On Duality Principles and Related Convex Dual Formulations Suitable for Local and Global Non-Convex Variational Optimization

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On Duality Principles and Related Convex Dual Formulations Suitable for Local and Global Non-Convex Variational Optimization

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A peer-reviewed article of this preprint also exists.

Fabio Botelho^*

This version is not peer-reviewed

Submitted:

12 December 2023

Posted:

12 December 2023

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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:

Subject: Computer Science and Mathematics - Applied Mathematics

MSC: 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 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 large systems 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 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 J.J. Telega and W.R. Bielski [1,2,3,4] and on a D.C. optimization approach developed in Toland [5].

About the other references, details on the Sobolev spaces involved are found in [6]. Related and more recent results on convex analysis and duality theory are addressed in [7,8,9,10,11]. In particular, the results in the present work are extensions and improvements of those results found in the recent book [12] and recent article [13], which by the way, are also based on the articles [1,2,3,4]. Similar models on the superconductivity physics may be found in [14,15].

Finally, we also emphasize 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.

It is worth highlighting, we may generically denote

\int_{Ω} [{(- γ \nabla^{2} + K I_{d})}^{- 1} v^{*}] v^{*} d x

simply by

\int_{Ω} \frac{{(v^{*})}^{2}}{- γ \nabla^{2} + K} d x,

where

I_{d}

denotes a concerning identity operator. Here it is also worth clarifying in general we will denote

- γ \nabla^{2} + K I_{d}

simply as

- γ \nabla^{2} + K .

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

Finally,

\nabla^{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

{〈 \cdot, \cdot 〉}_{V} : V \times V^{*} \to R .

More specifically, for each

f \in V^{'}

, we suppose there exists a unique

u^{*} \in V^{*}

such that

f (u) = {〈 u, u^{*} 〉}_{V}, \forall u \in V .

Moreover, we define the norm of f, denoted by

{∥ f ∥}_{V^{*}}

{∥ f ∥}_{V^{*}} = sup {| {〈 u, u^{*} 〉}_{V} : u \in {V and ∥ u ∥}_{V} \leq 1} \equiv ∥ u^{*} ∥_{V^{*}} .

For an open, bounded and connected set

Ω \subset R^{N}

and

Y = Y^{*} = L^{2} (Ω)

we recall that

{〈 u, u^{*} 〉}_{L^{2}} = \int_{Ω} u u^{*} d x .

More specifically, for each continuous and linear functional

f : Y \to R

there exists a unique

u^{*} \in Y^{*} = L^{2} (Ω)

such that

f (u) = \int_{Ω} u u^{*} d x, \forall u \in Y = L^{2} (Ω) .

Definition 1.3

(The first variation). Let V be a Banach space. Let

F : V \to R

be a functional.

We define the first variation of F at

u \in V

on the direction

φ \in V

, denoted by

δ F (u; φ)

δ F (u; φ) = lim_{ε \to 0} \frac{F (u + ε φ) - F (u)}{ε},

if such a limit exists.

If there exists

u^{*} \in V^{*}

such that

δ F (u; φ) = {〈 φ, u^{*} 〉}_{V}, \forall φ \in V

we say that F is Gâteaux differentiable at u. Moreover, in such a case,

u^{*} \in V^{*}

is said to be the Gâteaux derivative of F at u.

We may also denote

u^{*} = δ F (u) or u^{*} = \frac{\partial F (u)}{\partial u} .

Definition 1.4

(The second variation). Let V be a Banach space. Let

F : V \to R

be a functional.

We define the second variation of F at

u \in V

on the directions

φ \in V

and

φ_{1} \in V

, denoted by

δ^{2} F (u; φ, φ_{1})

, as

δ^{2} F (u; φ, φ_{1}) = lim_{ε \to 0} \frac{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 \to R

be a functional.

We define the polar functional of F, denoted by

F^{*} : V^{*} \to R

, by

F^{*} (u^{*}) = sup_{u \in V} {{〈 u, u^{*} 〉}_{V} - F (u)}, \forall u^{*} \in 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 \to R

be a twice continuously Fréchet differentiable functional.

Let

u^{*} \in V^{*}

. Assume there exists a unique

\hat{u} \in V

such that

u^{*} = \frac{\partial F (\hat{u})}{\partial u} .

Suppose also

det \{\frac{\partial^{2} F (u)}{\partial u^{2}}\} \neq 0,

in a neighborhood of

\hat{u} .

Under such hypotheses, defining the Legendre tranform of F at

u^{*}

F_{L}^{*} (u^{*})

where

F_{L}^{*} (u^{*}) = {〈 \hat{u}, u^{*} 〉}_{V} - F (\hat{u})

we have that

\hat{u} = \frac{\partial F_{L}^{*} (u^{*})}{\partial u^{*}} .

Remark 1.7.

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

u^{*} = \frac{\partial F (\hat{u})}{\partial u},

corresponds to globally maximize

H (u) = {〈 u, u^{*} 〉}_{V} - F (u)

on V, so that, in such a case,

F^{*} (u^{*}) = H (\hat{u}) = {〈 \hat{u}, u^{*} 〉}_{V} - F (\hat{u}) = 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^{*}), \forall u^{*} \in 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 \subset R^{3}

be an open, bounded, connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω .

Consider a functional

J : V \to R

where

\begin{matrix} J (u) & = & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x \\ + \frac{α}{2} \int_{Ω} {(u^{2} - β)}^{2} d x - {〈 u, f 〉}_{L^{2}} . \end{matrix}

(1)

Here

γ > 0,

α > 0

β > 0

and

f \in L^{2} (Ω) \cap L^{\infty} (Ω) .

Moreover,

V = W_{0}^{1, 2} (Ω)

and we denote

Y = Y^{*} = L^{2} (Ω) .

Define the functionals

F_{1} : V \times Y \to R

F_{2} : V \to R

\begin{matrix} F_{1} (u, v_{0}^{*}) & = & - \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x - {〈 u^{2}, v_{0}^{*} 〉}_{L^{2}} \\ + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + 2 v_{0}^{*} u - f)}^{2} d x + \frac{K_{2}}{2} \int_{Ω} u^{2} d x \\ + {〈 u, f 〉}_{L^{2}} + \frac{1}{2 α} \int_{Ω} {(v_{0}^{*})}^{2} d x \\ + β \int_{Ω} v_{0}^{*} d x, \end{matrix}

(2)

and

F_{2} (u) = \frac{K_{2}}{2} \int_{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 \nabla^{2} + 2 v_{0}^{*})}^{2} \geq c_{0} I_{d},

\forall v_{0}^{*} \in B^{*}

, where

B^{*} = {v_{0}^{*} \in Y^{*} : ∥ 2 v_{0}^{*} ∥_{\infty} < K / 2},

for an appropriate real constant

K > 0

We define also

J_{1} : V \times Y \to R

J_{1} (u, v_{0}^{*}) = F_{1} (u, v_{0}^{*}) - F_{2} (u),

and

F_{1}^{*} : Y^{*} \times B^{*} \to R

F_{2}^{*} : Y^{*} \to R

\begin{matrix} F_{1}^{*} (v_{2}^{*}, v_{0}^{*}) \\ = & sup_{u \in V} {{〈 u, v_{2}^{*} 〉}_{L^{2}} - F_{1} (u, v_{0}^{*})} \\ = & \frac{1}{2} \int_{Ω} \frac{{(v_{2}^{*} + f + K_{1} (- γ \nabla^{2} + 2 v_{0}^{*}) f)}^{2}}{K_{2} + γ \nabla^{2} - 2 v_{0}^{*} + K_{1} {(- γ \nabla^{2} + 2 v_{0}^{*})}^{2}} d x \\ - \frac{1}{2 α} \int_{Ω} {(v_{0}^{*})}^{2} d x - β \int_{Ω} v_{0}^{*} d x - \frac{K_{1}}{2} \int_{Ω} f^{2} d x \end{matrix}

(3)

and,

\begin{matrix} F_{2}^{*} (v_{2}^{*}) & = & sup_{u \in V} {{〈 u, v_{2}^{*} 〉}_{L^{2}} - F_{2} (u)} \\ = & \frac{1}{2 K_{2}} \int_{Ω} {(v_{2}^{*})}^{2} d x, \end{matrix}

(4)

respectively.

