∈ V such that

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

and

F 1 * ( v ˆ 2 * , v ˆ 0 * ) = H ( v ˆ 2 * , v ˆ 0 * , u ˆ ) ,

so that

(6) ∂ F 1 * ( v ˆ 2 * , v ˆ 0 * ) ∂ v 2 * = ∂ H ( v ˆ 2 * , v ˆ 0 * , u ˆ ) ∂ v 2 * + ∂ H ( v ˆ 2 * , v ˆ 0 * , u ˆ ) ∂ u ∂ u ˆ ∂ v 2 * = u ˆ .

Summarizing, we obtain

u 0 = ∂ F 1 * ( v ˆ 2 * , v ˆ 0 * ) ∂ v 2 * = u ˆ .

Also, denoting

A ( u 0 , v ˆ 0 * ) = − γ ∇ 2 u 0 + 2 v ˆ 0 * u 0 − f ,

from

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

we have

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

so that

(7) − A ( u 0 , v ˆ 0 * ) + K 1 ( − γ ∇ 2 + 2 v ˆ 0 * ) A ( u 0 , v ˆ 0 * ) = 0 .

From such results, we may infer that

A ( u 0 , v ˆ 0 * ) = − γ ∇ 2 u 0 + 2 v ˆ 0 * − f = 0 , in Ω .

Moreover, from

∂ J 1 * ( v ˆ 2 * , v ˆ 0 * ) ∂ v 0 * = 0 ,

we have

K 1 A ( u 0 , v ˆ 0 * ) 2 u 0 + v ˆ 0 * α − u 0 2 + β = 0 ,

so that

v 0 * = α ( u 0 2 − β ) .

From such last results, we obtain

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

and thus,

δ J ( u 0 ) = 0 .

Furthermore, also from such last results and the Legendre transform properties, we have

F 1 * ( v ˆ 2 * , v ˆ 0 * ) = ⟨ u 0 , v ˆ 2 * ⟩ L 2 − F 1 ( u 0 , v ˆ 0 * ) ,

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

so that

(8) J 1 * ( v ˆ 2 * , v ˆ 0 * ) = − F 1 * ( v ˆ 2 * , v ˆ 0 * ) + F 2 * ( v ˆ 2 * ) = F 1 ( u 0 , v ˆ 0 * ) − F 2 ( u 0 ) = J 1 ( u 0 , v ˆ 0 * ) = − J ( u 0 ) .

Finally, observe that from a concerning convexity,

− J ( u 0 ) = J 1 ( u 0 , v ˆ 0 * ) = inf u ∈ V J 1 ( u , v ˆ 0 * ) .

Joining the pieces, we obtain

(9) − J ( u 0 ) = J 1 ( u 0 , v ˆ 0 * ) = inf u ∈ V { J 1 ( u , v ˆ 0 * ) } = inf ( v 2 * , v 0 * ) ∈ D * × B * J 1 * ( v 2 * , v 0 * ) = J 1 * ( v ˆ 2 * , v ˆ 0 * ) .

The proof is complete.

4 A primal dual formulation for a local optimization of the primal one

In this section, we develop a primal dual formulation corresponding to a non-convex primal formulation.

We start by describing the primal formulation.

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

For the primal formulation, consider a functional J : V → R , where

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

Here, γ > 0 , α > 0 , β > 0 and f ∈ L 2 ( Ω ) ∩ L ∞ ( Ω ) .

Moreover, V = W 0 1 , 2 ( Ω ) , and we denote Y = Y * = L 2 ( Ω ) .

Define the functional J 1 * : V × [ Y * ] 2 → R , by

(11) J 1 * ( u , v 3 * , v 0 * ) = γ 2 ∫ Ω ∇ u ⋅ ∇ u d x + ⟨ u 2 , v 0 * ⟩ L 2 + K 1 2 ∫ Ω ( − γ ∇ 2 u + 2 v 3 * u − f ) 2 d x + K 1 2 ∫ Ω ( v 3 * − α ( u 2 − β ) ) 2 d x − ⟨ u , f ⟩ L 2 − 1 2 α ∫ Ω ( v 0 * ) 2 d x − β ∫ Ω v 0 * d x .

We define also

B * = { v 0 * ∈ Y * : ‖ 2 v 0 * ‖ ∞ < K ∕ 2 } ,

for an appropriate real constant K > 0 .

Furthermore, we define

D * = { v 3 * ∈ Y * : ‖ v 3 * ‖ ∞ ≤ K 2 } A + = { u ∈ V : u f ≥ 0 , a.e. in Ω } , V 2 = { u ∈ V : ‖ u ‖ ∞ ≤ K 3 }

for an appropriate real constant K 3 > 0 and

V 1 = A + ∩ V 2 .

Now observe that denoting φ 1 = v 3 * − α ( u 2 − β ) , we have

(12) ∂ 2 J 1 * ( u , v 3 * , v 0 * ) ∂ u 2 = K 1 ( − γ ∇ 2 + 2 v 3 * ) 2 + 4 K 1 α 2 u 2 − 2 K 1 α φ 1 − γ ∇ 2 + 2 v 0 *

and

(13) ∂ 2 J 1 * ( u , v 3 * , v 0 * ) ∂ ( v 3 * ) 2 = K 1 + 4 K 1 u 2 .

Denoting φ = − γ ∇ 2 u + 2 v 0 * u − f , we have also that

(14) ∂ 2 J 1 * ( u , v 3 * , v 0 * ) ∂ u ∂ v 3 * = K 1 ( − 2 γ ∇ 2 u + 2 v 3 * u ) + 2 K 1 φ − 2 K 1 α u .

