The Euler equations in planar nonsmooth convex domains
As a model problem for the barotropic mode of the primitive equations of the oceans and atmosphere, we consider the Euler system on a bounded convex planar domain , endowed with non-penetrating boundary conditions. For , and initial and forcing data with vorticity we show the existence of a weak solution, enriching and extending the results of Taylor .
In the physical case of a rectangular domain , a similar result holds for all as well. Moreover, by means of a new -type regularity estimate for the Dirichlet problem on a planar domain with corners, we prove uniqueness of solutions with bounded initial vorticity.
Key words and phrases:Euler system, nonsmooth domains, endpoint elliptic regularity
1991 Mathematics Subject Classification:Primary: 35Q31; Secondary: 35J57.
In print on
Journal of Mathematical Analysis and Applications, Volume 407, Issue 1, Nov 2013, Pages 69 to 89
Received 1 Dec 2012, available online 9 May 2013.
Submitted by Pierre Lemarie-Rieusset
Let be a bounded open set. We are concerned with the Euler equations, describing the motion of a perfect inviscid fluid inside ,
where stands for the velocity of the fluid particle located at at time , stands for the pressure gradient, and is an external forcing term. We endow (P) with initial and boundary conditions
we remark that the impermeability boundary condition (1.2) has to be properly reformulated when is not regular enough to admit a normal vector almost everywhere.
This work is primarily motivated by the study of the well-posedness of the barotropic mode of the inviscid primitive equations of the atmosphere and the oceans . As explained in [9, 10, 28], a certain vertical modal expansion of the primitive equations leads to an infinite system of coupled barotropic - baroclinic modes. In a first approximation, one can neglect the baroclinic modes and we obtain for the barotropic mode a system of equations very similar to the two-dimensional inviscid Euler equations in a rectangle. Henceforth this system is called the barotropic system. The study of the well-posedness of the barotropic system is thus very similar to the study of the well-posedness of the (inviscid) incompressible Euler equations in a rectangle In this article we consider the particular case of exactly the Euler system; the more general case will be considered elsewhere.
The study of the well-posedness of the Euler equations of incompressible fluid has a long history regarding both weak and strong solutions, starting with [25, 37], the latter article of Wolibner containing the first proof of global in time existence and uniqueness of regular solutions in two dimensions. This result was simplified and extended by Kato in ; when is sufficiently smooth, with some additional work, one obtains up to solutions, see e. g. [33, 34], or even analytic solutions .
The notion of weak solution to the two-dimensional Euler equations has been introduced by Yudovich . Yudovich, and later Bardos with a different (vanishing viscosity) approach  consider initial data with vorticity, and show existence of weak solutions and, in the case of bounded initial vorticity, uniqueness. Among many references on the well-posedness of the incompressible Euler equations in a bounded smooth domain, let us quote the classical articles [2, 5, 39] (see also [33, 34]). Other weaker notions of solutions (e. g. initial datum in , with no assumption on the initial vorticity) have been considered by several authors. See for instance [13, 29, 30] and Remark 6.1 below.
With the motivations indicated above, our aim in this article is twofold. Firstly, for general bounded convex domains , and for divergence-free initial datum with vorticity, we prove the existence of solutions of (P) in , in the range . Our proof follows the general scheme first devised in  (which considers the range ), developing several points not explicitly addressed therein: a weak solution is obtained as the limit of smooth solutions to (P) set on smooth convex subdomains increasing to . We remark that this approach has also been used in  to construct (possibly weaker notions of) Euler solutions in more general domains (complements of a finite number of compact connected sets with positive Sobolev capacity); however, for the class of domains under our consideration, our proof is simpler than the one in .