Furthermore, we define

D^{*} = {v_{2}^{*} \in Y^{*} : ∥ v_{2}^{*} ∥_{\infty} \leq 5 K_{2} / 4}

and

J_{1}^{*} : D^{*} \times B^{*} \to R,

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 ∥}_{\infty}, 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}^{*})

D^{*} \times 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

({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) \in D^{*} \times B^{*}

be such that

δ J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) = 0

and

u_{0} \in V

be such that

u_{0} = \frac{\partial F_{2}^{*} ({\hat{v}}_{2}^{*})}{\partial v_{2}^{*}} .

Under such hypotheses, we have

δ J (u_{0}) = 0,

and

\begin{matrix} - J (u_{0}) & = & J_{1} (u_{0}, {\hat{v}}_{0}^{*}) \\ = & inf_{u \in V} J_{1} (u, {\hat{v}}_{0}^{*}) \\ = & inf_{(v_{2}^{*}, v_{0}^{*}) \in D^{*} \times B^{*}} J_{1}^{*} (v_{2}^{*}, v_{0}^{*}) \\ = & J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) . \end{matrix}

(5)

Proof.

Observe that

δ J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) = 0

so that, since

J_{1}^{*}

is convex in

(v_{2}^{*}, v_{0}^{*})

D^{*} \times B^{*}

, we obtain

J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) = inf_{(v_{2}^{*}, v_{0}^{*}) \in D^{*} \times B^{*}} J_{1}^{*} (v_{2}^{*}, v_{0}^{*}) .

Now we will show that

δ J (u_{0}) = 0 .

From

\frac{\partial J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*})}{\partial v_{2}^{*}} = 0,

and

\frac{\partial F_{2}^{*} ({\hat{v}}_{2}^{*})}{\partial v_{2}^{*}} = u_{0}

we have

- \frac{\partial F_{1}^{*} ({\hat{v}}_{2}^{*}, v_{0}^{*})}{\partial v_{2}^{*}} + u_{0} = 0

and

{\hat{v}}_{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

\hat{u} \in V

such that

\frac{\partial H ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}, \hat{u})}{\partial u} = 0,

and

F_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) = H ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}, \hat{u}),

so that

\begin{matrix} \frac{\partial F_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*})}{\partial v_{2}^{*}} & = & \frac{\partial H ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}, \hat{u})}{\partial v_{2}^{*}} \\ + \frac{\partial H ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}, \hat{u})}{\partial u} \frac{\partial \hat{u}}{\partial v_{2}^{*}} \\ = & \hat{u} . \end{matrix}

(6)

Summarizing, we have got

u_{0} = \frac{\partial F_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*})}{\partial v_{2}^{*}} = \hat{u} .

Also, denoting

A (u_{0}, {\hat{v}}_{0}^{*}) = - γ \nabla^{2} u_{0} + 2 {\hat{v}}_{0}^{*} u_{0} - f,

from

\frac{\partial H ({\hat{v_{2}}}^{*}, {\hat{v}}_{0}^{*}, u_{0})}{\partial u} = 0,

we have

- (γ \nabla^{2} u_{0} - 2 {\hat{v}}_{0}^{*} u_{0} + f + K_{1} (- γ \nabla^{2} + 2 {\hat{v}}_{0}^{*}) A (u_{0}, {\hat{v}}_{0}^{*}) - {\hat{v}}_{2}^{*} + K_{2} u_{0}) = 0,

so that

- A (u_{0}, {\hat{v}}_{0}^{*}) + K_{1} (- γ \nabla^{2} + 2 {\hat{v}}_{0}^{*}) A (u_{0}, {\hat{v}}_{0}^{*}) = 0 .

(7)

From such results, we may infer that

A (u_{0}, {\hat{v}}_{0}^{*}) = - γ \nabla^{2} u_{0} + 2 {\hat{v}}_{0}^{*} - f = 0, in Ω .

Moreover, from

\frac{\partial J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*})}{\partial v_{0}^{*}} = 0,

we have

K_{1} A (u_{0}, {\hat{v}}_{0}^{*}) 2 u_{0} + \frac{{\hat{v}}_{0}^{*}}{α} - u_{0}^{2} + β = 0,

so that

v_{0}^{*} = α (u_{0}^{2} - β) .

From such last results we get

- γ \nabla^{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}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) = {〈 u_{0}, {\hat{v}}_{2}^{*} 〉}_{L^{2}} - F_{1} (u_{0}, {\hat{v}}_{0}^{*}),

F_{2}^{*} ({\hat{v}}_{2}^{*}) = {〈 u_{0}, {\hat{v}}_{2}^{*} 〉}_{L^{2}} - F_{2} (u_{0}),

so that

\begin{matrix} J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) \\ = & - F_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) + F_{2}^{*} ({\hat{v}}_{2}^{*}) \\ = & F_{1} (u_{0}, {\hat{v}}_{0}^{*}) - F_{2} (u_{0}) \\ = & J_{1} (u_{0}, {\hat{v}}_{0}^{*}) \\ = & - J (u_{0}) . \end{matrix}

(8)

Finally, observe that from a concerning convexity,

- J (u_{0}) = J_{1} (u_{0}, {\hat{v}}_{0}^{*}) = inf_{u \in V} J_{1} (u, {\hat{v}}_{0}^{*}) .

Joining the pieces, we have got

\begin{matrix} - J (u_{0}) & = & J_{1} (u_{0}, {\hat{v}}_{0}^{*}) \\ = & inf_{u \in V} \{J_{1} (u, {\hat{v}}_{0}^{*})\} \\ = & inf_{(v_{2}^{*}, v_{0}^{*}) \in D^{*} \times B^{*}} J_{1}^{*} (v_{2}^{*}, v_{0}^{*}) \\ = & J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) . \end{matrix}

(9)

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 \subset R^{3}

be an open, bounded, connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω .

For the primal formulation, consider a functional

J : V \to R

where

\begin{matrix} J (u) & = & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x \\ + \frac{α}{2} \int_{Ω} {(u^{2} - β)}^{2} d x - {〈 u, f 〉}_{L^{2}} . \end{matrix}

(10)

Here

γ > 0,

α > 0

β > 0

and

f \in L^{2} (Ω) \cap L^{\infty} (Ω) .

Moreover,

V = W_{0}^{1, 2} (Ω)

and we denote

Y = Y^{*} = L^{2} (Ω) .

Define the functional

J_{1}^{*} : V \times {[Y^{*}]}^{2} \to R

, by

\begin{matrix} J_{1}^{*} (u, v_{3}^{*}, v_{0}^{*}) & = & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + {〈 u^{2}, v_{0}^{*} 〉}_{L^{2}} \\ + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + 2 v_{3}^{*} u - f)}^{2} d x + \frac{K_{1}}{2} \int_{Ω} {(v_{3}^{*} - α (u^{2} - β))}^{2} d x \\ - {〈 u, f 〉}_{L^{2}} - \frac{1}{2 α} \int_{Ω} {(v_{0}^{*})}^{2} d x \\ - β \int_{Ω} v_{0}^{*} d x . \end{matrix}

(11)

We define also

B^{*} = {v_{0}^{*} \in Y^{*} : ∥ 2 v_{0}^{*} ∥_{\infty} < K / 2},

for an appropriate real constant

K > 0

Furthermore, we define

D^{*} = {v_{3}^{*} \in Y^{*} : ∥ v_{3}^{*} ∥_{\infty} \leq K_{2}}

A^{+} = {u \in V : u f \geq 0, a . e . in Ω},

V_{2} = {{u \in V : ∥ u ∥}_{\infty} \leq K_{3}}

for an appropriate real constant

K_{3} > 0

and

V_{1} = A^{+} \cap V_{2} .

