Citation: Yasir Nadeem Anjam. Singularities and regularity of stationary Stokes and Navier-Stokes equations on polygonal domains and their treatments[J]. AIMS Mathematics, 2020, 5(1): 440-466. doi: 10.3934/math.2020030
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In this paper, let J=(0−II0) and N=(−I00I) with I being the identity matrix on Rn. Denote by Ls(R2n) the space of all symmetric matrices in R2n and Sp(2n) the symplectic group of 2n×2n matrices.
We call B∈C(S1,Ls(R2n)) satisfies condition (BS1) if B(−t)=NB(t)N for all t∈R and B is 12-periodic, where S1=R/Z.
Let H∈C1(R×R2n,R) and H′ denote the gradient of H with respect to the last 2n variables. We assume H satisfies the following conditions:
(H1)H′(t,x)=B0(t)x+o(|x|) as |x|→0 uniformly in t,
(H2)H′(t,x)=B∞x+o(|x|) as |x|→∞ uniformly in t,
(H3) H(−t,Nx)=H(t,x)=H(t,−x)=H(t+12,−x), ∀(t,x)∈(R×R2n).
In [1] of 2008, the second author of this paper considered the multiplicity of 1-periodic brake orbits of asymptotically linear symmetric reversible Hamiltonian systems. Since condition (H3) holds, it is natural to consider 1-periodic solution of the asymptotically linear Hamiltonian systems
˙x=JH′(t,x), | (1.1) |
x(t+1)=x(t),x(12+t)=Nx(12−t),x(12+t)=−x(t),∀t∈R. | (1.2) |
We call the above 1-periodic solutions symmetric brake orbits. If H(−t,Nx)=H(t,x) for all (t,x)∈(R×R2n), we say H is reversible.
Note that conditions (H1)-(H3) yield that both B0 and B∞ belong to C(S1,Ls(R2n)) and satisfy the above (BS1) condition.
In 1948, Seifert firstly studied brake orbits in Hamiltonian system in [2]. For the existences and multiple existence results and more details on brake orbits one can refer the paper [3,4] and the references therein. In [1] the second author of this paper obtained multiple existence of brake orbits of the asymptotically linear Hamiltonian systems (1.1)-(1.2) under certain conditions. In this paper we will study multiplicity of symmetric brake orbits of Hamiltonian systems (1.1)-(1.2).
In [5], the difference between B0 and B∞, i.e., the different behaviors of H at zero and infinity plays an important role in the study of 1-periodic solutions of (1.1). For this reason we also define the relative Morse index to measure the "true" difference between B0 and B∞ under symmetric brake orbit boundary value. We shall study the relation between the relative Morse index and the Maslov-type index iL0√−1 and νL0√−1 defined below. As application, we obtain a multiplicity of symmetric brake orbits (1.1)-(1.2).
For any symplectic path γ in Sp(2n), the Maslov-type index for symmetric brake orbits boundary values of γ is defined in [6] to be a pair of integers (iL0(γ),νL0(γ))∈Z×{0,1,2,...,n}.
For any continuous path B in Ls(R2n) satisfying condition (BS1), as in [6], we define
(iL0√−1(B),νL0√−1(B))=(iL0√−1(γB,[0,14]),νL0√−1(γB(14))), | (1.3) |
where the symplectic path γB is the fundamental solution of the following linear Hamiltonian system
ddtγB(t)=JB(t)γB(t)andγB(0)=I2n. | (1.4) |
We will briefly introduce such Malov-type index theory in Section 2.
In order to consider the multiplicity of symmetric brake orbits, define
˜E={x∈W1/2,2(S1,R2n)∣x(−t)=Nx(t),x(t+12)=−x(t)a.e.t∈R}. | (1.5) |
Equip ˜E with the usual W1/2,2 norm. Then ˜E is a Hilbert space with the associated inner product ⟨,⟩. We define two self adjoint liner operators ˜A, ˜B from ˜E to ˜E by
⟨˜Ax,y⟩=∫10(−J˙x,y)dt,⟨˜Bx,y⟩=∫10(B(t)x,y)dt,∀x,y∈˜E, | (1.6) |
where B is a continuous path in Ls(R2n) satisfying condition (BS1).
