The nonlinear singular Burgers equation with small parameter and p-regularity theory

In this paper, we study a solutions existence problem of the following nonlinear singular Burgers equation F(u,ε)=ut′-uxx′′+uux′+εu2=f(x,t),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} F(u,\varepsilon )=u_{t}'-u_{xx}''+uu_{x}'+\varepsilon u^{2}=f(x,t), \end{aligned}$$\end{document}where F:Ω→C([0,π]×[0,∞))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F: \Omega \rightarrow \mathcal {C}([0,\pi ]\times [0,\infty ))$$\end{document}, Ω=C2([0,π]×[0,∞))×R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega = \mathcal {C}^{2}([0,\pi ]\times [0,\infty ))\times \mathbb {R}$$\end{document}, u(0,t)=u(π,t)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u(0,t)=u(\pi ,t) =0$$\end{document}, u(x,0)=g(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u(x,0)=g(x)$$\end{document}, and F, f(x, t), g(x) will be describe in the text. The first derivative of operator F at the solution point is degenerate. By virtue of p-regularity theory and Michael selection theorem, we prove the existence of continuous solution for this nonlinear problem.

The problems of the form F(u, ε) = 0, where F : U × M → Z is a sufficiently smooth nonlinear mapping from a Banach space U × M to a Banach space Z , we separate into two classes, called regular and irregular. Roughly speaking, regular problems are those to which implicit function theorem arguments can be applied and the irregular ones are those to which it cannot, at least not directly.
The basis for our practical applications will be the following analogue of Lyusternik theorem on tangent cone (see [10]).
where M is finitedimensional space, and U and Z are Banach spaces. Let the mappings F i (v, y), i = 1, . . . , p be defined by (8). Assume that F(v * , y * ) = 0 and ∀ȳ ∈ M, ȳ = 1, and F is strongly p-regular with respect to M along every element (0,ȳ),ȳ ∈ M, that is where C > 0-independent constant.
Above theorem is proved in [10]. This is an analogue of the Lyusternik theorem on the tangent cone, which concerns the existence of continuous solutions of the equation In its proof, we applied the Michael selection theorem (see [11]), which we provide in the modified form: Let us note that from the Banach theorem about surjective operator, we have A −1 ≤ K (see Definition 6).
Theorem 1 allows the important conclusion about existence solution to Burgers equation with respect to the boundary conditions. The existence of continuous solutions is interesting, because there are no many results connected with singular problems (see, for example, [1,3,5]).

Main constructions in p-regularity theory
In this section, we present some important definitions and theorems of p-regularity theory to be used in what follows [6,7,14,15].
We are interested in the following nonlinear problem: where the mapping F : W × Y → Z and W , Y and Z are Banach spaces. Assume that for some point where Z 1 = cl(ImF (v * , y * )) and V 1 = Z . For V 2 , we use one of the closed complement of Z 1 in Z (if such one there exists). Let P V 2 : Z → V 2 be the projector onto V 2 along Z 1 . By Z 2 , we denote the closure of the linear span of the image of the quadratic mapping P V 2 F (v * , y * )[·] 2 . Then, inductively, where V i is a choice of closed complement of Z 1 ⊕ · · · ⊕ Z i−1 , i = 2, . . . , p with respect to Z , and P V i : Z → V i is a projector onto V i along Z 1 ⊕ · · · ⊕ Z i−1 , i = 2, . . . , p with respect to Z . Finally, Z p = V p . The order p is the minimal number (if it exists) for which the decomposition (6) holds.
In what follows, we will denote ϕ (0) = ϕ for any mapping ϕ. Define the following mappings: where Then, the mapping F can be represented as: or

Definition 1 The linear operator
such that for any z = (v, y) is called a p-factor operator depending of h or shortly a p-factor operator if it is clear from the context.

Remark 1
In the completely degenerate case, the p-factor operator reduces to

Remark 2 For each mapping
Remark 3 For each mapping F i , we have in the completely degenerate case This means that F i−1 are i-factor operators corresponding to completely degenerate mappings F i up to order i. Therefore, the general degeneration of F can be reduced to the study of completely degenerated mappings F i , i = 1, . . . , p and their compositions.
Let's introduce the nonlinear operator p [·] p such that

Definition 3
The p-kernel of the operator p is a set Note that the following relation holds: Let A ∈ L(U , Z ) and AU = Z . Let {·} −1 denote right inverse operator, i.e., Obviously the operator {·} −1 is multivalued.

Definition 6 Define
Define the solution set for the mapping F as the set and let T (v * ,y * ) S denote the tangent cone to the set S at the point (v * , y * ), i.e., The following theorems describe the tangent cone to the solution set of equation (5) in the p-regular case.