Now observe that denoting

φ_{1} = v_{3}^{*} - α (u^{2} - β)

, we have

\begin{matrix} \frac{\partial^{2} J_{1}^{*} (u, v_{3}^{*}, v_{0}^{*})}{\partial u^{2}} & = & K_{1} {(- γ \nabla^{2} + 2 v_{3}^{*})}^{2} + 4 K_{1} α^{2} u^{2} \\ - 2 K_{1} α φ_{1} - γ \nabla^{2} + 2 v_{0}^{*} \end{matrix}

(12)

and

\begin{matrix} \frac{\partial^{2} J_{1}^{*} (u, v_{3}^{*}, v_{0}^{*})}{\partial {(v_{3}^{*})}^{2}} & = & K_{1} + 4 K_{1} u^{2} . \end{matrix}

(13)

Denoting

φ = - γ \nabla^{2} u + 2 v_{0}^{*} u - f

we have also that

\begin{matrix} \frac{\partial^{2} J_{1}^{*} (u, v_{3}^{*}, v_{0}^{*})}{\partial u \partial v_{3}^{*}} & = & K_{1} (- 2 γ \nabla^{2} u + 2 v_{3}^{*} u) + 2 K_{1} φ - 2 K_{1} α u . \end{matrix}

(14)

In such a case, we obtain

\begin{matrix} det \{\frac{\partial^{2} J_{1}^{*} (u, v_{3}^{*}, v_{0}^{*})}{\partial u \partial v_{3}^{*}}\} & = & \frac{\partial^{2} J_{1}^{*} (u, v_{3}^{*}, v_{0}^{*})}{\partial {(v_{3}^{*})}^{2}} \frac{\partial^{2} J_{1}^{*} (u, v_{3}^{*}, v_{0}^{*})}{\partial u^{2}} - {(\frac{\partial^{2} J_{1}^{*} (u, v_{3}^{*}, v_{0}^{*})}{\partial v_{3}^{*} \partial u})}^{2} \\ = & K_{1}^{2} {(- γ \nabla^{2} + 2 v_{3}^{*} + 4 α u^{2})}^{2} \\ + (- γ \nabla^{2} + 2 v_{0}^{*}) O (K_{1}) \\ - 4 K_{1}^{2} φ^{2} - 4 K_{1}^{2} φ [(- γ \nabla^{2} u + 2 v_{0}^{*} u) - 2 α u] \\ - 2 K_{1}^{2} α φ_{1} (1 + 4 u^{2}) . \end{matrix}

(15)

Observe that at a critical point

φ = 0

and

φ_{1} = 0 .

From such results we may infer that

det \{\frac{\partial^{2} J_{1}^{*} (u, v_{3}^{*}, v_{0}^{*})}{\partial u \partial 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}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) \in V_{1} \times D^{*} \times B^{*}

be such that

δ J_{1}^{*} (u_{0}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) = 0 .

Under such hypotheses, we have

δ J (u_{0}) = - γ \nabla^{2} u_{0} + 2 α (u_{0}^{2} - β) u_{0} - f = 0

and there exists

r > 0

such that

\begin{matrix} J (u_{0}) & = & sup_{v_{0}^{*} \in B^{*}} \{inf_{(u, v_{3}^{*}) \in B_{r} (u_{0}, {\hat{v}}_{3}^{*})} J_{1}^{*} (u, v_{3}^{*}, v_{0}^{*})\} \\ = & J_{1}^{*} (u_{0}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) . \end{matrix}

(16)

Proof.

The proof that

δ J (u_{0}) = 0

and

J (u_{0}) = J_{1}^{*} (u_{0}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*})

may be done similarly as in the previous sections.

Observe that, as previously obtained, there exists

r > 0

such that

det \{\frac{\partial^{2} J_{1}^{*} (u, v_{3}^{*}, {\hat{v}}_{0}^{*})}{\partial u \partial v_{3}^{*}}\} > 0, \forall (u, v_{3}^{*}) \in B_{r} (u_{0}, {\hat{v}}_{3}^{*})

and

\frac{\partial^{2} J_{1}^{*} (u_{0}, {\hat{v}}_{3}^{*}, v_{0}^{*})}{\partial {(v_{0}^{*})}^{2}} < 0, \forall v_{0}^{*} \in B^{*} .

Since for a sufficiently large

K_{1} > 0

we have

\frac{\partial^{2} J_{1}^{*} (u, v_{3}^{*}, {\hat{v}}_{0}^{*})}{\partial u^{2}} > 0, in B_{r} (u_{0}, {\hat{v}}_{3}^{*}),

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

J (u_{0}) = J_{1}^{*} (u_{0}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) = sup_{v_{0}^{*} \in B^{*}} \{inf_{(u, v_{3}^{*}) \in B_{r} (u_{0}, {\hat{v}}_{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 \subset R^{3}

be an open, bounded and connected set with a regular (Lipschitzian) boundary.

Consider the functional

J : V \to R

defined by

\begin{matrix} J (u) & = & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x \\ + \frac{α}{2} \int_{Ω} {(u^{2} - β)}^{2} d x - {〈 u, f 〉}_{L^{2}}, \end{matrix}

(17)

where

V = W_{0}^{1, 2} (Ω),

f \in L^{2} (Ω)

γ > 0, α > 0

and

β > 0 .

Denoting

Y = Y^{*} = L^{2} (Ω)

, define also

J_{1} : V \times {[Y^{*}]}^{3} \to R

\begin{matrix} J_{1} (u, z^{*}, v_{0}^{*}, v_{1}^{*}) & = & - \frac{1}{2} \int_{Ω} \frac{{(v_{1}^{*})}^{2}}{- γ \nabla^{2} + K_{1}} d x + {〈 u, v_{1}^{*} 〉}_{L^{2}} \\ - \frac{1}{2} \int_{Ω} K_{1} u^{2} d x + {〈 u^{2}, v_{0}^{*} 〉}_{L^{2}} - {〈 u, f 〉}_{L^{2}} \\ + \frac{1}{2} \int_{Ω} K u^{2} d x - {〈 u, z^{*} 〉}_{L^{2}} + \frac{1}{2} \int_{Ω} \frac{{(z^{*})}^{2}}{K} d x \\ + \frac{1}{2} \int_{Ω} (- γ \nabla^{2} + K_{1}) {(\frac{z^{*}}{K} - \frac{v_{1}^{*}}{- γ \nabla^{2} + K_{1}})}^{2} d x \\ - \frac{1}{2 α} {(v_{0}^{*})}^{2} d x - β \int_{Ω} v_{0}^{*} d x . \end{matrix}

(18)

Observe that

\frac{\partial^{2} J_{1} (u, z^{*}, v_{0}^{*}, v_{1}^{*})}{\partial u^{2}} = K - K_{1} + 2 v_{0}^{*},

\frac{\partial^{2} J_{1} (u, z^{*}, v_{0}^{*}, v_{1}^{*})}{\partial {(z^{*})}^{2}} = \frac{1}{K} + \frac{- γ \nabla^{2} + K_{1}}{K^{2}},

and

\frac{\partial^{2} J_{1} (u, z^{*}, v_{0}^{*}, v_{1}^{*})}{\partial u \partial z^{*}} = - 1 .