As in [5], we denote by M+(⋅), M−(⋅), and M0(⋅) the positive definite, negative definite, and null subspaces of the self adjoint linear operator defining it respectively.
Definition 1.1. Let B1 and B2 be continuous paths in Ls(R2n) satisfying condition (BS1). We define the relative Morse index of ˜B1 and ˜B2 by
I(˜B1,˜B2)=dim(M+(˜A−˜B1)∩M−(˜A−˜B2))−dim((M−(˜A−˜B1)⊕M0(˜A−˜B1))⋂(M+(˜A−˜B2)⊕M0˜A−˜B2)), | (1.7) |
where we also denote by ˜B1 and ˜B2 the operators defined by (1.6) respectively. We call I(˜B1,˜B2) the relative Morse index, following [5]. By Theorem 1.2 below this relative More index is well defined. Note that such definition of relative Morse index is different from those defined in [7], [8], and [9] etc, which is difference of Morse index or spectral flow or definition from Garlerkin approximation, the definition if dimker(A−B1).
Using the iteration theory of Maslov-type index theory and result in [1], we obtain the relation between the Maslov-type index (iL0√−1, νL0√−1) and the relative Morse index as the following
Theorem 1.1. Let B1 and B2 be be continuous paths in Ls(R2n) satisfying condition (BS1). We have
I(˜B1,˜B2)=iL0√−1(B2)−iL0√−1(B1)−νL0√−1(B1). | (1.8) |
As application, we obtain the main result of this paper.
Theorem 1.2. Suppose that H satisfies (H1), (H2), (H3), and νL0√−1(B0)=νL0√−1(B∞)=0. Then (1.1)-(1.2) has at least |iL0√−1(B1)−iL0√−1(B∞)| pairs of nontrivial 1-periodic symmetric brake orbits.
Organization: In Section 2, we will briefly introduce the Maslov-type index and its iteration theory for symplectic path under brake orbit boundary value. Based on this index theory we give he proof of Theorem 1.1. As application, in Section 3, we give he proof of Theorem 1.2.
Throughout this paper, let N, Z, R, C and U denote the set of natural integers, integers, rational numbers, real numbers, complex numbers and the unit circle in C, respectively.
In this section we will prove Theorem 1.1. We first simply recall the Maslov-type index and its iteration theory for brake orbits.
As we know, in 1984, Conley and Zehnder in their celebrated paper [10] introduced an index theory for the non-degenerate symplectic paths in the real symplectic matrix group Sp(2n) for n≥2. Since then, there are tremendous works about this kind of index theory developed or generalized in various directions. In 2006, combined with the Maslov index formulated in [11], Long, Zhang and Zhu developed an index theory called μ-index in [12] and obtained an result on the existence of multiple brake orbits. The difference of the Maslove-type μ-index in [12] and the Maslov-type L-index in that paper is constant n (half of the dimension). In [13], Liu and Zhang established the Bott-type iteration formulas and some precise iteration formula of the L-index theory and proved the multiplicity of brake orbits on every C2 compact convex symmetric hypersurface in R2n.
Set
PT(2n)={γ∈C([0,T],Sp(2n))∣γ(0)=I2n}, |
where we omit T from the notation of PT if [0,T] is replaced by [0,+∞). Let J be the standard almost complex in (R2n,ω0) and J is a compatible with ω0, i.e.,
ω0(x,y)=Jx⋅y,ω0(Jx,Jy)=ω0(x,y)andω0(x,Jx)>0forx≠0. |
A n-dimensional subspace Λ⊆R2n is called a Lagrangian subspace if ω0(x,y)=0, for any x,y∈Λ. Let F=R2n⨁R2n be equipped with symplectic form (−ω0)⨁ω0. Then J=(−J)⨁J is an almost complex structure on F and J is compatible with (−ω0)⨁ω0. Denote by Lag(F) the set of Lagrangian subspaces of F. Then for any M∈Sp(2n), its graph
Gr(M)={(xMx)|x∈R2n}∈Lag(F). |
Denote by L0={0}×Rn and L1=Rn×{0} the two fixed Lagrangian subspaces of R2n and let
V0=L0×L0, V1=L1×L1,Gr(M)|Vj={(xMy)|x,y∈Lj}. | (2.1) |
Then both Vj,Gr(M)|Vj∈Lag(F) for M∈Sp(2n) and j=0,1.