Proceeding as in , we make a more systematic study of the approximated problems P. In particular, we impose a uniform Lipschitz character to the domains . This uniformity reflects on the uniform boundedness of the Leray projectors associated to each subdomain, which we exploit in our compactness arguments. As a result, our solution is slightly more regular than the one constructed in , being continuous in time with values in and belonging to for each time , and satisfying (P) almost everywhere in . The range of exponents can also be dealt with, by working with a weaker notion of solution than the one given in Subsection 4.1.2, as for example in . However, the restriction corresponds to the sharp range of exponents for the regularity of the Biot-Savart law (see Theorem 2.2), giving the gradient of the velocity in terms of the vorticity, in a general convex domain, and seems unavoidable if one aims for a reversible existence result (in the sense that for belongs to the same space where the data is required to be, so that we can solve the backward Euler equations with initial data The proof of existence of solutions is the object of Proposition 4.1 and Theorem 5.1.
Our second aim in the article is to consider domains of specific interest to us, in particular the rectangle . In this case we are able, in Theorem 5.2, to extend the existence result of Theorem 5.1 to the range . Furthermore, and this is the main object of Theorem 5.2, we prove uniqueness of solutions in the case . To the best of our knowledge, this is the first uniqueness result for Euler solutions on a domain with corners without requiring additional assumptions on the initial vorticity other than being bounded. In , the author proves a uniqueness result for a class of domains somehow complementary to ours, that is domains with a finite number of corners having angles greater than , assuming that the initial vorticity is bounded and has definite signum. Our proof follows Yudovich’s energy method, and relies on the endpoint regularity result for the solution to the Dirichlet problem on a rectangle, Proposition 3.1, which appears to be new, to the best of our knowledge. We do not dwell on the latter point, but unbounded initial vorticities with -type blowup of the -norms as , like in , would also suffice for uniqueness.
Theorem 5.2 holds verbatim for a more general class of domains, that is bounded domains with piecewise smooth boundary and with corners of aperture of the form , for some integer We briefly discuss this extension in Subsection 6.3.
Plan of the paper.
In Section 2, we develop the necessary tools for the analysis of (P) on a bounded convex non-smooth domain . Section 3 contains the endpoint-type regularity result for the Dirichlet problem on a rectangle, which will be instrumental in establishing uniqueness of solutions. In Section 4 we give a weak formulation of (P), and construct a weak solution to (P) on a bounded convex domain (Proposition 4.1). Section 5 contains the statements and the proofs of the main results, Theorems 5.1 and 5.2. In Section 6, we make some additional remarks: in particular, we briefly outline, for comparison’s sake, the analogue of Theorem 5.2 in the space-periodic case We also discuss some extensions of Theorem 5.2 to a more general class of domains with corners.
Given a domain , and scalar functions , vector valued functions , we denote
Throughout, for , we use the notations
respectively for the Hölder and Sobolev conjugate exponents of .
We set up our notation for bump functions. Let and be a smooth nonnegative radial function supported in and with . We will make use of the -normalized bump functions
This work was supported in part by NSF Grants DMS 0906440, and DMS 1206438 and by the Research Fund of Indiana University. The authors acknowledge very useful discussions on the subject with Madalina Petcu and wish to thank Vlad Vicol for bringing to their attention very useful references. Furthermore, the authors are grateful to the anonymous referee for his/her valuable comments to the first draft of this article, leading to an improvement of the final presentation.
2. Elliptic regularity in a bounded convex domain
In this section, we set the foundation for our analysis of the Euler system (P). Throughout the section, is an open, bounded, convex subset of which contains the origin. In Section 3, we will specialize to the case of a rectangular domain and develop further elliptic regularity results.
We first recall the analytic properties of the boundary and construct an approximation of by an increasing sequence of convex smooth subdomains with uniformly Lipschitz boundary. Then, we describe the normal trace operator on , introduce the class of tangential vector fields, and establish the Helmholtz decomposition of , for . Finally, we discuss some regularity results for the Dirichlet problem in , which we exploit to define the spaces in which the evolution of the Euler system (P) will take place.
2.1. Regularity and approximation of bounded convex domains
We begin with a proposition.
Let be a bounded convex open set containing the origin. There exist positive constants and a finite collection of open squares of diameter such that:
whenever , there exists a function , in the coordinates with origin the center of and oriented along the sides of , with the properties
is convex and Lipschitz with constant ,
has the strong local Lipschitz property with constants .