Hence,

\begin{matrix} det \{δ_{u z^{*}}^{2} J (u, z^{*}, v_{0}^{*}, v_{1}^{*})\} & = & \frac{\partial^{2} J_{1} (u, z^{*}, v_{0}^{*}, v_{1}^{*})}{\partial u^{2}} \frac{\partial^{2} J_{1} (u, z^{*}, v_{0}^{*}, v_{1}^{*})}{\partial {(z^{*})}^{2}} - {(\frac{\partial^{2} J_{1} (u, z^{*}, v_{0}^{*}, v_{1}^{*})}{\partial u \partial z^{*}})}^{2} \\ = & \frac{K (- γ \nabla^{2} + 2 v_{0}^{*}) - K_{1} (- γ \nabla^{2} - 2 v_{0}^{*}) - K_{1}^{2} - γ \nabla^{2} (2 v_{0}^{*})}{K^{2}} \\ \equiv & H (v_{0}^{*}, K_{1}) . \end{matrix}

(19)

The best possible

K_{1}

is obtained through the optimal equation

\frac{\partial H (v_{0}^{*}, K_{1})}{\partial K_{1}} = 0,

that is,

- 2 K_{1} - (- γ \nabla^{2} - 2 v_{0}^{*}) = 0

so that we set

K_{1} = \frac{γ \nabla^{2} + 2 v_{0}^{*}}{2} .

Replacing such a

K_{1}

into (19), we obtain

\begin{matrix} det \{δ_{u z^{*}}^{2} J (u, z^{*}, v_{0}^{*}, v_{1}^{*})\} & = & \frac{K (- γ \nabla^{2} + 2 v_{0}^{*}) + \frac{1}{4} {(- γ \nabla^{2} + 2 v_{0}^{*})}^{2}}{K^{2}} \\ \equiv & H_{1} (v_{0}^{*}) . \end{matrix}

(20)

Setting now

K = 5 (- γ \nabla^{2})

, define

C^{*} = A_{1}^{*} \cap A_{2}^{*} \cap B^{*}

where

A_{1}^{*} = {v_{0}^{*} \in Y^{*} : K - K_{1} + 2 v_{0}^{*} > 0},

A_{2}^{*} = {v_{0}^{*} \in Y^{*} : H_{1} (v_{0}^{*}) > 0},

and

B^{*} = {v_{0}^{*} \in Y^{*} : ∥ 2 v_{0}^{*} ∥_{\infty} < K_{2}}

for some appropriate

K_{2} > 0

Now redefining

J_{1}

\begin{matrix} J_{1} (u, z^{*}, v_{0}^{*}, v_{1}^{*}) & = & - \frac{1}{2} \int_{Ω} \frac{{(v_{1}^{*})}^{2}}{- γ \nabla^{2} + K_{1}} d x + {〈 u, v_{1}^{*} 〉}_{L^{2}} \\ - \frac{1}{2} \int_{Ω} K_{1} u^{2} d x + {〈 u^{2}, v_{0}^{*} 〉}_{L^{2}} - {〈 u, f 〉}_{L^{2}} \\ + \frac{1}{2} \int_{Ω} K u^{2} d x - {〈 u, z^{*} 〉}_{L^{2}} + \frac{1}{2} \int_{Ω} \frac{{(z^{*})}^{2}}{K} d x \\ + \frac{1}{2} \int_{Ω} (- γ \nabla^{2} + K_{1} - ε) {(\frac{z^{*}}{K} - \frac{v_{1}^{*}}{- γ \nabla^{2} + K_{1}})}^{2} d x \\ - \frac{1}{2 α} {(v_{0}^{*})}^{2} d x - β \int_{Ω} v_{0}^{*} d x . \end{matrix}

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for some small parameter

0 < ε ≪ 1

, we may prove the following theorem.

Theorem 5.1.

Let

(u_{0}, z_{0}^{*}, {\hat{v}}_{0}^{*}, {\hat{v}}_{1}^{*}) \in V \times Y^{*} \times C^{*} \times Y^{*}

be such that

δ J_{1} (u_{0}, z_{0}^{*}, {\hat{v}}_{0}^{*}, {\hat{v}}_{1}^{*}) = 0 .

Under such hypotheses,

δ J (u_{0}) = 0

and

\begin{matrix} J (u_{0}) & = & inf_{u \in V} J (u) \\ = & sup_{(v_{0}^{*}, v_{1}^{*}) \in C^{*} \times Y^{*}} \{inf_{(u, z^{*}) \in V \times Y^{*}} J_{1} (u, z^{*}, v_{0}^{*}, v_{1}^{*})\} \\ = & J_{1} (u_{0}, z_{0}^{*}, {\hat{v}}_{0}^{*}, {\hat{v}}_{1}^{*}) . \end{matrix}

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Proof.

The proof that

δ J (u_{0}) = 0

and

J (u_{0}) = J_{1} (u_{0}, z_{0}^{*}, {\hat{v}}_{0}^{*}, {\hat{v}}_{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}^{*}, {\hat{v}}_{0}^{*}, {\hat{v}}_{1}^{*}) = 0

and

{\hat{v}}_{0}^{*} \in C^{*}

, so that we may infer that

J_{1} (u_{0}, z_{0}^{*}, {\hat{v}}_{0}^{*}, {\hat{v}}_{1}^{*}) = inf_{(u, z^{*}) \in V \times Y^{*}} J_{1} (u, z^{*}, {\hat{v}}_{0}^{*}, {\hat{v}}_{1}^{*})

and

J_{1} (u_{0}, z_{0}^{*}, {\hat{v}}_{0}^{*}, {\hat{v}}_{1}^{*}) = sup_{(v_{0}^{*}, v_{1}^{*}) \in C^{*} \times 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}^{*}, {\hat{v}}_{0}^{*}, {\hat{v}}_{1}^{*}) = sup_{(v_{0}^{*}, v_{1}^{*}) \in C^{*} \times Y^{*}} \{inf_{(u, z^{*}) \in V \times Y^{*}} J_{1} (u_{0}, z_{0}^{*}, v_{0}^{*}, v_{1}^{*})\} .

Finally, observe that

\begin{matrix} J (u_{0}) & = & J_{1} (u_{0}, z_{0}^{*}, {\hat{v}}_{0}^{*}, {\hat{v}}_{1}^{*}) \\ \leq & J_{1} (u, K u, {\hat{v}}_{0}^{*}, {\hat{v}}_{1}^{*}) \\ \leq & sup_{(v_{0}^{*}, v_{1}^{*}) \in Y^{*} \times Y^{*}} J_{1} (u, K u, v_{0}^{*}, v_{1}^{*}) \\ = & J (u), \forall u \in V . \end{matrix}

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Summarizing, we have got

J (u_{0}) = J_{1} (u_{0}, z_{0}^{*}, {\hat{v}}_{0}^{*}, {\hat{v}}_{1}^{*}) = inf_{u \in V} J (u) .

Joining the pieces, we may infer that

\begin{matrix} J (u_{0}) & = & inf_{u \in V} J (u) \\ = & sup_{(v_{0}^{*}, v_{1}^{*}) \in C^{*} \times Y^{*}} \{inf_{(u, z^{*}) \in V \times Y^{*}} J_{1} (u, z^{*}, v_{0}^{*}, v_{1}^{*})\} \\ = & J_{1} (u_{0}, z_{0}^{*}, {\hat{v}}_{0}^{*}, {\hat{v}}_{1}^{*}) . \end{matrix}

(24)

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 \equiv 1

Ω

we have solved the Ginzburg-Landau type equation

- γ \nabla^{2} u + α (u^{2} - β) 2 u - f = 0, in Ω

with

u = 0, on \partial Ω .

