Abstract
In this paper, we consider the steady incompressible Navier–Stokes equations in a smooth bounded domain \(\Omega \subset \mathbb R^n\) with the dimension \(n\ge 3\). We first establish asymptotic expansion formulae of Sobolev regular finite energy solutions in \(\Omega\). In the second part of this paper, explicit representation formulae of Sobolev regular solutions are showed in the regular polyhedron \(\Omega :=[0,T]^n\).
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction and main results
In this paper, we consider the steady incompressible Navier–Stokes equations:
where \(x\in \Omega\), and \(\Omega \subset \mathbb R^n\) is a smooth bounded domain, \(U:\mathbb R^+\times \Omega \rightarrow \mathbb R^n\) is the fluid velocity, and it is of the form \(U(x)=(U_1(x),U_2(x),\ldots ,U_n(x))\), \(P(x):\mathbb R^+\times \Omega \rightarrow \mathbb R\) stands for the pressure in the fluid, and the constant \(\nu\) is the viscosity. We denote by \(f:=(f_1,f_2,\ldots ,f_n)\) an external force. The divergence free condition in second equations of (1.1) guarantees the incompressibility of the fluid.
We supplement the steady incompressible Navier–Stokes equations (1.1) with the Dirichlet boundary condition
and the pressure takes the form
In particular, when the external force \(f=0\) in equations (1.1), then our problem reduces to the steady incompressible Navier–Stokes equations
The question of whether a solution of the 3D incompressible Navier–Stokes equations can develop a finite time singularity from smooth initial data with finite energy is one of the Millennium Prize problems, see [8]. In 1934, Leray [22] showed that the 3D incompressible Navier–Stokes equations (1.1) admit global-forward-in-time weak solutions of the initial value problem. Caffarelli, Kohn and Nirenberg [6] established a \(\varepsilon\)-regularity criterion for equations (1.1). After that, Lin [23] gave a new and simpler proof for the result of Caffarelli, Kohn and Nirenberg. Koch and Tataru [21] proved the global well-posedness for the equations (1.1) in a space of arbitrary dimension with small initial data in \(BMO^{-1}\) space. Recently, Buckmaster and Vicol [5] proved that the Leray weak solutions of the 3D Navier–Stokes equations are not unique in the class of weak solutions with finite kinetic energy. We refer the readers to [2,3,4, 9, 19, 28, 34,35,34] for more related results on this equations.
Much attention attracted the existence and the regularity properties of stationary solutions for the incompressible Navier–Stokes equations. These solutions depend on the force f and the domain \(\Omega\). Gerhardt [15] proved that the steady four-dimensional problem admits a solution in \(W^{2,p}\) by assumption the force \(f\in L^p\). Frehse and Ruzicka [11] and Struwe [31] got the existence and regularity of the solutions in the five-dimensional case, respectively. Frehse and Ruzicka [10] also obtained the existence of regular solutions in a bounded domain of six-dimension. The fifteen dimensional torus case was given in [12]. Maz’ya and Rossmann [25] showed the existence of weak solutions in the three-dimensional case for a polyhedral domain. Kim [20] considered the existence of very weak solutions in a bounded domain of dimension \(d=2,3,4\). Farwig and Sohr [7] proved the existence, uniqueness and very low regularity of solutions to the inhomogeneous Navier–Stokes equations with special external force in a bounded domain of \(d\ge 3\) dimension. Recently, Hou and Pei [18] got the existence of weak solutions in a bounded connected polygon or polyhedron of two or three dimension. Luo [24] obtained the non-uniqueness of weak solutions for this kind of problems in the case of the d-torus with \(d\ge 4\). We cannot list all the contributions to this field, but we refer the readers to [12,13,14] and the references therein.
As pointed out in [24], the question of uniqueness of regular solutions to the steady incompressible Navier–Stokes equations (1.1) remains mostly an open problem. The non-uniqueness of the weak solution has been given in [24]. In this paper, under the assumption of the external force f being small and \(f\not \equiv 0\), we give asymptotic expansion formulae of Sobolev regular solution with finite energy for the steady incompressible Navier–Stokes equations (1.1) in a smooth bounded domain \(\Omega\) of dimension \(n\ge 3\). Next, we give explicit representation formulae of the Sobolev regular solution in a special domain, namely if \(\Omega :=([0,T])^n\).