It is known (see [20, Corollary 126.96.36.199] that bounded and convex implies that has a Lipschitz boundary (in the sense of [20, Definition 188.8.131.52]), which is exactly what is described in the first two assertions. The fact that is convex is a consequence of the fact that its epigraph is convex. Finally, the first two assertions imply the strong local Lipschitz property with constants as described in [1, IV.4.2]. ∎
2.1.1. Approximation by smooth convex subdomains
We construct a sequence of smooth convex domains increasing to , that is
and with the property that
We introduce the Minkowski functional of
The function is a convex function on (see [22, pp. 57-59]). A convex function on a compact subset of any normed space is globally Lipschitz (see  for a simple proof): thus, call the Lipschitz constant of . The function is convex and Lipschitz with the same constant , and
For let be the -neighborhood of . It is easy to verify that the mollification (see Section 1 for notation)
is smooth and convex, and moreover that the Lipschitz constants of are uniformly bounded by . Finally, we have that, for in the uniform Lipschitz norm, that is
Choose a subsequence with
it follows that is a convex open set, , and that . Moreover the are smooth, and uniformly Lipschitz (with respect to ). Thus each has the strong local Lipschitz property with Lipschitz constant uniformly bounded in . The construction (2.2) is thus completed.
Several important consequences of (2.1)-(2.2) will be derived in the next subsections. Here, we mention that (2.2) guarantees that the implicit constants appearing in the Sobolev embeddings and trace theorems on , which depend on the constants in the strong local Lipschitz property (see Theorem IV.4.1 and its proof in  for instance) are uniform in (they do depend on however, through and ).
2.2. Normal vector, normal traces, tangential vector fields
Maintaining the notation of Proposition 2.1, we observe that for each , is defined almost everywhere on and wherever defined. We can thus define the normal vector almost everywhere on by
This definition can be extended to all of by , and using a partition of unity subordinated to the covering of by the to a bounded vector field on all of .
is defined almost everywhere on . We are interested in the spaces
The classical trace theorem due to Gagliardo  yields the lemma below, arguing along the same lines of [35, Theorem I.1.2]. For a definition of Sobolev spaces (of fractional order) on Lipschitz submanifolds of , see Subsection 1.3.3 in . Note that if (that is, is restriction to of a function in ), then the restriction of to is a Lipschitz function on the Lipschitz submanifold .
Let be a bounded convex domain and . The map extends as a linear bounded map
and the generalized Stokes formula
holds for every and .
2.2.1. -tangential vector fields
We say that is a tangential divergence free vector field if
As a consequence of Lemma 2.1, it follows that
2.3. Helmholtz decomposition
Let . It is well known (see for example Theorem I.1.4 in ) that for Lipschitz
Let us denote by the corresponding orthogonal projector. The following result, due to Geng and Shen  allows us to obtain (2.6) for all and extend boundedly to . Note that Theorem I.1.4 in  (i.e. the case of Theorem 2.1 below) holds whenever is Lipschitz; actually, the range is known to be sharp for general Lipschitz domains .
Let be a bounded convex domain and . The following hold:
extends to a bounded linear operator
with operator norm only depending on . Moreover, for each , there exists , unique up to an additive constant such that
The second and third statements appear almost verbatim in [17, Theorem 1.3]. Let us show how they imply (2.7); denote by the space on the right hand side of (2.7). The backward inclusion is easy (see the proof of [35, Theorem I.1.4] for example). To get equality, we begin by characterizing the annihilator of in as
Proof of (2.9).
It is known (see  for a simple proof) that
Now from the Riesz representation theorem, for each there exists a (possibly nonunique) such that for all . From the above characterization, it follows that The Deny-Lions characterization (see for example [35, Proposition I.1.1]) gives that ; since , the Poincaré inequality finally yields . ∎
2.4. The Dirichlet problem and the Biot-Savart law
We recall the following consequence of the Lax-Milgram lemma.