To obtain such numerical results, refereing 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

\begin{matrix} F_{1} (u, v_{0}^{*}) & = & - \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x - {〈 u^{2}, v_{0}^{*} 〉}_{L^{2}} \\ + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + 2 v_{0}^{*} u - f)}^{2} d x + \frac{K_{2}}{2} \int_{Ω} u^{2} d x \\ + {〈 u, f 〉}_{L^{2}} + \frac{1}{2 α} \int_{Ω} {(v_{0}^{*})}^{2} d x \\ + β \int_{Ω} v_{0}^{*} d x, \end{matrix}

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and,

\begin{matrix} F_{2}^{*} (v_{2}^{*}) & = & sup_{u \in V} {{〈 u, v_{2}^{*} 〉}_{L^{2}} - F_{2} (u)} \\ = & \frac{1}{2 K_{2}} \int_{Ω} {(v_{2}^{*})}^{2} d x . \end{matrix}

(26)

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 reference [16].

For the corresponding solution

u_{0}

, please see Figure 1.

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 \subset R^{3}

be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω

Consider the Burger’s type equation in

u \in V

given by

- γ \nabla^{2} u + u u_{x} + u u_{y} - f = 0, in Ω,

where

γ > 0

f \in L^{2} (Ω)

and

V = {u \in W^{1, 2} (Ω) : u = u_{0} on \partial Ω} .

At this point we define the functional

J : V \times Y \times Y \to R

where

J (u, v_{2}^{*}, v_{3}^{*}) = \frac{1}{2} \int_{Ω} {(- γ \nabla^{2} u + v_{2}^{*} u_{x} + v_{3}^{*} u_{y} - f)}^{2} d x + \frac{1}{2} \int_{Ω} {(v_{2}^{*} - u)}^{2} d x + \frac{1}{2} \int_{Ω} {(v_{3}^{*} - u)}^{2} d x .

Here

Y = Y^{*} = L^{2} (Ω) .

Let

φ, φ_{2}, φ_{3} \in C_{c}^{\infty} (Ω) .

Observe that

δ_{u u}^{2} J ((u, v_{2}^{*}, v_{3}^{*}), φ, φ) = \int_{Ω} {(- γ \nabla^{2} φ + v_{2}^{*} φ_{x} + v_{3}^{*} φ_{y})}^{2} d x + \int_{Ω} {(φ)}^{2} d x + \int_{Ω} {(φ)}^{2} d x,

δ_{v_{2}^{*} v_{2}^{*}}^{2} J ((u, v_{2}^{*}, v_{3}^{*}), φ_{2}, φ_{2}) = \int_{Ω} u_{x}^{2} φ_{2}^{2} d x + \int_{Ω} φ_{2}^{2} d x,

δ_{v_{3}^{*} v_{3}^{*}}^{2} J ((u, v_{2}^{*}, v_{3}^{*}), φ_{3}, φ_{3}) = \int_{Ω} u_{y}^{2} φ_{3}^{2} d x + \int_{Ω} φ_{3}^{2} d x,

and denoting

W = - γ \nabla^{2} u + v_{2}^{*} u_{x} + v_{3}^{*} u_{y} - f,

we have

\begin{matrix} δ_{u v_{2}^{*}}^{2} J ((u, v_{2}^{*}, v_{3}^{*}), φ, φ_{1}) & = & \int_{Ω} W φ_{x} φ_{2} d x \\ + \int_{Ω} (- γ \nabla^{2} φ + 2 v_{2}^{*} φ_{x} + v_{3}^{*} φ_{y}) u_{x} φ_{2} d x - \int_{Ω} φ_{2} φ d x . \end{matrix}

(27)

\begin{matrix} δ_{u v_{3}^{*}}^{2} J ((u, v_{2}^{*}, v_{3}^{*}), φ, φ_{1}) & = & \int_{Ω} W φ_{y} φ_{3} d x \\ + \int_{Ω} (- γ \nabla^{2} φ + 2 v_{2}^{*} φ_{x} + v_{3}^{*} φ_{y}) u_{y} φ_{3} d x - \int_{Ω} φ_{3} φ d x . \end{matrix}

(28)

δ_{v_{2}^{*} v_{3}^{*}}^{2} J ((u, v_{2}^{*}, v_{3}^{*}), φ_{2}, φ_{3}) = \int_{Ω} u_{x} u_{y} φ_{2} φ_{3} d x .

Therefore

\begin{matrix} \frac{1}{2} δ_{u u}^{2} J ((u, v_{2}^{*}, v_{3}^{*}), φ, φ) + \frac{1}{2} δ_{v_{2}^{*} v_{2}^{*}}^{2} J ((u, v_{2}^{*}, v_{3}^{*}), φ_{2}, φ_{2}) + \frac{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}) \\ = & \frac{1}{2} \int_{Ω} {(- γ \nabla^{2} φ + v_{2}^{*} φ_{x} + v_{3}^{*} φ_{y} + u_{x} φ_{2} + u_{y} φ_{3})}^{2} d x \\ + \frac{1}{2} \int_{Ω} {(φ_{2} - φ)}^{2} d x + \frac{1}{2} \int_{Ω} {(φ_{3} - φ)}^{2} d x \\ + \int_{Ω} W (φ_{x} φ_{2} + φ_{y} φ_{3}) d x . \end{matrix}

(29)

Observe that at a critical point we have

W = - γ \nabla^{2} u + v_{2}^{*} u_{x} + v_{3}^{*} u_{y} - f = 0, in Ω

From this and (29) 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] \subset 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.

For the case in which

γ = 0.05

, please see Figure 3.

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

**********

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 $(b 14 > 10^{- 5})$ and $(k 2 < 100)$

k2=k2+1;

k1=1;

b12=1.0;

while $(b 12 > 10^{- 5})$ 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 \subset 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 \to R

the velocity field in the direction x of the Cartesian system

(x, y)

, by

v : Ω \to R

, the velocity field in the direction y and by

p : Ω \to 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,

\begin{matrix} \{\begin{matrix} μ \nabla^{2} u - ρ u u_{x} - ρ v u_{y} - p_{x} + ρ g_{x} = 0, & in Ω, \\ μ \nabla^{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, \end{matrix} \end{matrix}

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\begin{matrix} \{\begin{matrix} u = v = 0, & on Γ_{0}, \\ u = u_{\infty}, v = 0, p = p_{\infty}, & on Γ_{1} \end{matrix} \end{matrix}

(31)

About the references, we emphasize that related existence, numerical and theoretical results for similar systems may be found in [17,18,19,20] and [21], respectively. In particular [21] addresses extensively both theoretical and numerical methods and an interesting interplay between them.

Finally, it is worth mentioning these two first paragraphs of this article have been published as a preprint, reference [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

\begin{matrix} \{\begin{matrix} ν \nabla^{2} u - u \partial_{x} u - v \partial_{y} u - \partial_{x} P + g_{x} = 0, & in Ω, \\ ν \nabla^{2} v - u \partial_{x} v - v \partial_{y} v - \partial_{y} P + g_{y} = 0, & in Ω, \\ \partial_{x} u + \partial_{y} v = 0, & in Ω, \end{matrix} \end{matrix}

(32)

\begin{matrix} \{\begin{matrix} u = v = 0, & on Γ_{0}, \\ u = u_{\infty}, v = 0, P = P_{\infty}, & on Γ_{1} \end{matrix} \end{matrix}

(33)

As previously mentioned, at first we look for solutions

(u, v, P) \in W^{1, 2} (Ω; R^{3})

in a distributional sense.