We now state the main result in this paper.
Theorem 1.1
Let the viscous constant \(\nu \ge 1\) and \(s\ge 1\). Asssume that the external force \(f\in H^s(\Omega )\) with \(\Vert f\Vert _{H^s(\Omega )}\lesssim \varepsilon\) and \(f\not \equiv 0\). Then the steady incompressible Navier–Stokes equations (1.1) with the Dirichlet boundary conditions (1.2) admit a Sobolev regular solution with finite energy \(U\in H^s(\Omega )\). Here, the pressure is given by (1.3).
Moreover, there exists a small constant \(0<\varepsilon \ll 1\) such that
for any \(x\in \Omega \subset \mathbb R^n\) \((n\ge 3)\).
Remark 1.1
The main feature of Theorem 1.1 is that it gives explicit representation formulae as follows
where the function \(U^{(0)}(x)\) satisfies the assumption
and
and \(\mathbf{h} ^{(m)}( x)\) (\(m=1,2,3,\ldots\)) is obtained by solving the linearized problem with the Dirichlet boundary condition in Sobolev space \(H^s(\Omega )\) with \(s>1\)
and \(E^{(m-1)}(x)\) denotes the error term, while the linear operator \(\mathcal L[U^{m-1}]\mathbf{h} ^{(m)}\) is defined in (2.7). The index s of the Sobolev regularity depends on the higher derivative estimate of solution for the linearized equations. From (1.4), we remark that the solution depends strongly on the initial approximation function \(U^{(0)}(x)\). Our proof is based on the Nash-Moser iteration scheme, by using some ideas developed in [37, 38]. For the general Nash-Moser implicit function theorem, we refer to the seminal papers of Nash [27], Moser [26] and Hörmander [17], and to Rabinowitz [29] for a singular perturbation problem of elliptic equations by using the Nash-Moser implicit function theorem.
In particular, if we consider the domain \(\Omega :=([0,T])^n\) (a regular polyhedron) with the finite constant \(T>0\), then we have the following explicit representation formulae.
Corollary 1.1
Let the integers \(p>1\) and \(q>2\), and the parameter \(0<\varepsilon \ll 1\). Then the steady incompressible Navier–Stokes equations (1.1) admit an explicit expansion of the Sobolev regular solution with finite energy as follows
for all \(x=(x_1,x_2,\ldots ,x_n)\in \Omega :=([0,T])^n\), and
where
and the remainder term \(\mathcal R(x)\in H^s(\Omega )\) satisfies
Moreover, the pressure is determined by
Notations Throughout this paper, we assume that \(\Omega \subset \mathbb R^n\) with \(n\ge 3\) and we denote the usual norms of \(\mathbb L^2(\Omega )\) and \(\mathbb H^s(\Omega )\) by \(\Vert \cdot \Vert _{\mathbb L^2}\) and \(\Vert \cdot \Vert _{\mathbb H^s}\), respectively. The norm of the Sobolev space \(H^s(\Omega ):=(\mathbb H^s(\Omega ))^n\) is denoted by \(\Vert \cdot \Vert _{H^s}\). The symbol \(a\lesssim b\) means that there exists a positive constant C such that \(a\le Cb\). We denote by \((x_1,x_2,x_3,\ldots ,x_n)^T\) the column vector in \(\mathbb R^n\). The letter C with subscripts to denote dependencies stands for a positive constant that might change its value at each occurrence.
The paper is organized as follows. In Sect. 2, we first give a class of initial approximation functions, then the Carleman-type estimate of solution for the linearized equations about the initial approximation functions is shown. Next, we prove the existence of the Sobolev regular solution for the linearized equations. In Sect. 3, we establish the general approximation step for the construction of the Nash-Moser iteration scheme. In the final section of this paper, we show how to construct a small Sobolev regular solution for the incompressible steady Navier–Stokes equations (1.1) by the proof of convergence for the Nash-Moser iteration scheme.
2 The first approximation step
We introduce a family of smooth operators possessing the following properties.
Lemma 2.1
[1, 17] There is a family \(\{\Pi _{\theta }\}_{\theta \ge 1}\) of smoothing operators in the space \(H^s(\Omega )\) acting on the class of functions such that
where C is a positive constant and \((s_1-s_2)_+:=\max (0,s_1-s_2)\).