In such a case, we obtain

(15) det ∂ 2 J 1 * ( u , v 3 * , v 0 * ) ∂ u ∂ v 3 * = ∂ 2 J 1 * ( u , v 3 * , v 0 * ) ∂ ( v 3 * ) 2 ∂ 2 J 1 * ( u , v 3 * , v 0 * ) ∂ u 2 − ∂ 2 J 1 * ( u , v 3 * , v 0 * ) ∂ v 3 * ∂ u 2 = K 1 2 ( − γ ∇ 2 + 2 v 3 * + 4 α u 2 ) 2 + ( − γ ∇ 2 + 2 v 0 * ) O ( K 1 ) − 4 K 1 2 φ 2 − 4 K 1 2 φ [ ( − γ ∇ 2 u + 2 v 0 * u ) − 2 α u ] − 2 K 1 2 α φ 1 ( 1 + 4 u 2 ) .

Observe that at a critical point

φ = 0

and

φ 1 = 0 .

From such results, we may infer that

det ∂ 2 J 1 * ( u , v 3 * , v 0 * ) ∂ u ∂ v 3 * > 0

around any critical point.

With such results in mind, at this point and on assuming a related not relabeled finite dimensional model version, in a finite differences or finite elements context, we may prove the following theorem.

Theorem 4.1

Let ( u 0 , v ˆ 3 * , v ˆ 0 * ) ∈ V 1 × D * × B * be such that

δ J 1 * ( u 0 , v ˆ 3 * , v ˆ 0 * ) = 0 .

Under such hypotheses, we have

δ J ( u 0 ) = − γ ∇ 2 u 0 + 2 α ( u 0 2 − β ) u 0 − f = 0 ,

and there exists r > 0 such that

(16) J ( u 0 ) = sup v 0 * ∈ B * { inf ( u , v 3 * ) ∈ B r ( u 0 , v ˆ 3 * ) J 1 * ( u , v 3 * , v 0 * ) } = J 1 * ( u 0 , v ˆ 3 * , v ˆ 0 * ) .

Proof

The proof that

δ J ( u 0 ) = 0

and

J ( u 0 ) = J 1 * ( u 0 , v ˆ 3 * , v ˆ 0 * )

may be done similarly as in the previous sections.

Observe that, as previously obtained, there exists r > 0 such that

det ∂ 2 J 1 * ( u , v 3 * , v ˆ 0 * ) ∂ u ∂ v 3 * > 0 , ∀ ( u , v 3 * ) ∈ B r ( u 0 , v ˆ 3 * )

and

∂ 2 J 1 * ( u 0 , v ˆ 3 * , v 0 * ) ∂ ( v 0 * ) 2 < 0 , ∀ v 0 * ∈ B * .

Since for a sufficiently large K 1 > 0 , we have

∂ 2 J 1 * ( u , v 3 * , v ˆ 0 * ) ∂ u 2 > 0 , in B r ( u 0 , v ˆ 3 * ) ,

from these last results and the standard Saddle point theorem, we have

J ( u 0 ) = J 1 * ( u 0 , v ˆ 3 * , v ˆ 0 * ) = sup v 0 * ∈ B * { inf ( u , v 3 * ) ∈ B r ( u 0 , v ˆ 3 * ) J 1 * ( u , v 3 * , v 0 * ) } .

The proof is complete.

5 One more primal dual formulation and related convex (in fact concave) dual formulation

In this section, we develop one more primal dual formulation for the model in question.

The novelty here is that a critical point of such a primal dual formulation corresponds to a global optimal point for the concerning original primal one.

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

Consider the functional J : V → R defined by

(17) 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 ) , f ∈ L 2 ( Ω ) , γ > 0 , α > 0 , and β > 0 .

Denoting Y = Y * = L 2 ( Ω ) , define also J 1 : V × [ Y * ] 3 → R by

(18) J 1 ( u , z * , v 0 * , v 1 * ) = − 1 2 ∫ Ω ( v 1 * ) 2 − γ ∇ 2 + K 1 d x + ⟨ u , v 1 * ⟩ L 2 − 1 2 ∫ Ω K 1 u 2 d x + ⟨ u 2 , v 0 * ⟩ L 2 − ⟨ u , f ⟩ L 2 + 1 2 ∫ Ω K u 2 d x − ⟨ u , z * ⟩ L 2 + 1 2 ∫ Ω ( z * ) 2 K d x + 1 2 ∫ Ω × ( − γ ∇ 2 + K 1 ) z * K − v 1 * − γ ∇ 2 + K 1 2 d x − 1 2 α ( v 0 * ) 2 d x − β ∫ Ω v 0 * d x .

Observe that

∂ 2 J 1 ( u , z * , v 0 * , v 1 * ) ∂ u 2 = K − K 1 + 2 v 0 * ,

∂ 2 J 1 ( u , z * , v 0 * , v 1 * ) ∂ ( z * ) 2 = 1 K + − γ ∇ 2 + K 1 K 2 ,

and

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

Hence,

(19) det { δ u z * 2 J ( u , z * , v 0 * , v 1 * ) } = ∂ 2 J 1 ( u , z * , v 0 * , v 1 * ) ∂ u 2 ∂ 2 J 1 ( u , z * , v 0 * , v 1 * ) ∂ ( z * ) 2 − ∂ 2 J 1 ( u , z * , v 0 * , v 1 * ) ∂ u ∂ z * 2 = K ( − γ ∇ 2 + 2 v 0 * ) − K 1 ( − γ ∇ 2 − 2 v 0 * ) − K 1 2 − γ ∇ 2 ( 2 v 0 * ) K 2 ≡ H ( v 0 * , K 1 ) .

The best possible K 1 is obtained through the optimal equation

∂ H ( v 0 * , K 1 ) ∂ K 1 = 0 ,

that is,

− 2 K 1 − ( − γ ∇ 2 − 2 v 0 * ) = 0 ,

so that we set

K 1 = γ ∇ 2 + 2 v 0 * 2 .

By replacing such a K 1 into (19), we obtain

(20) det { δ u z * 2 J ( u , z * , v 0 * , v 1 * ) } = K ( − γ ∇ 2 + 2 v 0 * ) + 1 4 ( − γ ∇ 2 + 2 v 0 * ) 2 K 2 ≡ H 1 ( v 0 * ) .