At this point we define the functional

J : V \times Y \times Y \to R

where

V = {(u, v, P) \in W^{1, 2} (Ω; R^{3}) : u = v = 0, on Γ_{0}, u = u_{\infty}, v = 0, P = P_{\infty}, on Γ_{1}}

and

\begin{matrix} J (u, v, P, v_{2}^{*}, v_{3}^{*}) & = & \frac{1}{2} \int_{Ω} {(- ν \nabla^{2} u + v_{2}^{*} u_{x} + v_{3}^{*} u_{y} + P_{x} - g_{x})}^{2} d x \\ + \frac{1}{2} \int_{Ω} {(- ν \nabla^{2} v + v_{2}^{*} v_{x} + v_{3}^{*} v_{y} + P_{y} - g_{y})}^{2} d x \\ + \frac{1}{2} \int_{Ω} {(v_{2}^{*} - u)}^{2} d x + \frac{1}{2} \int_{Ω} {(v_{3}^{*} - v)}^{2} d x \\ + \frac{1}{2} \int_{Ω} {(u_{x} + v_{y})}^{2} d x . \end{matrix}

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Here

Y = Y^{*} = L^{2} (Ω) .

Let

φ, φ_{2}, φ_{3}, φ_{4}, φ_{5} \in C_{c}^{\infty} (Ω) .

We define also

u = (u, v, P, v_{2}^{*}, v_{3}^{*}),

J_{1} (u, v, P, v_{2}^{*}, v_{3}^{*}) = \frac{1}{2} \int_{Ω} {(- ν \nabla^{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}^{*}) = \frac{1}{2} \int_{Ω} {(- ν \nabla^{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}^{*}) = \frac{1}{2} \int_{Ω} {(v_{2}^{*} - u)}^{2} d x + \frac{1}{2} \int_{Ω} {(v_{3}^{*} - v)}^{2} d x + \frac{1}{2} \int_{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) = \int_{Ω} {(- ν \nabla^{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}) = \int_{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}) = \int_{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}) = \int_{Ω} {(φ_{4})}_{x}^{2} d x,

and denoting

W = - γ \nabla^{2} u + v_{2}^{*} u_{x} + v_{3}^{*} u_{y} + P_{x} - g_{x},

we have

\begin{matrix} δ_{u v_{2}^{*}}^{2} J_{1} ((u, v, P, v_{2}^{*}, v_{3}^{*}), φ, φ_{1}) & = & \int_{Ω} W φ_{x} φ_{2} d x \\ + \int_{Ω} (- ν \nabla^{2} φ + 2 v_{2}^{*} φ_{x} + v_{3}^{*} φ_{y}) u_{x} φ_{2} d x d x . \end{matrix}

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\begin{matrix} δ_{u v_{3}^{*}}^{2} J_{1} ((u, v, P, v_{2}^{*}, v_{3}^{*}), φ, φ_{1}) & = & \int_{Ω} W φ_{y} φ_{3} d x \\ + \int_{Ω} (- ν \nabla^{2} φ + 2 v_{2}^{*} φ_{x} + v_{3}^{*} φ_{y}) u_{y} φ_{3} d x . \end{matrix}

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δ_{u P}^{2} J_{1} ((u, v, P, v_{2}^{*}, v_{3}^{*}), φ, φ_{4}) = \int_{Ω} (- ν \nabla^{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}) = \int_{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}) = \int_{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}) = \int_{Ω} u_{y} φ_{3} {(φ_{4})}_{x} d x .

Therefore, defining

\begin{matrix} {δ^{2} J_{1} (u, φ)} \\ = & \frac{1}{2} δ_{u u}^{2} J_{1} ((u, v, P, v_{2}^{*}, v_{3}^{*}), φ, φ) + \frac{1}{2} δ_{v_{2}^{*} v_{2}^{*}}^{2} J_{1} ((u, v, P, v_{2}^{*}, v_{3}^{*}), φ_{2}, φ_{2}) \\ + \frac{1}{2} δ_{v_{3}^{*} v_{3}^{*}}^{2} J_{1} ((u, v, P, v_{2}^{*}, v_{3}^{*}), φ_{3}, φ_{3}) + \frac{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}), \end{matrix}

(37)

we have

\begin{matrix} {δ^{2} J_{1} (u, φ)} & = & \int_{Ω} {(- ν \nabla^{2} φ + v_{2}^{*} φ_{x} + v_{3}^{*} φ_{y} + φ_{2} u_{x} + + φ_{3} u_{y} + {(φ_{4})}_{x})}^{2} d x \\ + \int_{Ω} W (φ_{2} φ_{x} + φ_{3} φ_{y}) d x . \end{matrix}

(38)

Similarly, we may obtain

δ_{v v}^{2} J_{2} ((u, v, P, v_{2}^{*}, v_{3}^{*}), φ_{5}, φ_{5}) = \int_{Ω} {(- ν \nabla^{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}) = \int_{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}) = \int_{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}) = \int_{Ω} {(φ_{4})}_{y}^{2} d x,

and denoting

W_{1} = - γ \nabla^{2} v + v_{2}^{*} v_{x} + v_{3}^{*} v_{y} + P_{y} - g_{y},

we have

\begin{matrix} δ_{v v_{2}^{*}}^{2} J_{2} ((u, v, P, v_{2}^{*}, v_{3}^{*}), φ, φ_{1}) & = & \int_{Ω} W_{1} {(φ_{5})}_{x} φ_{2} d x \\ + \int_{Ω} (- ν \nabla^{2} φ_{5} + 2 v_{2}^{*} {(φ_{5})}_{x} + v_{3}^{*} {(φ_{5})}_{y}) v_{x} φ_{2} d x . \end{matrix}

(39)

\begin{matrix} δ_{v v_{3}^{*}}^{2} J_{2} ((u, v, P, v_{2}^{*}, v_{3}^{*}), φ, φ_{1}) & = & \int_{Ω} W_{1} {(φ_{5})}_{y} φ_{3} d x \\ + \int_{Ω} (- ν \nabla^{2} φ_{5} + 2 v_{2}^{*} {(φ_{5})}_{x} + v_{3}^{*} {(φ_{5})}_{y}) v_{y} φ_{3} d x . \end{matrix}

(40)

δ_{v P}^{2} J_{2} ((u, v, P, v_{2}^{*}, v_{3}^{*}), φ, φ_{4}) = \int_{Ω} (- ν \nabla^{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}) = \int_{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}) = \int_{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}) = \int_{Ω} v_{y} φ_{3} {(φ_{4})}_{y} d x .

Therefore, defining

\begin{matrix} {δ^{2} J_{2} (u, φ)} \\ = & \frac{1}{2} δ_{v v}^{2} J_{2} ((u, v, P, v_{2}^{*}, v_{3}^{*}), φ_{5}, φ_{5}) + \frac{1}{2} δ_{v_{2}^{*} v_{2}^{*}}^{2} J_{2} ((u, v, P, v_{2}^{*}, v_{3}^{*}), φ_{2}, φ_{2}) \\ + \frac{1}{2} δ_{v_{3}^{*} v_{3}^{*}}^{2} J_{2} ((u, v, P, v_{2}^{*}, v_{3}^{*}), φ_{3}, φ_{3}) + \frac{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}), \end{matrix}

(41)

we have

\begin{matrix} {δ^{2} J_{2} (u, φ)} & = & \int_{Ω} {(- ν \nabla^{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 \\ + \int_{Ω} W_{1} (φ_{2} {(φ_{5})}_{x} + φ_{3} {(φ_{5})}_{y}) d x . \end{matrix}

(42)

Moreover, we may have

δ_{u u}^{2} J_{3} ((u, v, P, v_{2}^{*}, v_{3}^{*}), φ, φ) = \int_{Ω} φ^{2} d x + \int_{Ω} φ_{x}^{2} d x,

δ_{v v}^{2} J_{3} ((u, v, P, v_{2}^{*}, v_{3}^{*}), φ, φ) = \int_{Ω} φ_{5}^{2} d x + \int_{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}) = \int_{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}) = \int_{Ω} φ_{3}^{2} d x,

δ_{u v_{2}^{*}}^{2} J_{3} ((u, v, P, v_{2}^{*}, v_{3}^{*}), φ, φ_{1}) = - \int_{Ω} φ φ_{2} d x,

δ_{v v_{3}^{*}}^{2} J_{3} ((u, v, P, v_{2}^{*}, v_{3}^{*}), φ, φ_{1}) = - \int_{Ω} φ_{5} φ_{3} d x,

δ_{u v}^{2} J_{3} ((u, v, P, v_{2}^{*}, v_{3}^{*}), φ, φ_{5}) = \int_{Ω} φ_{x} {(φ_{5})}_{y} d x .