In our iteration scheme, we set
where \(N_0\) is a fixed positive constant, then by (2.1), it follows that
We consider the approximation problem of the steady incompressible Navier–Stokes equations (1.1) as follows
with the Dirichlet boundary condition (1.2) and the incompressible condition
2.1 The initial approximation function
Let \(s\ge 1\) be a fixed finite constant and \(0<\varepsilon _0<\varepsilon ^2\ll 1\). For any \(x\in \Omega\), we choose the initial approximation functions
Meanwhile, we require
Moreover, for any fixed constant \(s\ge 1\) and \(x\in \Omega\) and \(i,j=1,2,\ldots ,n\), we also need the condition
and the initial error term
where \(E^{(0)}\) denotes the error term taking the form
with
and
A family of explicit examples In fact, many vector functions can be chosen to satisfy (2.4)–(2.6). We now give a family of exact examples of the initial approximation function satisfying (2.4)–(2.6).
Let the integers \(p>1\) and \(q>2\). We choose the initial approximation functions of the form
where
and
By direct computations, it follows that
hence
and
Moreover, we observe that these functions decay to 0 as \(|x|\rightarrow +\infty\), and for every fixed integer p, the assumptions (2.4)–(2.6) are fulfilled.
Here, the initial approximation pressure satisfies
2.2 The Carleman estimate of linearized equations
We now construct the first approximation solution denoted by \(U^{(1)}(x)\) of (2.3). The first approximation step between the initial approximation function and first approximation solution is denoted by
Then we linearize the nonlinear system (2.3) around \(U^{(0)}\) to get the linearized operators as follows
where \(\mathcal D_{U^{(0)}}\) denotes the Fréchet derivatives on \(U^{(0)}\). By (1.3), we obtain
We now consider the linear system
and the boundary condition
The solution of this problem gives the first approximation step of the steady incompressible Navier–Stokes equations (1.1).
Before we carry out some a priori estimates, for \(j=1,2,3,\ldots ,n\), we rewrite equations of (2.9) into a coupled system as follows
with the Dirichlet boundary condition
We now derive the Carleman-type estimate of the solution to the linear system (2.11).
Lemma 2.2
Let \(\nu \ge 1\). Assume the initial approximation function \(U^{(0)}\) satisfying (2.4)–(2.6). Then the solution \(\mathbf{h} ^{(1)}(x)\) of the linear system (2.11) satisfies
Proof
Let \(\psi (x_1)\) be a function with \(x_1\in \Omega\) such that
and \(e^{-\psi (x_1)}\) is bounded for \(x_1\in \Omega\). Here the constant \(\kappa\) belongs to \((0,{1\over 4})\). The condition (2.14) implies \(\psi ''(x_1)>{1\over 4}>0\). In fact, there are many functions satisfying these conditions. We give in what follows a simple example. Since \(x_1\in \Omega\), there exists a suitable positive constant \(b>1\) such that the function
is well defined. Here, we take the constant \(a:=({1\over 4}-\kappa )^{-{1\over 2}}\).
Direct computations give that
thus the ODE inequality (2.14) holds for a suitable \(b>1\). Meanwhile, there exists a positive constant \(C_0\) such that
Multiplying both sides of equations in (2.11) by \(e^{-\psi (x_1)}h_j^{(1)}\), respectively, then integrating over \(\Omega\), by noticing the Dirichlet boundary condition (2.10), for \(j=1,2,3,\ldots ,n\), it follows that
We sum up (2.15) from \(j=1\) to \(j=n\), then it follows that
On the one hand, note that we have chosen the initial approximation function \(U^{(0)}\) satisfying (2.4)–(2.6). We integrate by parts to get
since the initial approximation function \(U^{(0)}\) satisfies \(\nabla \cdot U^{(0)}\), inequality (2.17) is reduced into
and direct computation gives that
and noticing the incompressible condition
it follows that
Furthermore, by (2.8), using the standard Calderon-Zygmund theory and Young’s inequality, it follows that
On the other hand, by the Young inequality, we obtain
and
Thus we substitute (2.17)–(2.23) into (2.16) and we find
where the coefficients are given by
Since the weighted function \(\psi (x_1)\) satisfies (2.14), the main term of \(A_1(x)\) \((i=1,2,3,\ldots ,n)\) is
Thus, it follows that
Combining this estimate with (2.14) and the fact that \(\nu \ge 1\), we deduce that there exists a positive constant \(C_{\nu ,\varepsilon ,n}\) depending on \(\nu ,\varepsilon ,n\) such that
where \(\kappa \in (0,{1\over 4})\). Similarly, it follows that
and
Thus, it follows from (2.24) that
Combining this estimate with the fact that \(e^{-\psi (x_1)}\) is a bounded function, we obtain
The proof is now complete. \(\square\)
Furthermore, we derive the higher order derivatives estimates of elliptic equations. For a fixed constant \(s\ge 1\), applying \(D_i^s:=\partial _{x_i}^s\) \((\forall i=1,2,3,\ldots ,n)\) to both sides of (2.11), it follows that
with the boundary condition
where \(1\le l\le s\) and
Next, we derive higher derivative estimates of solutions to the coupled system (2.11).