Setting now K = 5 ( − γ ∇ 2 ) , define

C * = A 1 * ∩ A 2 * ∩ B * ,

where

A 1 * = { v 0 * ∈ Y * : K − K 1 + 2 v 0 * > 0 } , A 2 * = { v 0 * ∈ Y * : H 1 ( v 0 * ) > 0 } ,

and

B * = { v 0 * ∈ Y * : ‖ 2 v 0 * ‖ ∞ < K 2 }

for some appropriate K 2 > 0 .

Now redefining J 1 as

(21) J 1 ( u , z * , v 0 * , v 1 * ) = − 1 2 ∫ Ω ( v 1 * ) 2 − γ ∇ 2 + K 1 d x + ⟨ u , v 1 * ⟩ L 2 − 1 2 ∫ Ω K 1 u 2 d x + ⟨ u 2 , v 0 * ⟩ L 2 − ⟨ u , f ⟩ L 2 + 1 2 ∫ Ω K u 2 d x − ⟨ u , z * ⟩ L 2 + 1 2 ∫ Ω ( z * ) 2 K d x + 1 2 ∫ Ω ( − γ ∇ 2 + K 1 − ε ) z * K − v 1 * − γ ∇ 2 + K 1 2 d x − 1 2 α ( v 0 * ) 2 d x − β ∫ Ω v 0 * d x ,

for some small parameter 0 < ε ≪ 1 , we may prove the following theorem.

Theorem 5.1

Let ( u 0 , z 0 * , v ˆ 0 * , v ˆ 1 * ) ∈ V × Y * × C * × Y * be such that

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

Under such hypotheses, δ J ( u 0 ) = 0 and

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

Proof

The proof that δ J ( u 0 ) = 0 and J ( u 0 ) = J 1 ( u 0 , z 0 * , v ˆ 0 * , v ˆ 1 * ) may be done similarly as in the previous sections and will not be repeated.

Also, from the hypotheses, δ J 1 ( u 0 , z 0 * , v ˆ 0 * , v ˆ 1 * ) and v ˆ 0 * ∈ C * , so that we may infer that

J 1 ( u 0 , z 0 * , v ˆ 0 * , v ˆ 1 * ) = inf ( u , z * ) ∈ V × Y * J 1 ( u , z * , v ˆ 0 * , v ˆ 1 * )

and

J 1 ( u 0 , z 0 * , v ˆ 0 * , v ˆ 1 * ) = sup ( v 0 * , v 1 * ) ∈ C * × Y * J 1 ( u 0 , z 0 * , v 0 * , v 1 * ) .

Thus, from a standard saddle point theorem, we have

J 1 ( u 0 , z 0 * , v ˆ 0 * , v ˆ 1 * ) = sup ( v 0 * , v 1 * ) ∈ C * × Y * { inf ( u , z * ) ∈ V × Y * J 1 ( u 0 , z 0 * , v 0 * , v 1 * ) } .

Finally, observe that

(23) J ( u 0 ) = J 1 ( u 0 , z 0 * , v ˆ 0 * , v ˆ 1 * ) ≤ J 1 ( u , K u , v ˆ 0 * , v ˆ 1 * ) ≤ sup ( v 0 * , v 1 * ) ∈ Y * × Y * J 1 ( u , K u , v 0 * , v 1 * ) = J ( u ) , ∀ u ∈ V .

Summarizing, we obtain

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

Joining the pieces, we may infer that

(24) J ( u 0 ) = inf u ∈ V J ( u ) = sup ( v 0 * , v 1 * ) ∈ C * × Y * { inf ( u , z * ) ∈ V × Y * J 1 ( u , z * , v 0 * , v 1 * ) } = J 1 ( u 0 , z 0 * , v ˆ 0 * , v ˆ 1 * ) .

The proof is complete.

6 A numerical example

In order to illustrate the applicability of such results, we have developed the following numerical example.

For Ω = [ 0 , 1 ] , γ = 0.1 , α = β = 1 , and f ≡ 1 on Ω , we have solved the Ginzburg–Landau type equation

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

with u = 0 , on ∂ Ω .

To obtain such numerical results, referring to those previous ones of Section 3, we have used the following primal dual functional J 2 ( u , v 0 * , v 2 * ) , where

J 2 ( u , v 0 * , v 2 * ) = F 1 ( u , v 0 * ) − ⟨ u , v 2 * ⟩ L 2 + F 2 * ( v 2 * ) ,

where

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

and,

(26) F 2 * ( v 2 * ) = sup u ∈ V { ⟨ u , v 2 * ⟩ L 2 − F 2 ( u ) } = 1 2 K 2 ∫ Ω ( v 2 * ) 2 d x .

Observe that a critical point of J 2 corresponds to a critical of the dual functional J 1 * . From the convexity of J 1 * , such a critical point corresponds a to a global optimal one for J 1 * .

We have obtained results through finite differences combined with a MAT-LAB optimization tool. For an extensive approach on finite differences schemes, please see the study by Strikwerda [16].

For the corresponding solution u 0 , please see Figure 1.

$Figure 1 Solution u 0 ( x ) {u}_{0}\left(x) for the primal formulation.$

Figure 1

Solution u 0 ( x ) for the primal formulation.

7 A primal dual variational formulation for a Burger’s type equation

In this section, we develop a primal dual variational formulation for a Burger’s type equation.

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

Consider the Burger’s type equation in u ∈ V given by

− γ ∇ 2 u + u u x + u u y − f = 0 , in Ω ,

where γ > 0 , f ∈ L 2 ( Ω ) and

V = { u ∈ W 1 , 2 ( Ω ) : u = u 0 on ∂ Ω } .

At this point, we define the functional J : V × Y × Y → R , where

J ( u , v 2 * , v 3 * ) = 1 2 ∫ Ω ( − γ ∇ 2 u + v 2 * u x + v 3 * u y − f ) 2 d x + 1 2 ∫ Ω ( v 2 * − u ) 2 d x + 1 2 ∫ Ω ( v 3 * − u ) 2 d x .