Therefore, defining

\begin{matrix} {δ^{2} J_{3} (u, φ)} \\ = & \frac{1}{2} δ_{u u}^{2} J_{3} ((u, v, P, v_{2}^{*}, v_{3}^{*}), φ, φ) + \frac{1}{2} δ_{v v}^{2} J_{3} ((u, v, P, v_{2}^{*}, v_{3}^{*}), φ_{5}, φ_{5}) \\ + \frac{1}{2} δ_{v_{2}^{*} v_{2}^{*}}^{2} J_{3} ((u, v, P, v_{2}^{*}, v_{3}^{*}), φ_{2}, φ_{2}) + \frac{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}), \end{matrix}

(43)

we have

{δ^{2} J_{3} (u, φ)} = \int_{Ω} {(φ_{2} - φ)}^{2} d x + \int_{Ω} {(φ_{3} - φ_{5})}^{2} d x + \int_{Ω} {(φ_{x} + {(φ_{5})}_{y})}^{2} d x .

Finally, joining the pieces we may infer that

\begin{matrix} {δ^{2} J (u, φ)} & = & {δ^{2} J_{1} (u, φ)} + {δ^{2} J_{2} (u, φ)} + {δ^{2} J_{3} (u, φ)} \\ = & \int_{Ω} {(- ν \nabla^{2} φ + v_{2}^{*} φ_{x} + v_{3}^{*} φ_{y} + φ_{2} u_{x} + φ_{3} u_{y} + {(φ_{4})}_{x})}^{2} d x \\ + \int_{Ω} W (φ_{2} φ_{x} + φ_{3} φ_{y}) d x \\ + \int_{Ω} {(- ν \nabla^{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 \\ + \int_{Ω} W_{1} (φ_{2} {(φ_{5})}_{x} + φ_{3} {(φ_{5})}_{y}) d x \\ + \int_{Ω} {(φ_{2} - φ)}^{2} d x + \int_{Ω} {(φ_{3} - φ_{5})}^{2} d x + \int_{Ω} {(φ_{x} + {(φ_{5})}_{y})}^{2} d x . \end{matrix}

(44)

Observe that at a critical point we have

W = - ν \nabla^{2} u + v_{2}^{*} u_{x} + v_{3}^{*} u_{y} + P_{x} - g_{x} = 0, in Ω

and

W_{1} = - ν \nabla^{2} v + v_{2}^{*} v_{x} + v_{3}^{*} v_{y} + P_{y} - g_{y} = 0, in Ω .

From this and (29) 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 \subset R^{n}

be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω = Γ .

For

1 < p < + \infty

, consider a functional

J : V \to R

where

J (u) = G (\nabla u) + F (u) - {〈 u, f 〉}_{L^{2}},

where

V = \{u \in W^{1, p} (Ω; R^{N}) : u = u_{0} on \partial Ω\}

and

f \in L^{2} (Ω; R^{N}) .

We assume

G : Y \to R

and

F : V \to R

are Fréchet differentiable and F is also convex.

Also

G (\nabla u) = \int_{Ω} g (\nabla u) d x,

where

g : R^{N \times n} \to R

it is supposed to be Fréchet differentiable. Here we have denoted

Y = L^{p} (Ω; R^{N \times n})

We define also

J_{1} : V \times Y_{1} \to R

J_{1} (u, ϕ) = G_{1} (\nabla u + \nabla_{y} ϕ) + F (u) - {〈 u, f 〉}_{L^{2}},

where

Y_{1} = W^{1, p} (Ω \times Ω; R^{N})

and

G_{1} (\nabla u + \nabla_{y} ϕ) = \frac{1}{| Ω |} \int_{Ω} \int_{Ω} g (\nabla u (x) + \nabla_{y} ϕ (x, y)) d x d y .

Moreover, we define the relaxed functional

J_{2} : V \to R

J_{2} (u) = inf_{ϕ \in V_{0}} J_{1} (u, ϕ),

where

V_{0} = {ϕ \in Y_{1} : ϕ (x, y) = 0, on Ω \times \partial Ω} .

Now observe that

\begin{matrix} J_{1} (u, ϕ) & = & G_{1} (\nabla u + \nabla_{y} ϕ) + F (u) - {〈 u, f 〉}_{L^{2}} \\ = & - \frac{1}{| Ω |} \int_{Ω} \int_{Ω} v^{*} (x, y) \cdot (\nabla u + \nabla_{y} ϕ (x, y)) d y d x + G_{1} (\nabla u + \nabla_{y} ϕ) \\ + \frac{1}{| Ω |} \int_{Ω} \int_{Ω} v^{*} (x, y) \cdot (\nabla u + \nabla_{y} ϕ (x, y)) d y d x + F (u) - {〈 u, f 〉}_{L^{2}} \\ \geq & inf_{v \in Y_{2}} \{- \frac{1}{| Ω |} \int_{Ω} \int_{Ω} v^{*} (x, y) \cdot v (x, y) d y d x + G_{1} (v)\} \\ + inf_{(v, ϕ) \in V \times V_{0}} \{\frac{1}{| Ω |} \int_{Ω} \int_{Ω} v^{*} (x, y) \cdot (\nabla u + \nabla_{y} ϕ (x, y)) d y d x + F (u) - {〈 u, f 〉}_{L^{2}}\} \\ = & - G_{1}^{*} (v^{*}) - F^{*} ({div}_{x} (\frac{1}{| Ω |} \int_{Ω} v^{*} (x, y) d y) + f) \\ + \frac{1}{| Ω |} \int_{\partial Ω} (\int_{Ω} v^{*} (x, y) d y) \otimes n u_{0} d Γ, \end{matrix}

(45)

\forall (u, ϕ) \in V \times V_{0}, v^{*} \in A^{*}

, where

A^{*} = {v^{*} \in Y_{2}^{*} : {div}_{y} v^{*} (x, y) = 0, in Ω} .

Here we have denoted

G_{1}^{*} (v^{*}) = sup_{v \in Y_{2}} \{\frac{1}{| Ω |} \int_{Ω} \int_{Ω} v^{*} (x, y) \cdot v (x, y) d y d x - G_{1} (v)\},

where

Y_{2} = L^{p} (Ω \times Ω; R^{N \times n}),

Y_{2}^{*} = L^{q} (Ω \times Ω; R^{N \times n}),

and where

\frac{1}{p} + \frac{1}{q} = 1 .