Lemma 2.3
Let \(\nu \ge 1\). Assume the initial approximation function \(U^{(0)}\) satisfying (2.4)–(2.6). Then the solution \(\mathbf{h} ^{(1)}(x)\) of the linear system (2.11) satisfies
Proof
This proof is based on the induction. Let \(s=1\), by (2.27), it follows that
with the boundary condition
Let us choose the weighted function satisfies (2.14). We multiply both sides of (2.30) by \(D_i^1h_j^{(1)}e^{-\psi (x_1)}\), respectively, then integrating over \(\Omega\) by noticing (2.31), and summing up those equalities from \(j=1\) to \(j=n\), it follows that
where
We now estimate each term in (2.32). On the one hand, since we have chosen the initial approximation function \(U^{(0)}\) satisfying (2.4)–(2.6), using the similar method of getting (2.17)–(2.20), we obtain
By the incompressible condition \(\nabla \cdot \mathbf{h} ^{(1)}=0\) and integrating by parts, we find
from which, by the standard Calderon-Zygmund theory and Young’s inequality we find
and
On the other hand, it follows that
Thus, summing up (2.32) from \(i=1\) to \(i=n\), we use (2.33)–(2.40) to derive
where
We notice that the main term of \(\overline{A}_i(x)\) \((i=1,2,3,\ldots ,n)\) is given by
Combining this fact with the assumption (2.5), we deduce that
Thus, by (2.14), there exists a positive constant \(C_{\nu ,\varepsilon ,n}\) depending on \(\nu ,\varepsilon ,n\) such that
where \(\kappa \in (0,{1\over 4})\). Similarly, it follows that
and
Therefore
Furthermore, one can see that the last two terms in the right-hand side of (2.42) can be controlled by using (2.13). Thus, it follows that
Assume that the \(2\le l\le s-1\) derivative case holds, that is,
We now prove that the sth derivative case holds. Multiplying both sides of equations (2.27) by \(D_i^sh^{(1)}_je^{-\psi (x_1)}\), then integrating over \(\Omega\) by using the boundary condition (2.28), and summing up those equalities from \(j=1\) to \(j=n\), it follows that
We notice that
and
Thus, with similar arguments as for getting (2.41), we can obtain
where
By assumptions of (2.4)–(2.5), one can see that the coefficients \(C_i(x)\) \((i=1,2,3,\ldots ,n)\) have the same main terms with \(\overline{A}_i(x)\). It follows that
By this estimate and (2.14), one can deduce that there exists a positive constant \(C_{\nu ,\varepsilon ,n}\) depending on \(\nu ,\varepsilon ,n\) such that
where \(\kappa \in (0,{1\over 4})\). Similarly, it follows that
Thus, we can reduce (2.48) into
Hence, with similar arguments as for getting (2.43), we use (2.44) to derive
Combining this estimate with the fact that \(e^{-\psi (x_1)}\) is bounded, we obtain (2.29). \(\square\)
2.3 The existence of first approximation step
Based on the previous a priori estimates, we are ready to prove the existence of the first approximation step, by using the classical theory of elliptic equations; see [16, 30].
Proposition 2.1
Assume the initial approximation function \(U^{(0)}\) satisfying (2.4)–(2.6). Then the linearized elliptic system
admits a Sobolev regular solution \(\mathbf{h} ^{(1)}(x)\in H^s(\Omega )\).