Denote by μCLMF(V,W,[a,b]) the Maslov-type index for (ordered) pair of paths of Lagrangian subspaces (V,W) in F on [a,b], which is defined by Cappel, Lee and Miller in [11].
Definition 2.1. (cf. [6,12,13]) For γ∈Pτ(2n), define
iLjω(γ)={μCLMF(Gr(eθJ)|Vj,Gr(γ(t)),t∈[0,τ]),ω=e√−1θ∈U∖{1}, μCLMF(Vj,Gr(γ(t)),t∈[0,τ])−n,ω=1, | (2.2) |
νLjω(γ)=νLjω(γ(τ))=dimC(γ(τ)Lj∩eθJLj), ω=e√−1θ∈U. | (2.3) |
For j=0,1, we define (iLj(γ),νLj(γ))=(iLjω(γ),νLjω(γ)) if ω=1. Note that, for any continuous path Ψ∈Pτ, the following Maslov-type indices of Ψ is defined by (cf [1,12])
μ1(Ψ,[a,b])=μCLMF(V0,Gr(Ψ),[0,τ]),ν(Ψ,[0,τ])=dimΨ(τ)L0∩L0. | (2.4) |
When there is no confusion we will omit the intervals in the above definitions. Hence we have
iL0(γ)=μ1(γ)−n,νL0(γ)=ν1(γ), | (2.5) |
For B∈C(ST,Ls(R2n)), the fundamental solution γB of the linear Hamiltonian system
{˙γ(t)=JB(t)γ(t),γ(0)=I2n. | (2.6) |
satisfies γB∈PT(2n), and is called the associated symplectic path of B. For ω∈U, we define the Maslov-type indices of B via the restriction γB|[0,T/2]∈PT/2(2n) :
(iLjω(B,T2),νLjω(B,T2)):=(iLjω(γB|[0,T/2]),νLjω(γB(T2))). |
In 1956, Bott in [14] established the famous iteration formula of the Morse index for closed geodesics on Riemannian manifolds. For convex Hamiltonian systems, Ekeland developed the similar Bott-type iteration index formulas for the Ekeland index theory (cf. [15] of 1990). In 1999 (cf. [16]), Long established the Bott-type iteration formulas for the Maslov-type index theory. Motivated by the above results, in [13] of Liu and Zhang in 2014, the following Bott-type iteration formulas for the L0-index was established.
Definition 2.2. (cf. [13]) Given an τ>0, a positive integer k and a path γ∈Pτ(2n), the k-th iteration γk of γ in brake orbit boundary sense is defined by ˜γ|[0,kτ] with
˜γ(t)={γ(t−2jτ)(γ(2τ))j,t∈[2jτ,(2j+1)τ],j∈N∪{0}, Nγ((2j+2)τ−t)N(γ(2τ))j+1,t∈[(2j+1)τ,(2j+2)τ],j∈N∪{0}, |
where γ(2τ):=Nγ(τ)−1Nγ(τ). $
Theorem 2.1. (cf. [13] of Liu and Zhang in 2014) Suppose γ∈Pτ(2n), for the iteration symplectic paths γk, when k is odd, there hold
iL0(γk)=iL0(γ1)+k−12∑i=1iω2ik(γ2),νL0(γk)=νL0(γ1)+k−12∑i=1νω2ik(γ2); |
when k is even, there hold
iL0(γk)=iL0(γ1)+iL0√−1(γ1)+k2−1∑i=1iω2ik(γ2),νL0(γk)=νL0(γ1)+νL0√−1(γ1)+k2−1∑i=1νω2ik(γ2), |
where ωk=eπ√−1/k, and (iω,νω) is the ω-index pair defined by Long(cf. [16]).