Furthermore,

\begin{matrix} F^{*} ({div}_{x} (\frac{1}{| Ω |} \int_{Ω} v^{*} (x, y) d y) + f) - \frac{1}{| Ω |} \int_{\partial Ω} (\int_{Ω} v^{*} (x, y) d y) \otimes n u_{0} d Γ \\ = & sup_{(v, ϕ) \in V \times V_{0}} \{- \frac{1}{| Ω |} \int_{Ω} \int_{Ω} v^{*} (x, y) \cdot (\nabla u + \nabla_{y} ϕ (x, y)) d y d x - F (u) + {〈 u, f 〉}_{L^{2}}\}, \end{matrix}

(46)

Therefore, denoting

J_{3}^{*} : Y_{2}^{*} \to R

J_{3}^{*} (v^{*}) = - G_{1}^{*} (v^{*}) - F^{*} ({div}_{x} (\int_{Ω} v^{*} (x, y) d y) + f) + \frac{1}{| Ω |} \int_{\partial Ω} (\int_{Ω} v^{*} (x, y) d y) \otimes n u_{0} d Γ,

we have got

inf_{u \in V} J_{2} (u) \geq sup_{v^{*} \in A^{*}} J_{3}^{*} (v^{*}) .

Finally, we highlight such a dual functional

J_{3}^{*}

is convex (in fact concave).

10. A general concave primal dual formulation for an originally non-convex primal one

In this section we develop a general concave dual formulation related to a primal one which is originally non-convex.

Let

Ω \subset R^{3}

be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω .

Consider a Fréchet differentiable functional

J : V \to R

where

V = W_{0}^{1, 2} (Ω) .

Assume there exists

K_{1}, K_{3}, K > 0

such that

- K_{1} I_{d} \leq δ^{2} J (u) \leq K_{1} I_{d}, \forall {∥ u ∥}_{\infty} \leq K_{3},

where

K ≫ max {K_{1}, K_{3}, 1} .

Observe that

\begin{matrix} J (u) & = & - {〈 u, e^{K v_{1}^{*}} 〉}_{L^{2}} + \frac{1}{2} \int_{Ω} {(\sqrt[4]{K} u + 2 K)}^{2} d x \\ + {〈 u, e^{K v_{1}^{*}} 〉}_{L^{2}} - \frac{1}{2} \int_{Ω} {(\sqrt[4]{K} u + 2 K)}^{2} d x + J (u) \\ \geq & inf_{u \in L^{2}} \{- {〈 u, e^{K v_{1}^{*}} 〉}_{L^{2}} + \frac{1}{2} \int_{Ω} {(\sqrt[4]{K} u + 2 K)}^{2} d x\} \\ + {〈 u, e^{K v_{1}^{*}} 〉}_{L^{2}} - \frac{1}{2} \int_{Ω} {(\sqrt[4]{K} u + 2 K)}^{2} d x + J (u) \\ = & - \frac{1}{2} \int_{Ω} \frac{e^{2 K v_{1}^{*}}}{\sqrt{K}} d x + 2 K \int_{Ω} \frac{e^{K v_{1}^{*}}}{\sqrt[4]{K}} d x \\ + {〈 u, e^{K v_{1}^{*}} 〉}_{L^{2}} - \frac{1}{2} \int_{Ω} {(\sqrt[4]{K} u + 2 K)}^{2} d x + J (u) \\ \equiv & J_{1}^{*} (u, v_{1}^{*}), \end{matrix}

(47)

where we have denoted

\begin{matrix} J_{1}^{*} (u, v_{1}^{*}) & = & - \frac{1}{2} \int_{Ω} \frac{e^{2 K v_{1}^{*}}}{\sqrt{K}} d x + 2 K \int_{Ω} \frac{e^{K v_{1}^{*}}}{\sqrt[4]{K}} d x \\ + {〈 u, e^{K v_{1}^{*}} 〉}_{L^{2}} - \frac{1}{2} \int_{Ω} {(\sqrt[4]{K} u + 2 K)}^{2} d x + J (u) . \end{matrix}

(48)

We observe that the calculation of the concerning polar functional is made through the extremal equation

\frac{\partial}{\partial u} \{- {〈 u, e^{K v_{1}^{*}} 〉}_{L^{2}} + \frac{1}{2} \int_{Ω} {(\sqrt[4]{K} u + 2 K)}^{2} d x\} = 0,

which stands for

- e^{K v_{1}^{*}} + (\sqrt[4]{K} u + 2 K) \sqrt[4]{K} = 0,

so that

e^{K v_{1}^{*}} \approx 2 K \sqrt[4]{K},

and thus

K v_{1}^{*} \approx ln (2 K \sqrt[4]{K}),

that is

v_{1}^{*} \approx \frac{ln (2 K \sqrt[4]{K})}{K} .

With such results in mind, we define the non-active restriction

v_{1}^{*} \geq \frac{3}{4} \frac{ln (2 K \sqrt[4]{K})}{K}, in Ω .

Defining now

A^{*} = \{(u, v_{1}^{*}) \in V \times Y^{*} : v_{1}^{*} \geq \frac{3}{4} \frac{ln (2 K \sqrt[4]{K})}{K}, in Ω\}

where

Y = Y^{*} = L^{2} (Ω),

by just directly computing

δ^{2} J_{1}^{*} (u, v_{1}^{*})

we may easily obtain that such a Hessian is positive definite in

A^{*}

so that

J_{1}^{*}

is concave on the convex set

A^{*}

Defining

B^{*} = \{(u, v_{1}^{*}) \in V \times Y^{*} : {∥ u ∥}_{\infty} \leq K_{3}\},

suppose now

(u_{0}, {\hat{v}}_{1}^{*}) \in A^{*} \cap B^{*}

is such that

δ J_{1}^{*} (u_{0}, {\hat{v}}_{1}^{*}) = 0 .

From the variation of

J_{1}^{*}

v_{1}^{*}

, we have

- K \frac{e^{2 K {\hat{v}}_{1}^{*}}}{\sqrt{K}} + 2 K^{2} \frac{e^{K {\hat{v}}_{1}^{*}}}{\sqrt[4]{K}} + K e^{K {\hat{v}}_{1}^{*}} u_{0} = 0,

so that

- e^{K {\hat{v}}_{1}^{*}} + (\sqrt[4]{K} u_{0} + 2 K) \sqrt[4]{K} = 0 .

From the variation of

J_{1}^{*}

in u, we obtain

e^{K {\hat{v}}_{1}^{*}} - (\sqrt[4]{K} u_{0} + 2 K) \sqrt[4]{K} + δ J (u_{0}) = 0

so that

δ J (u_{0}) = 0 .

Remark 10.1.

Summarizing, the functional

J_{1}^{*}

is concave on

A^{*} \cap B^{*}

(which corresponds to a non-active restriction) and a critical point of

J_{1}^{*}

corresponds to a critical point of the original primal functional J. Therefore,

J_{1}^{*}

corresponds to a concave primal dual formulation for a originally non-convex primal functional

J .

11. 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 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 non-convex 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.

Data Availability Statement

Details on the software for numerical results available upon request. e-mail: fabio.botelho@ufsc.br

Conflicts of Interest

The author declares no conflict of interest concerning this article.

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Figure 1. Solution

u_{0} (x)

for the primal formulation.

Figure 1. Solution

u_{0} (x)

for the primal formulation.

Figure 2. Solution

u_{0} (x)

for a Burger’s type equation with

γ = 0.5

Figure 2. Solution

u_{0} (x)

for a Burger’s type equation with

γ = 0.5

Figure 3. Solution

u_{0} (x)

for a Burger’s type equation with

γ = 0.05

Figure 3. Solution

u_{0} (x)

for a Burger’s type equation with

γ = 0.05

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On Duality Principles and Related Convex Dual Formulations Suitable for Local and Global Non-Convex Variational Optimization

Abstract

1. Introduction

2. The primal variational formulation and the dual functional definitions

3. The main duality principle and a concerning convex dual formulation

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

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

6. A numerical example

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

7.1. A numerical example concerning a Burger’s type equation

8. A primal dual variational formulation for a Navier-Stokes system

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

10. A general concave primal dual formulation for an originally non-convex primal one

11. Conclusion

Data Availability Statement

Conflicts of Interest

References

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