Moreover, it follows that
Proof
Let \(\mathbb P\) be the Leray projector onto the space of divergence free functions. We apply the Leray projector to equations (2.9), it follows that
By (2.13) in Lemma 2.2 and (2.29) in Lemma 2.3, we can get the uniform bound estimate
From the standard theory of elliptic equations of the general order [16, 30], the linear elliptic equations (2.51) admit a unique weak solution \(\mathbf{h} ^{(1)}\in H^1\) if \(E^{(0)}\in H^1\). Since the error term \(E^{(0)}\in H^s(\Omega )\) for \(s>1\), we conclude that \(\mathbf{h} ^{(1)}\in H^s(\Omega )\). \(\square\)
3 The mth approximation step
We define
with the integer \(2\le k\le m-1\) and the constant \(s\ge 1\).
Assume that the m-th approximation solutions of (2.3) is denoted by \(\mathbf{h} ^{(m)}( x)\) with \(m=2,3,\ldots\). Let
then we have
We linearize nonlinear equations (2.3) around \(U^{(m-1)}(x)\) to get the following boundary value problem
with the boundary conditions
where the error term is given by
and
where
This is also the nonlinear term in approximation problem (2.3) at \(U^{(m-1)}(x)\).
The following result establishes how to construct the m-th approximation solution.
Proposition 3.1
Let \(\nu \ge 1\). Assume the initial approximation function \(U^{(0)}\) satisfying (2.4)–(2.6), \(U^{(m-1)}( x)\in \mathcal B_{\varepsilon }\) and \(\sum _{i=1}^{m-1}\Vert \mathbf{h} ^{(i)}\Vert ^2_{H^s}\lesssim \varepsilon ^2\). Then the linearized problem (3.2) with the boundary condition (3.3) admits a Sobolev regular solution \(\mathbf{h} ^{(m)}( x)\in H^s(\Omega )\), which satisfies
where the error term satisfies
Proof
Direct computation gives that
By the assumption \(\sum _{i=1}^{m-1}\Vert \mathbf{h} ^{(i)}\Vert ^2_{H^s}\lesssim \varepsilon ^2\), we observe that
Thus, noticing that \(U^{(0)}(x)\) satisfies (2.4)–(2.6), by small modification of \(\partial _{x_i}U^{(0)}_j(x)\), it follows that
Moreover, we notice that the \((m-1)\)-th approximation solution is
and
Thus, it follows that
Then we will find the m-th \((m\ge 2)\) approximation solution \(U^{(m)}(x)\), which is equivalent to find \(\mathbf{h} ^{(m)}(x)\) such that
Substituting (3.10) into (2.3), it follows that
Setting
we supplement it with the boundary conditions (3.3).
Since we assume \(U^{(m-1)}(x)\in \mathcal B_{\varepsilon }\), there is the same structure between the linear system (2.9) and the linear system of mth approximation solutions. Thus, by means of the same proof process in Proposition 2.1, we can show that the above problem admits a solution \(\mathbf{h} ^{(m)}( x)\in H^s(\Omega )\). For this purpose, we should use (2.2). Furthermore, similar to (2.50), we can use (3.8)–(3.9) to derive
where one can see the \((m-1)\)-th error term \(E^{(m-1)}\) such that
Moreover, by (3.5) and the standard Calderon-Zygmund theory, it follows that
The proof is now complete. \(\square\)
4 Convergence of the approximation scheme
Our target is to prove that \(U^{(\infty )}(x)\) is a global solution of nonlinear equations (1.1). This is equivalent to show that the series \(\sum \limits _{i=1}^m\mathbf{h} ^{(i)}(x)\) is convergent.
For a fixed constant \(s\ge 1\), let \(1\le s=\bar{k}<k_0\le k\) and
which gives that
Proposition 4.1
Let \(\nu \ge 1\). Assume the initial approximation function \(U^{(0)},\) satisfying (2.4)–(2.6). Then the steady incompressible Navier–Stokes equations (1.1) with the Dirichlet boundary condition (1.2) admit a global Sobolev solution
Moreover, the following estimate holds
Proof
The proof is based on the induction. Note that \(N_m=N_0^m\) with \(N_0>1\). For all \(m=1,2,\ldots\), we claim that there exists a sufficient small positive constant \(\varepsilon\) such that
For the case of \(m=1\), we recall the assumptions (2.4)–(2.6) on the initial approximation function \(U^{(0)}(x)\). By (2.50), taking \(0<\varepsilon <N_0^{-(8+k-\bar{k})}\varepsilon ^2\ll 1\), it follows that
Moreover, by (3.7) and the above estimate, we have
and
which means that \(U^{(1)}\in \mathcal B_{\varepsilon }\).