Theorem 3 Let W , Y and Z be the Banach spaces, and let the mapping F
Theorem 4 (Generalized Lyusternik Theorem, [6]) Let W , Y and Z be the Banach spaces, and let the mapping Let us explain that here (for Banach spaces U and Z ) F ∈ C p (U , Z ) means that F : U → Z is p times continuously Frechét differentiable.
The following Lemma will be important in the study of the surjectivity of p-factor operators.
Lemma 1 Suppose that Z = Z 1 ⊕ Z 2 , where Z 1 and Z 2 are closed subspaces in Z , A, B ∈ L(U , Z ), and Im A = Z 1 . Let P 2 also be the projection onto Z 2 along Z 1 .
This lemma is a consequence of the following.

Lemma 2 Suppose that Z
The proof is obvious. Lemma 1 follows from Lemma 2 if we put A 1 = A and Lemma 3 is the generalization of lemma 1.

Auxiliary p-factor implicit function theorems
Now, we consider two theorems, which are the modification of analogical theorems in [6].
Theorem 5 (The p-order implicit function theorem for nontrivial p-kernel) Let W , Y and Z be Banach spaces, . . , p be defined by (8) and the p-factor operator p (h) be given by (11).
The proof of the above theorem is similar to the proof of analogous theorem in [6].

Remark 4 Estimate (19) can be replaced by the following
The following theorem is some generalization of Theorem 5. , y), i = 1, . . . , p be defined by (8) and the p-factor operator p (h) be given by (11). Assume that there exists an elementh ∈ p r =1 Ker r F (r ) h y ),h y = 0 such that Im p (h) · (0 × Y ) = Z , that is the mapping F is p-regular along the elementh. Then for a sufficiently small α > 0, ν > 0 and δ = αν p there exists the continuous mapping ϕ(y) : U δ (y * ) → U ν (v * ) and constant K > 0 such that the following hold:

Theorem 6 Let W , Y and Z be Banach spaces, F
arbitrary, continuous function such that The difference between Theorems 5 and 6 is that in Theorem 5 we take derivation with respect to v, but in Theorem 6 with respect to (v, y).

Solutions to Burgers equations
In this section, we will present the main result of this work.
Consider the Burgers equation F : where F is sufficiently smooth (at least up to order p + 1) and u(0, t) = u(π, t) = 0.
We will apply Theorem 6 denoting The following result will hold , ν there exists continuous solution of (24) in the following form where γ (ε) is some continuous function such that γ (ε) = O(k(ε)) and y(x, t, ε) Proof We introduce the mapping (u, ε, k) and equation with boundary conditions u(0, t) = u(π, t) = 0.
Here (u * , ε * , k * ) = (0, 0, 0) is the trivial solution of this equation and the tangent cone goes out from this point. Note that the analysis of the first derivative of the mapping comes to the analysis of 3-regularity of the mapping F.
Denoting u t − u x x by Lu (parabolic operator) and u x by ∂ ∂ x u (operator of differentiation), we obtain Denote by F u , F ε partial derivatives of F with respect to u, ε and analogically higher-order partial derivatives by F uu , F uε , F εu , F εε , F uuu , F uuε , . . ., etc.
We have where and Note that Taking into account the Fourier method of solving second-order partial differential equations and bearing in mind that we are looking for at least one solution we can determine The image of the operator F u (0, 0) is defined as follows: We will look for a solution to the equation F(u, ε) = 0 in the form Then One can show that the boundary value problem i.e.,ū x x +ū = − sin x,ū(0) =ū(π ) = 0, does not have a solutions.
Therefore, the operator F u (0, 0) is not surjective and This implies that Z = Z 1 ⊕ V 2 , where Z 1 = ImF u (0, 0) and V 2 = Z ⊥ 1 . The projector P V 2 : Z → V 2 can be described as This implies that Let us evaluate the second derivative of the mapping F where From this, we obtain and Substituting h u = se −t sin τ , where s ∈ R, we get We continue Let us evaluate the third derivative of the mapping F: and F (u, ε) = F (0, 0).
Note that F (4) (u, ε) = 0, Therefore, we will show that the mapping F is 3-regular on some elements that belong to the 3-kernel of the 3-factor operator and next we will describe the solutions of Burgers equation. Now let us take p(τ, t) = (h u , h ε ). For such defined vector p(τ, t) the following relations hold: Substituting h u = se −t sin τ , where s ∈ R and bearing in mind that π 0 sin 3 τ dτ = 4 3 we get

Conclusion
Our research was inspired by works [9,10]. We obtain analytical formulas for solving the nonlinear Burgers equation of the form (1) based on the p-regularity theory. Additionally by Theorem 1 and Michael selection theorem 2, we conclude that there exists a continuous solution.