Proof of Theorem 1.1. For any B∈Ls(R2n) satisfying condition (BS1), as in [1], we define
E={x∈W1/2,2(S1,R2n)∣x(−t)=Nx(t),a.e.t∈R}. | (2.7) |
Equip E with the usual W1/2,2 norm. Then E is a Hilbert space with the associated inner product ⟨,⟩. We define two self adjoint liner operators A, B from E to E by
⟨Ax,y⟩=∫10(−J˙x,y)dt,⟨Bx,y⟩=∫10(B(t)x,y)dt,∀x,y∈E. | (2.8) |
We also define
ˆE={x∈W1/2,2(S1,R2n)∣x(−t)=Nx(t),x(t+12)=x(t),a.e.t∈R}. | (2.9) |
Equip ˆE with the usual W1/2,2 norm. We define two self adjoint liner operators ˆA, ˆB from ˆE to ˆE by
⟨ˆAx,y⟩=∫10(−J˙x,y)dt,⟨ˆBx,y⟩=∫10(B(t)x,y)dt,∀x,y∈ˆE. | (2.10) |
Then ˜E and ˆE are both subspaces of E and A invariant, we have both the A orthogonal and B orthogonal decomposition
E=˜E⊕ˆE,˜A=A|˜E,ˆA=A|ˆE. |
Since B satisfies condition (BS1), one can verify the following orthogonal decomposition
B=˜B⊕ˆB,˜B=B|˜E,ˆB=B|ˆE. |
Then we have the orthogonal decomposition
M∗(A−B)=M∗(˜A−˜B)⊕M∗(ˆA−^B),for∗=±,0, |
where M∗(A−B)⊂E, M∗(˜A−˜B)⊂˜E, M∗(ˆA−ˆB)⊂ˆE. So by the definitions of I(B1,B2), I(˜B1,˜B2), I(ˆB1,ˆB2) we have
I(B1,B2)=I(˜B1,˜B2)+I(ˆB1,ˆB2), | (2.11) |
where I(B1,B2) and I(ˆB1,ˆB2) are defined similarly as (1.3).
Since B satisfies condition (BS1), one has B∈C(S1/2,Ls(R2n)). Thus
⟨ˆAx,y⟩=2∫120(−J˙x,y)dt,⟨ˆBx,y⟩=2∫120(B(t)x,y)dt,∀x,y∈ˆE. |
So by Theorem 1.2 of [1] and (2.8) we have
I(ˆB1,ˆB2)=iL0(γB2(t),[0,14])−iL0(γB1(t),[0,14])−ν1(γB1(14)) | (2.12) |
Also by Theorem 1.2 of [1] and (2.8) we have
I(B1,B2)=iL0(γB2(t),[0,12])−iL0(γB1(t),[0,12])−ν1(γB1(12)) | (2.13) |
Since both B1 and B2 satisfy condition (BS1), for j=1,2, by Theorem 2.3 we have
iL0(γBj(t),[0,12])=iL0(γBj(t),[0,14])+iL0√−1(γBj(t),[0,14]), | (2.14) |
νL0(γBj(t),[0,12])=νL0(γBj(t),[0,14])+νL0√−1(γBj(t),[0,14]). | (2.15) |
By (2.11)–(2.15) one has
I(˜B1,˜B2)=iL0√−1(γB2(t),[0,14])−iL0√−1(γB2(t),[0,14])−νL0√−1(γB2(t),[0,14]). | (2.16) |
Thus Theorem 1.1 holds by the definitions of (iL0√−1(γB),νL0√−1(γB)) in (1.3).
In this section we prove Theorem 1, 2.
We study the 1-periodic brake orbit solution of Hamiltonian system (1.1)-(1.2)
˙x=JH′(t,x),x(t+1)=x(t),x(12+t)=Nx(12−t). |
It is well know that x is a solution of (1.1)-(1.2) if and only if it is a critical point of the functional f defined on ˜E as follows
f(x)=12⟨˜Ax,x⟩+˜Φ(x),x∈˜E, | (3.1) |
where ˜E is defined by (1.5), ˜A is defined in (1.6), ˜Φ(x)=∫10−H(t,x)dt. It is easy to check that ˜Φ′(x) is compact.
In [17], Benci proved the following important abstract theorem:
Theorem 3.1. Let f∈C1(E,R) have the form (3.1) and satisfy
(f1) Every sequence {uj} such that f(uj)→c<˜Φ(0) and ||f′(uj)||→0 as j→+∞ is bounded.