Assume that the case of \((m-1)\) holds, that is,
We now prove that the case of m holds. Using (2.2), (3.6) and the second inequality of (4.3), we obtain
Combining this estimate with (2.2), (3.7) and (4.1), we obtain
We choose a sufficiently small positive constant \(\varepsilon _0\) such that
Thus, by (4.5) we have
and
So, the error term goes to 0 as \(m\rightarrow \infty\), that is,
On the other hand, note that \(N_m=N_0^m\), by (4.3)–(4.4). It follows that
This means that \(U^{(m)}\in \mathcal B_{\varepsilon }\). Hence we conclude that (4.2) holds.
Therefore, the steady incompressible Navier–Stokes equations (1.1) with the Dirichlet boundary condition admit global solutions
from which, one can see the solution depends on the initial approximation function \(U^{(0)}(x)\) strongly.
Finally, by (1.3) and the standard Calderon-Zygmund theory (that is, for the Riesz operator \(\mathcal R\)), we have \(\Vert \mathcal RU\Vert _{\mathbb L^{s_0}}\le \Vert U\Vert _{\mathbb L^{s_0}}\) with \(1<s_0<\infty\). We conclude that
and this completes the proof. \(\square\)
References
Alinhac, S.: Existence d’ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels. Commun. Partial Differ. Equ. 14(2), 173–230 (1989)
Al Baba, H.: Maximal \(L^p-L^q\) regularity to the Stokes problem with Navier boundary conditions. Adv. Nonlinear Anal. 8, 743–761 (2019)
Beirao da Veiga, H., Yang, J.Q.: Regularity criteria for Navier–Stokes equations with slip boundary conditions on non-flat boundaries via two velocity components. Adv. Nonlinear Anal. 9, 633–643 (2020)
Brandolese, L.: Fine properties of self-similar solutions of the Navier–Stokes equations. Arch. Ration. Mech. Anal. 192, 375–401 (2009)
Buckmaster, T., Vicol, V.: Nonuniqueness of weak solutions to the Navier–Stokes equation. Ann. Math. 189, 101–144 (2019)
Caffarelli, L., Kohn, R.V., Nirenberg, L.: Partial regularity of suitable weak solutions of the Navier–Stokes equations. Commun. Pure Appl. Math. 35, 771–831 (1982)
Farwig, R., Sohr, H.: Existence, uniqueness and regularity of stationary solutions to inhomogeneous Navier–Stokes equations in \({\mathbb{R}}^n\). Czechoslov. Math. J. 59, 61–79 (2009)
Fefferman, C.L.: Existence and smoothness of the Navier–Stokes equations. Millenn. Prize Probl. 57, 67 (2006)
Foiaş, C., Manley, O., Rosa, R., Temam, R.: Navier–Stokes Equations and Turbulence. Encyclopedia of Mathematics and Its Applications, vol. 83. Cambridge University Press, Cambridge (2001)
Frehse, J., Ruzicka, M.: On the regularity of the stationary Navier–Stokes equations. Ann. Sc. Norm. Super. Pisa Cl. Sci. 21, 63–95 (1994)
Frehse, J., Ruzicka, M.: Existence of regular solutions to the stationary Navier–Stokes equations. Math. Ann. 302, 699–717 (1995)
Frehse, J., Ruzicka, M.: Regular solutions to the steady Navier–Stokes equations. In: Sequeira, A. (ed.) Navier–Stokes Equations and Related Nonlinear Problems (Funchal, 1994), pp. 131–139. Plenum, New York (1995)
Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations II. Nonlinear Steady Problems. Springer Tracts in Natural Philosophy, vol. 39. Springer, New York (1994)
Galdi, G.P., Simader, C.G., Sohr, H.: A class of solutions to stationary stokes and Navier–Stokes equations with boundary data in \(W^{-{1\over q}, q}\). Math. Ann. 331, 41–74 (2005)
Gerhardt, C.: Stationary solutions to the Navier–Stokes equations in dimension four. Math. Z. 165, 193–197 (1979)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Grundlehren der Mathematischen Wissenschaften, vol. 224, 2nd edn. Springer, Berlin (1983)
Hörmander, L.: Implicit Function Theorems. Stanford Lecture Notes, University of Stanford (1977)