Let be given. Then there exists a unique satisfying
Referring to (2.10), we use the notation . The classical theory of elliptic equations (see for example ) tells us that, when is a bounded smooth domain, and , has two derivatives in for any . This is no longer true in general if the domain is merely bounded and convex. However, the above result still holds in the range : we state this precisely in the theorem below, due to Fromm .
Let be a bounded convex domain. Let and be given. Then, there exists a positive constant , depending only on and on the Lipschitz character of , such that
Let be the approximating sequence to constructed in (2.1). In this context, for a function , we denote by its extension by zero to . Note that
We state and prove a so-called -convergence type result. The restrictions can be lifted, but we only need the particular case contained in the lemma below.
Let and set Then
We first show that weakly in . By density of in it suffices to show that
Fix and choose such that . Then for and
and similarly . Thus
and the weak convergence follows. Moreover, we also see that
which allows the upgrade from weak to strong convergence in of to , thus finishing the proof of the lemma. ∎
We now make the connection between the Euler system and more explicit. Set
We have that
The next lemma shows that is the left inverse of on , .
Let and . Then
By density, it suffices to show that
Let now . We have
and the last term on the right hand side is zero. We integrated by parts in the last equality, which is legitimate since . ∎
2.5. The spaces
The evolution of our solution to the Euler system (P) will take place in the space of tangential vector-fields with vorticity. That is, we define, for ,
The embedding discussed above allows for an improvement of the boundary regularity of functions in , by further applying Gagliardo’s trace theorem to the components of . More precisely,
In view of (2.21), we can therefore make sense of as whenever .
3. Elliptic regularity in a rectangle
We assume throughout this section that . Hereafter, we develop an appropriate substitute of Theorem 2.2 in the range , with spaces of functions with bounded mean oscillation replacing .
3.1. Local bmo spaces
We denote by the strict subspace of normed by
where the suprema above are taken over squares , denotes the area of , and denotes the average of on the cube . See for example  for more details.
Let be a domain. For a function let be its extension to zero outside , i.e. . We define the Banach spaces
i.e. the space of functions on whose trivial extension is in , and
i.e. the space of functions on which are restrictions to of functions in . The continuous embeddings are immediate to verify; this, together with John-Nirenberg’s inequality
where the constant is only dependent on , hints at the relevance of and as a substitute for . For , we use the notation
The next theorem, which we quote from , tells us that the solution to the Dirichlet problem is in whenever .
Let be either a bounded domain of class or the halfspace . Then there exists a constant , depending only on , such that
3.2. Theorem 3.1 in a rectangle
We prove the following proposition, which is a (partial) extension of Theorem 3.1 to the (nonsmooth, convex) domain .
Let . There exists such that
In this proof, the almost inequality sign is hiding a positive constant, possibly varying from line to line and depending on only. We also refer to Remark 6.3 for notation.
Denote by . We preliminarily observe that, by Grisvard’s classical estimate for domains with corners of aperture less than or equal to (which we recall in Section 6, see (6.3)) applied for , and John-Nirenberg’s inequality (3.2)
By the Sobolev embedding , we have in particular that
The core of the argument begins now. Let be an open cover of as follows: each is an open ball centered at the corner , of radius chosen small enough so that when ; is an open set with smooth boundary and For , set . Let be a partition of unity subordinated to the cover , and write . Then solves the Dirichlet problem
We gather from (3.5) that
Since is a smooth domain, we apply Theorem 3.1 with , and get
We now deal with the cases and show that
which will be enough to obtain (3.4). The four corners are treated in the exact same way, so we fix , and . Note that solves the Dirichlet problem
Consider the odd reflections of and , defined on by:
One sees that is the solution to the Dirichlet problem on with data and zero boundary condition, i.e. . We note that indeed, this is the same as ; recall that the bar denotes extension by zero to . This is clear, since Therefore, we can apply Theorem 3.1 in the case of , and deduce that
Since restricted to coincides with , and is supported in , we have
3.3. The space
In view of the elliptic regularity result (3.4), for it makes sense to extend the scale of spaces to the endpoint by setting