Here, Y = Y * = L 2 ( Ω ) . Let

φ , φ 2 , φ 3 ∈ C c ∞ ( Ω ) .

Observe that

δ u u 2 J ( ( u , v 2 * , v 3 * ) , φ , φ ) = ∫ Ω ( − γ ∇ 2 φ + v 2 * φ x + v 3 * φ y ) 2 d x + ∫ Ω ( φ ) 2 d x + ∫ Ω ( φ ) 2 d x ,

δ v 2 * v 2 * 2 J ( ( u , v 2 * , v 3 * ) , φ 2 , φ 2 ) = ∫ Ω u x 2 φ 2 2 d x + ∫ Ω φ 2 2 d x ,

δ v 3 * v 3 * 2 J ( ( u , v 2 * , v 3 * ) , φ 3 , φ 3 ) = ∫ Ω u y 2 φ 3 2 d x + ∫ Ω φ 3 2 d x ,

and denoting W = − γ ∇ 2 u + v 2 * u x + v 3 * u y − f , we have

(27) δ u v 2 * 2 J ( ( u , v 2 * , v 3 * ) , φ , φ 1 ) = ∫ Ω W φ x φ 2 d x + ∫ Ω ( − γ ∇ 2 φ + 2 v 2 * φ x + v 3 * φ y ) u x φ 2 d x − ∫ Ω φ 2 φ d x .

(28) δ u v 3 * 2 J ( ( u , v 2 * , v 3 * ) , φ , φ 1 ) = ∫ Ω W φ y φ 3 d x + ∫ Ω ( − γ ∇ 2 φ + 2 v 2 * φ x + v 3 * φ y ) u y φ 3 d x − ∫ Ω φ 3 φ d x .

δ v 2 * v 3 * 2 J ( ( u , v 2 * , v 3 * ) , φ 2 , φ 3 ) = ∫ Ω u x u y φ 2 φ 3 d x .

Therefore,

(29) 1 2 δ u u 2 J ( ( u , v 2 * , v 3 * ) , φ , φ ) + 1 2 δ v 2 * v 2 * 2 J ( ( u , v 2 * , v 3 * ) , φ 2 , φ 2 ) + 1 2 δ v 3 * v 3 * 2 J ( ( u , v 2 * , v 3 * ) , φ 3 , φ 3 ) + δ u v 2 * 2 J ( ( u , v 2 * , v 3 * ) , φ , φ 2 ) + δ u v 3 * 2 J ( ( u , v 2 * , v 3 * ) , φ , φ 3 ) + δ v 2 * v 3 * 2 J ( ( u , v 2 * , v 3 * ) , φ 2 , φ 3 ) = 1 2 ∫ Ω ( − γ ∇ 2 φ + v 2 * φ x + v 3 * φ y + u x φ 2 + u y φ 3 ) 2 d x + 1 2 ∫ Ω ( φ 2 − φ ) 2 d x + 1 2 ∫ Ω ( φ 3 − φ ) 2 d x + ∫ Ω W ( φ x φ 2 + φ y φ 3 ) d x .

Observe that at a critical point, we have W = − γ ∇ 2 u + v 2 * u x + v 3 * u y − f = 0 , in Ω .

From this and (44), we may infer that δ 2 J is positive definite in a neighborhood of any critical point of J .

Thus, we may also conclude that the functional J has a large region of convexity around any of its critical points.

7.1 A numerical example concerning a Burger’s type equation

In this subsection, we present numerical results related to a solution of the one-dimensional Burger’s equation:

− γ u x x + u u x = 0 , in [ 0 , 1 ] ⊂ R ,

with the boundary conditions

u ( 0 ) = 1 and u ( 1 ) = 0 .

For a first example, we set γ = 0.5 , and for the second one, we set γ = 0.05 .

To obtain the numerical results, we have used an adaptation with small changes of the primal dual variational formulation presented in the previous section.

For the solution u 0 ( x ) for the case in which γ = 0.5 , please see Figure 2.

$Figure 2 Solution u 0 ( x ) {u}_{0}\left(x) for a Burger’s type equation with γ = 0.5 \gamma =0.5 .$

Figure 2

Solution u 0 ( x ) for a Burger’s type equation with γ = 0.5 .

For the case in which γ = 0.05 , please see Figure 3.

$Figure 3 Solution u 0 ( x ) {u}_{0}\left(x) for a Burger’s type equation with γ = 0.05 \gamma =0.05 .$

Figure 3

Solution u 0 ( x ) for a Burger’s type equation with γ = 0.05 .

Here, we present the software in MAT-LAB through which we have obtained such numerical results.

We highlight to have searched for a critical point of the primal dual formulation previously presented with some small changes and adaptations. Indeed, we have developed a procedure similar to the matrix version of the generalized method of lines.

Here, the software in a finite difference context, where A stands for γ :

***********

1. clear all
m8=100;
d=1/m8;
K=5.0;
A=0.5;
for i=1:m8
- uo(i,1)=1.0;
- u1(i,1)=1.0;
- v3(i,1)=K*uo(i,1);
end;
k2=1;
b14=1.0;
while ( b 14 > 1 0 − 5 ) and ( k 2 < 100 )
- k2=k2+1;
- k1=1;
- b12=1.0;
- while ( b 12 > 1 0 − 5 ) and ( k 1 < 100 )
k1=k1+1;
i=1;
m12=A*2+uo(i,1)*d+K* d 2 ;
m50(i)=A/m12;
z(i)=1/m12*(A+v3(i,1)* d 2 +uo(i,1)*d);
for i=2:m8-1
- m12=A*2-A*m50(i-1)+uo(i,1)*d-uo(i,1)*m50(i-1)*d+K* d 2 ;
- m50(i)=A/m12;
- z(i)=1/m12*(v3(i,1)* d 2 +A*z(i-1)+uo(i,1)*z(i-1)*d);
end;
u(m8,1)=0.0;
for i=1:m8-1
- u(m8-i,1)=m50(m8-i)*u(m8-i+1,1)+z(m8-i);
end;
b12=max(abs(u-uo));
uo=u;
- end;
- b14=max(abs(u-u1));
- u1=u;
for i=1:m8-1
- v3(i,1)=K*u(i,1);
end;
uo(50,1)
end;
for i=1:m8
- x(i)=i*d;
end;
plot(x,u)

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

8 A primal dual variational formulation for a Navier–Stokes system

In this section, we develop a primal dual variational formulation for the time-independent incompressible Navier–Stokes system.