Hou, Y., Pei, S.: On the weak solutions to steady Navier–Stokes equations with mixed boundary conditions. Math. Z. 291, 47–54 (2019)
Jia, H., Šverák, V.: Local in space estimates near initial time for weak solutions of the Navier–Stokes equations and forward self-similar solutions. Invent. Math. 196, 233–265 (2014)
Kim, H.: Existence and regularity of very weak solutions of the stationary Navier–Stokes equations. Arch. Ration. Mech. Anal. 193, 117–152 (2009)
Koch, H., Tataru, D.: Well-posedness for the Navier–Stokes equations. Adv. Math. 157, 22–35 (2001)
Leray, J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934)
Lin, F.: A new proof of the Caffarelli–Kohn–Nirenberg theorem. Commun. Pure Appl. Math. 51, 241–257 (1998)
Luo, X.: Stationary solutions and nonuniqueness of weak solutions for the Navier–Stokes equations in high dimensions. Arch. Ration. Mech. Anal. 233, 701–747 (2019)
Maz’ya, V., Rossmann, J.: Mixed boundary value problems for the stationary Navier–Stokes system in polyhedral domains. Arch. Ration. Mech. Anal. 194, 669–712 (2009)
Moser, J.: A rapidly converging iteration method and nonlinear partial differential equations I-II. Ann. Scuola Norm. Sup. Pisa. 20, 265–313 (1966)
Nash, J.: The embedding for Riemannian manifolds. Am. J. Math. 63, 20–63 (1956)
Necas, J., Ruzicka, M., Šverák, V.: On Leray’s self-similar solutions of the Navier–Stokes equations. Acta Math. 176, 283–294 (1996)
Rabinowitz, P.H.: A rapid convergence method for a singular perturbation problem. Ann. Inst. H. Poincaré, Anal. Non Linéaire 1, 1–17 (1984)
Safonov, M.V.: Nonlinear Elliptic Equations of the Second Order. Lecture notes, University of Florence, Italy (1991)
Struwe, M.: Regular solutions of the stationary Navier–Stokes equations on \({\mathbb{R}}^5\). Math. Ann. 302, 719–741 (1995)
Temam, R.: Navier-Stokes Equations and Nonlinear Functional Analysis. In: CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 66, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1995)
Temam, R.: Navier–Stokes Equations, Theory and Numerical Analysis. AMS Chelsea Publishing, Providence (2001)
Wang, Y.Z., Jie, X.: Well/ill-posedness for the dissipative Navier–Stokes system in generalized Carleson measure spaces. Adv. Nonlinear Anal. 8, 203–224 (2019)
Yan, W.P.: The motion of closed hypersurfaces in the central force field. J. Differ. Equ. 261, 1973–2005 (2016)
Yan, W.P.: Dynamical behavior near explicit self-similar blow up solutions for the Born–Infeld equation. Nonlinearity 32, 4682–4712 (2019)
Yan, W.P., Zhang, B.L.: Long time existence of solution for the bosonic membrane in the light cone gauge. J. Geom. Anal. (2019). https://doi.org/10.1007/s12220-019-00269-1
Zhao, X., Yan, W.P.: Existence of standing waves for quasi-linear Schrödinger equations on \({\mathbb{T}}^n\). Adv. Nonlinear Anal. 9, 978–993 (2020)
Acknowledgements
W. Yan is supported by the National Natural Science Foundation of China (No. 11771359). V.D. Rădulescu acknowledges the support of the Slovenian Research Agency Grants P1-0292, J1-8131, J1-7025, N1-0064, and N1-0083.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This paper is dedicated to the memory of Professor Ciprian Foiaş (1933–2020), a Master of Navier–Stokes equations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Yan, W., Rădulescu, V.D. Sobolev regular solutions for the incompressible Navier–Stokes equations in higher dimensions: asymptotics and representation formulae. Rend. Circ. Mat. Palermo, II. Ser 70, 995–1021 (2021). https://doi.org/10.1007/s12215-020-00540-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12215-020-00540-3