Consider Ω ⊂ R 2 an open, bounded, and connected set with a regular (Lipschitzian) internal boundary denoted Γ 0 , and a regular external one denoted by Γ 1 . For a two-dimensional motion of a fluid on Ω , we denote by u : Ω → R the velocity field in the direction x of the Cartesian system ( x , y ) , by v : Ω → R , the velocity field in the direction y , and by p : Ω → R , the pressure one. Moreover, ρ denotes the fluid density, μ is the viscosity coefficient, and g denotes the gravity field. Under such notation and statements, the time-independent incompressible Navier–Stokes system of partial differential equations stands for,

(30) μ ∇ 2 u − ρ u u x − ρ v u y − p x + ρ g x = 0 , in Ω , μ ∇ 2 v − ρ u v x − ρ v v y − p y + ρ g y = 0 , in Ω , u x + v y = 0 , in Ω ,

(31) u = v = 0 , on Γ 0 , u = u ∞ , v = 0 , p = p ∞ , on Γ 1 .

About the references, we emphasize that related existence, numerical, and theoretical results for similar systems may be found in [17–21], respectively. In particular, Temam [21] addresses extensively both theoretical and numerical methods and an interesting interplay between them.

Finally, these two first paragraphs of this article have been published as a preprint [22], more specifically, reference:

F.S. Botelho, Approximate Numerical Procedures for the Navier–Stokes System Through the Generalized Method of Lines. Preprints.org 2023, 2023020422.

https://doi.org/10.20944/preprints202302.0422.v3.

Defining now P = p ∕ ρ and ν = μ ∕ ρ , consider again the Navier–Stokes system in the following format

(32) ν ∇ 2 u − u ∂ x u − v ∂ y u − ∂ x P + g x = 0 , in Ω , ν ∇ 2 v − u ∂ x v − v ∂ y v − ∂ y P + g y = 0 , in Ω , ∂ x u + ∂ y v = 0 , in Ω ,

(33) u = v = 0 , on Γ 0 , u = u ∞ , v = 0 , P = P ∞ , on Γ 1 .

As previously mentioned, at first, we look for solutions ( u , v , P ) ∈ W 1 , 2 ( Ω ; R 3 ) in a distributional sense.

At this point, we define the functional J : V × Y × Y → R , where

V = { ( u , v , P ) ∈ W 1 , 2 ( Ω ; R 3 ) : u = v = 0 , on Γ 0 , u = u ∞ , v = 0 , P = P ∞ , on Γ 1 }

and

(34) J ( u , v , P , v 2 * , v 3 * ) = 1 2 ∫ Ω ( − ν ∇ 2 u + v 2 * u x + v 3 * u y + P x − g x ) 2 d x 1 2 ∫ Ω ( − ν ∇ 2 v + v 2 * v x + v 3 * v y + P y − g y ) 2 d x + 1 2 ∫ Ω ( v 2 * − u ) 2 d x + 1 2 ∫ Ω ( v 3 * − v ) 2 d x + 1 2 ∫ Ω ( u x + v y ) 2 d x .

Here, Y = Y * = L 2 ( Ω ) . Let

φ , φ 2 , φ 3 , φ 4 , φ 5 ∈ C c ∞ ( Ω ) .

We define also

u = ( u , v , P , v 2 * , v 3 * ) ,

J 1 ( u , v , P , v 2 * , v 3 * ) = 1 2 ∫ Ω ( − ν ∇ 2 u + v 2 * u x + v 3 * u y + P x − g x ) 2 d x ,

J 2 ( u , v , P , v 2 * , v 3 * ) = 1 2 ∫ Ω ( − ν ∇ 2 v + v 2 * v x + v 3 * v y + P y − g y ) 2 d x ,

and

J 3 ( u , v , P , v 2 * , v 3 * ) = 1 2 ∫ Ω ( v 2 * − u ) 2 d x + 1 2 ∫ Ω ( v 3 * − v ) 2 d x + 1 2 ∫ Ω ( u x + v y ) 2 d x ,

so that

J ( u , v , P , v 2 * , v 3 * ) = J 1 ( u , v , P , v 2 * , v 3 * ) + J 2 ( u , v , P , v 2 * , v 3 * ) + J 3 ( u , v , P , v 2 * , v 3 * ) .

Observe that

δ u u 2 J 1 ( ( u , v , P , v 2 * , v 3 * ) , φ , φ ) = ∫ Ω ( − ν ∇ 2 φ + v 2 * φ x + v 3 * φ y ) 2 d x ,

δ v 2 * v 2 * 2 J 1 ( ( u , v , P , v 2 * , v 3 * ) , φ 2 , φ 2 ) = ∫ Ω u x 2 φ 2 2 d x ,

δ v 3 * v 3 * 2 J 1 ( ( u , v , P , v 2 * , v 3 * ) , φ 3 , φ 3 ) = ∫ Ω u y 2 φ 3 2 d x ,

δ P P 2 J 1 ( ( u , v , P , v 2 * , v 3 * ) , φ 4 , φ 4 ) = ∫ Ω ( φ 4 ) x 2 d x ,

and denoting W = − γ ∇ 2 u + v 2 * u x + v 3 * u y + P x − g x , we have

(35) δ u v 2 * 2 J 1 ( ( u , v , P , v 2 * , v 3 * ) , φ , φ 1 ) = ∫ Ω W φ x φ 2 d x + ∫ Ω ( − ν ∇ 2 φ + 2 v 2 * φ x + v 3 * φ y ) u x φ 2 d x d x .

(36) δ u v 3 * 2 J 1 ( ( u , v , P , v 2 * , v 3 * ) , φ , φ 1 ) = ∫ Ω W φ y φ 3 d x + ∫ Ω ( − ν ∇ 2 φ + 2 v 2 * φ x + v 3 * φ y ) u y φ 3 d x .

δ u P 2 J 1 ( ( u , v , P , v 2 * , v 3 * ) , φ , φ 4 ) = ∫ Ω ( − ν ∇ 2 φ + v 2 * φ x + v 3 * φ y ) ( φ 4 ) x d x .

δ v 2 * v 3 * 2 J 1 ( ( u , v , P , v 2 * , v 3 * ) , φ 2 , φ 3 ) = ∫ Ω u x u y φ 2 φ 3 d x .

δ v 2 * P 2 J 1 ( ( u , v , P , v 2 * , v 3 * ) , φ 2 , φ 3 ) = ∫ Ω u x φ 2 ( φ 4 ) x d x .

δ v 3 * P 2 J ( ( u , v , P , v 2 * , v 3 * ) , φ 3 , φ 4 ) = ∫ Ω u y φ 3 ( φ 4 ) x d x .

Therefore, defining

(37) { δ 2 J 1 ( u , φ ) } = 1 2 δ u u 2 J 1 ( ( u , v , P , v 2 * , v 3 * ) , φ , φ ) + 1 2 δ v 2 * v 2 * 2 J 1 ( ( u , v , P , v 2 * , v 3 * ) , φ 2 , φ 2 ) + 1 2 δ v 3 * v 3 * 2 J 1 ( ( u , v , P , v 2 * , v 3 * ) , φ 3 , φ 3 ) + 1 2 δ P P 2 J 1 ( ( u , v , P , v 2 * , v 3 * ) , φ 4 , φ 4 ) + δ u v 2 * 2 J 1 ( ( u , v , P , v 2 * , v 3 * ) , φ , φ 2 ) + δ u v 3 * 2 J 1 ( ( u , v , P , v 2 * , v 3 * ) , φ , φ 3 ) + δ u P 2 J 1 ( ( u , v , P , v 2 * , v 3 * ) , φ , φ 4 ) + δ v 2 * v 3 * 2 J 1 ( ( u , v 2 * , v 3 * ) , φ 2 , φ 3 ) + δ v 2 * P 2 J 1 ( ( u , v , P , v 2 * , v 3 * ) , φ 2 , φ 4 ) + δ v 3 * P 2 J 1 ( ( u , v , P , v 2 * , v 3 * ) , φ 3 , φ 4 ) ,

we have

(38) { δ 2 J 1 ( u , φ ) } = ∫ Ω ( − ν ∇ 2 φ + v 2 * φ x + v 3 * φ y + φ 2 u x + + φ 3 u y + ( φ 4 ) x ) 2 d x + ∫ Ω W ( φ 2 φ x + φ 3 φ y ) d x .

Similarly, we may obtain

δ v v 2 J 2 ( ( u , v , P , v 2 * , v 3 * ) , φ 5 , φ 5 ) = ∫ Ω ( − ν ∇ 2 φ 5 + v 2 * ( φ 5 ) x + v 3 * ( φ 5 ) y ) 2 d x ,

δ v 2 * v 2 * 2 J 2 ( ( u , v , P , v 2 * , v 3 * ) , φ 2 , φ 2 ) = ∫ Ω v x 2 φ 2 2 d x , δ v 3 * v 3 * 2 J 2 ( ( u , v , P , v 2 * , v 3 * ) , φ 3 , φ 3 ) = ∫ Ω v y 2 φ 3 2 d x , δ P P 2 J 2 ( ( u , v , P , v 2 * , v 3 * ) , φ 4 , φ 4 ) = ∫ Ω ( φ 4 ) y 2 d x ,

and denoting W 1 = − γ ∇ 2 v + v 2 * v x + v 3 * v y + P y − g y , we have

(39) δ v v 2 * 2 J 2 ( ( u , v , P , v 2 * , v 3 * ) , φ , φ 1 ) = ∫ Ω W 1 ( φ 5 ) x φ 2 d x + ∫ Ω ( − ν ∇ 2 φ 5 + 2 v 2 * ( φ 5 ) x + v 3 * ( φ 5 ) y ) v x φ 2 d x .

(40) δ v v 3 * 2 J 2 ( ( u , v , P , v 2 * , v 3 * ) , φ , φ 1 ) = ∫ Ω W 1 ( φ 5 ) y φ 3 d x + ∫ Ω ( − ν ∇ 2 φ 5 + 2 v 2 * ( φ 5 ) x + v 3 * ( φ 5 ) y ) v y φ 3 d x .

δ v P 2 J 2 ( ( u , v , P , v 2 * , v 3 * ) , φ , φ 4 ) = ∫ Ω ( − ν ∇ 2 φ 5 + v 2 * ( φ 5 ) x + v 3 * ( φ 5 ) y ) ( φ 4 ) y d x .

δ v 2 * v 3 * 2 J 2 ( ( u , v , P , v 2 * , v 3 * ) , φ 2 , φ 3 ) = ∫ Ω v x v y φ 2 φ 3 d x .

δ v 2 * P 2 J 2 ( ( u , v , P , v 2 * , v 3 * ) , φ 2 , φ 3 ) = ∫ Ω v x φ 2 ( φ 4 ) y d x .

δ v 3 * P 2 J 2 ( ( u , v , P , v 2 * , v 3 * ) , φ 3 , φ 4 ) = ∫ Ω v y φ 3 ( φ 4 ) y d x .

Therefore, defining

(41) { δ 2 J 2 ( u , φ ) } = 1 2 δ v v 2 J 2 ( ( u , v , P , v 2 * , v 3 * ) , φ 5 , φ 5 ) + 1 2 δ v 2 * v 2 * 2 J 2 ( ( u , v , P , v 2 * , v 3 * ) , φ 2 , φ 2 ) + 1 2 δ v 3 * v 3 * 2 J 2 ( ( u , v , P , v 2 * , v 3 * ) , φ 3 , φ 3 ) + 1 2 δ P P 2 J 2 ( ( u , v , P , v 2 * , v 3 * ) , φ 4 , φ 4 ) + δ v v 2 * 2 J 2 ( ( u , v , P , v 2 * , v 3 * ) , φ 5 , φ 2 ) + δ v v 3 * 2 J 2 ( ( u , v , P , v 2 * , v 3 * ) , φ 5 , φ 3 ) + δ v P 2 J 2 ( ( u , v , P , v 2 * , v 3 * ) , φ 5 , φ 4 ) + δ v 2 * v 3 * 2 J 2 ( ( u , v 2 * , v 3 * ) , φ 2 , φ 3 ) + δ v 2 * P 2 J 2 ( ( u , v , P , v 2 * , v 3 * ) , φ 2 , φ 4 ) + δ v 3 * P 2 J 2 ( ( u , v , P , v 2 * , v 3 * ) , φ 3 , φ 4 ) ,

we have

(42) { δ 2 J 2 ( u , φ ) } = ∫ Ω ( − ν ∇ 2 φ 5 + v 2 * ( φ 5 ) x + v 3 * ( φ 5 ) y + φ 2 v x + φ 3 v y + ( φ 4 ) y ) 2 d x + ∫ Ω W 1 ( φ 2 ( φ 5 ) x + φ 3 ( φ 5 ) y ) d x .

Moreover, we may have

δ u u 2 J 3 ( ( u , v , P , v 2 * , v 3 * ) , φ , φ ) = ∫ Ω φ 2 d x + ∫ Ω φ x 2 d x , δ v v 2 J 3 ( ( u , v , P , v 2 * , v 3 * ) , φ , φ ) = ∫ Ω φ 5 2 d x + ∫ Ω ( φ 5 ) y 2 d x , δ v 2 * v 2 * 2 J 3 ( ( u , v , P , v 2 * , v 3 * ) , φ 2 , φ 2 ) = ∫ Ω φ 2 2 d x , δ v 3 * v 3 * 2 J 3 ( ( u , v , P , v 2 * , v 3 * ) , φ 3 , φ 3 ) = ∫ Ω φ 3 2 d x , δ u v 2 * 2 J 3 ( ( u , v , P , v 2 * , v 3 * ) , φ , φ 1 ) = − ∫ Ω φ φ 2 d x , δ v v 3 * 2 J 1 ( ( u , v , P , v 2 * , v 3 * ) , φ , φ 1 ) = − ∫ Ω φ 5 φ 3 d x , δ u v 2 J 3 ( ( u , v , P , v 2 * , v 3 * ) , φ , φ 5 ) = ∫ Ω φ x ( φ 5 ) y d x .

Therefore, defining

(43) { δ 2 J 3 ( u , φ ) } = 1 2 δ u u 2 J 3 ( ( u , v , P , v 2 * , v 3 * ) , φ , φ ) + 1 2 δ v v 2 J 3 ( ( u , v , P , v 2 * , v 3 * ) , φ 5 , φ 5 ) + 1 2 δ v 2 * v 2 * 2 J 3 ( ( u , v , P , v 2 * , v 3 * ) , φ 2 , φ 2 ) + 1 2 δ v 3 * v 3 * 2 J 1 ( ( u , v , P , v 2 * , v 3 * ) , φ 3 , φ 3 ) + δ u v 2 * 2 J 3 ( ( u , v , P , v 2 * , v 3 * ) , φ , φ 2 ) + δ v v 3 * 2 J 3 ( ( u , v , P , v 2 * , v 3 * ) , φ 5 , φ 3 ) + δ u v 2 J 3 ( ( u , v , P , v 2 * , v 3 * ) , φ , φ 5 ) ,

we have

{ δ 2 J 3 ( u , φ ) } = ∫ Ω ( φ 2 − φ ) 2 d x + ∫ Ω ( φ 3 − φ 5 ) 2 d x + ∫ Ω ( φ x + ( φ 5 ) y ) 2 d x .

Finally, joining the pieces, we may infer that

(44) { δ 2 J ( u , φ ) } = { δ 2 J 1 ( u , φ ) } + { δ 2 J 2 ( u , φ ) } + { δ 2 J 3 ( u , φ ) } = ∫ Ω ( − ν ∇ 2 φ + v 2 * φ x + v 3 * φ y + φ 2 u x + φ 3 u y + ( φ 4 ) x ) 2 d x + ∫ Ω W ( φ 2 φ x + φ 3 φ y ) d x + ∫ Ω ( − ν ∇ 2 φ 5 + v 2 * ( φ 5 ) x + v 3 * ( φ 5 ) y + φ 2 v x + φ 3 v y + ( φ 4 ) y ) 2 d x + ∫ Ω W 1 ( φ 2 ( φ 5 ) x + φ 3 ( φ 5 ) y ) d x + ∫ Ω ( φ 2 − φ ) 2 d x + ∫ Ω ( φ 3 − φ 5 ) 2 d x + ∫ Ω ( φ x + ( φ 5 ) y ) 2 d x .

Observe that at a critical point we have

W = − ν ∇ 2 u + v 2 * u x + v 3 * u y + P x − g x = 0 , in Ω

and

W 1 = − ν ∇ 2 v + v 2 * v x + v 3 * v y + P y − g y = 0 , in Ω .

From this and (44), we may infer that δ 2 J is positive definite in a neighborhood of any critical point of J .

Thus, we may also conclude that the functional J has a large region of convexity around any of its critical points.

9 A duality principle for a related relaxed formulation concerning the vectorial approach in the calculus of variations

In this section, we develop a duality principle for a related vectorial model in the calculus of variations.

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

For 1 < p < + ∞ , consider a functional J : V → R , where

J ( u ) = G ( ∇ u ) + F ( u ) − ⟨ u , f ⟩ L 2 ,

where

V = { u ∈ W 1 , p ( Ω ; R N ) : u = u 0 on ∂ Ω }

and f ∈ L 2 ( Ω ; R N ) .

We assume G : Y → R and F : V → R are Fréchet differentiable and F is also convex.

Also

G ( ∇ u ) = ∫ Ω g ( ∇ u ) d x ,

where g : R N × n → R it is supposed to be Fréchet differentiable. Here, we have denoted Y = L p ( Ω ; R N × n ) .

We define also J 1 : V × Y 1 → R by

J 1 ( u , ϕ ) = G 1 ( ∇ u + ∇ y ϕ ) + F ( u ) − ⟨ u , f ⟩ L 2 ,

where

Y 1 = W 1 , p ( Ω × Ω ; R N )

and

G 1 ( ∇ u + ∇ y ϕ ) = 1 ∣ Ω ∣ ∫ Ω ∫ Ω g ( ∇ u ( x ) + ∇ y ϕ ( x , y ) ) d x d y .

Moreover, we define the relaxed functional J 2 : V → R by

J 2 ( u ) = inf ϕ ∈ V 0 J 1 ( u , ϕ ) ,

where

V 0 = { ϕ ∈ Y 1 : ϕ ( x , y ) = 0 , on Ω × ∂ Ω } .

Now observe that

(45) J 1 ( u , ϕ ) = G 1 ( ∇ u + ∇ y ϕ ) + F ( u ) − ⟨ u , f ⟩ L 2 = − 1 ∣ Ω ∣ ∫ Ω ∫ Ω v * ( x , y ) ⋅ ( ∇ u + ∇ y ϕ ( x , y ) ) d y d x + G 1 ( ∇ u + ∇ y ϕ ) + 1 ∣ Ω ∣ ∫ Ω ∫ Ω v * ( x , y ) ⋅ ( ∇ u + ∇ y ϕ ( x , y ) ) d y d x + F ( u ) − ⟨ u , f ⟩ L 2 ≥ inf v ∈ Y 2 − 1 ∣ Ω ∣ ∫ Ω ∫ Ω v * ( x , y ) ⋅ v ( x , y ) d y d x + G 1 ( v ) + inf ( v , ϕ ) ∈ V × V 0 1 ∣ Ω ∣ ∫ Ω ∫ Ω v * ( x , y ) ⋅ ( ∇ u + ∇ y ϕ ( x , y ) ) d y d x + F ( u ) − ⟨ u , f ⟩ L 2 = − G 1 * ( v * ) − F * div x 1 ∣ Ω ∣ ∫ Ω v * ( x , y ) d y + f + 1 ∣ Ω ∣ ∫ ∂ Ω ∫ Ω v * ( x , y ) d y ⊗ n u 0 d Γ ,

∀ ( u , ϕ ) ∈ V × V 0 , v * ∈ A * , where

A * = { v * ∈ Y 2 * : div y v * ( x , y ) = 0 , in Ω } .

Here, we have denoted

G 1 * ( v * ) = sup v ∈ Y 2 1 ∣ Ω ∣ ∫ Ω ∫ Ω v * ( x , y ) ⋅ v ( x , y ) d y d x − G 1 ( v ) ,

where Y 2 = L p ( Ω × Ω ; R N × n ) , Y 2 * = L q ( Ω × Ω ; R N × n ) , and where

1 p + 1 q = 1 .

Furthermore,

(46) F * div x 1 ∣ Ω ∣ ∫ Ω v * ( x , y ) d y + f − 1 ∣ Ω ∣ ∫ ∂ Ω ∫ Ω v * ( x , y ) d y ⊗ n u 0 d Γ = sup ( v , ϕ ) ∈ V × V 0 − 1 ∣ Ω ∣ ∫ Ω ∫ Ω v * ( x , y ) ⋅ ( ∇ u + ∇ y ϕ ( x , y ) ) d y d x − F ( u ) + ⟨ u , f ⟩ L 2 .

Therefore, denoting J 3 * : Y 2 * → R by

J 3 * ( v * ) = − G 1 * ( v * ) − F * div x ∫ Ω v * ( x , y ) d y + f + 1 ∣ Ω ∣ ∫ ∂ Ω ∫ Ω v * ( x , y ) d y ⊗ n u 0 d Γ ,

we obtain

inf u ∈ V J 2 ( u ) ≥ sup v * ∈ A * J 3 * ( v * ) .

Finally, we highlight such a dual functional J 3 * is convex (in fact concave).

10 Conclusion

In this article, we have developed convex dual and primal dual variational formulations suitable for the local optimization of non-convex primal formulations.

It is worth highlighting, the results may be applied to a large class of models in physics and engineering.

We also emphasize that the first duality principles here presented are applied to a Ginzburg–Landau type equation. In particular, we highlight the primal dual formulation presented in Section 5, which is suitable for the global optimization of a originally nonconvex primal formulation.

Among the results, we have included primal dual variational formulations for a Burger’s type equation and for a time independent incompressible Navier–Stokes system. Concerning the Burger’s equation model, we have presented numerical examples and a related software in MAT-LAB.

Finally, in the last section, we have developed a new duality principle suitable for the relaxed quasi-convex primal formulation for a general model in a vectorial calculus of variations context.

In a future research, we intend to extend such results for some models of plates and shells and other models in the elasticity theory.

Funding information: The author states no funding involved.
Author contributions: The author has accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The author declares no conflict of interest concerning this article.
Data availability statement: Details on the software for numerical results available upon request.

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Received: 2023-07-17

Revised: 2023-10-08

Accepted: 2023-10-09

Published Online: 2023-10-31

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/nleng-2022-0343

Keywords for this article

convex dual variational formulations; duality principles for non-convex primal local and global optimization; Ginzburg–Landau type equation

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