(f2) ˜Φ(u)=˜Φ(−u), u∈˜E.
(f3) There are two closed subspaces of ˜E, E+ and E−, and a constant ρ>0 such that
(a) f(u)>0 for u∈E+, where c0<c∞<˜Φ(0) be two constants.
(b) f(u)<c∞<˜Φ(0) for u∈E−∩Sρ, (Sρ={u∈E|||u||=ρ}).
Then the number of pairs of nontrivial critical points of f is greater than or equalto dim(E+∩E−)−cod(E−+E+). More over, the corresponding critical values belong toto [c0,c∞].
Proof of Theorem 1.2. We take the method in [1,5] to prove this theorem.
We set ˜E+=M+(˜A−˜B∞) and ˜E−=M−(˜A−B0). By Definition 1.1 and Theorem 1.2, we have
dim(˜E+∩˜E−)−cod(˜E−+˜E+)=dim(M+(˜A−˜B∞)∩M0(˜A−˜B0))−dim((M−(˜A−˜B∞)⊕M−(˜A−˜B∞))∩(M+(˜A−˜B0)⊕M0(˜A−˜B0)))=I(˜B∞,˜B0)=iL0√−1(˜B0)−iL0√−1(˜B∞). | (3.2) |
Here ˜B0 and ˜B∞ are compact operators from ˜E to ˜E defined by (1.6). Since 0 is an isolated eigenvalue of ˜A with n-dimensional eigenspace ˜E0, by (4-4′) of [17], there exist two real numbers α<0 and β>0 such that
⟨˜A−˜B0u,u⟩≤α||u||2,∀u∈˜E−, | (3.3) |
⟨˜A−˜B∞u,u⟩≥β||u||2,∀u∈˜E+. | (3.4) |
Define
V∞(t,x)=H(t,x)−12⟨˜B∞(t)x,x⟩,V0(t,x)=H(t,x)−12⟨˜B0(t)x,x⟩, | (3.5) |
and let g∞(x)=∫10V∞(t,x)dt and g0(x)=∫10V0(t,x)dt, then we have
f(x)=12⟨(˜A−˜B∞)x,x⟩−g∞(x),∀x∈˜E, | (3.6) |
f(x)=12⟨(˜A−˜B0)x,x⟩−g0(x),∀x∈˜E. | (3.7) |
By (H1)-(H2) and the same arguments in the proof of Lemma 5.5 of [17], we get
lim||x||→+∞||g′∞(x)||||x||=0, | (3.8) |
lim||x||→+0||g′0(x)||||x||=0. | (3.9) |
So by definition of g0 and (3.9), we have
g0(u)=−˜Φ(0)+o(||u||2),for||u||→0. | (3.10) |
By (3.3) and (3.10) we have
f(u)≤α||u||2+˜Φ(0)+o(||u||2),foru∈E−and||u||→0. | (3.11) |
Since α<0, there exist a constant ρ>0 and γ1<0 such that
f(u)<γ1+˜Φ(0),∀u∈E−∩Sρ. | (3.12) |
Setting c∞=γ12+˜Φ(0), (f3)(b) of Theorem 3.1 is satisfied.
By (H2) for there exist M>0 such that
|V∞(t,x)|≤β2|x|2+M|x|,∀x∈R2n. | (3.13) |
Thus
|g∞(u)|=|∫10V∞(t,u)dt|≤∫10|V∞(t,u)|dt≤∫10β2|u|2+M|u|≤β2||u||2+M||u||. | (3.14) |
Then by (3.4) and (3.14), for every u∈˜E+, we get
f(u)=12⟨(˜A−˜B∞)u,u⟩+g∞(u)≥β||u||2−|g∞(u)|≥β2||u||2−M||u||. | (3.15) |
This implies that f is bounded from below on ˜E+ and we can set
c0=infu∈E+f(u)−wwithw>0suchthatc0<c∞. |
Thus (f3)(a) of Theorem 3.1 is satisfied.
Since ν1(˜B∞)=0, M0(˜A−˜B∞)=0. Now we prove that (f1) is satisfied. other wise we can suppose ||uj||→+∞ as j→+∞, then by (3.6) and (3.8) we have
0=limj→+∞f′(uj)=limj→+∞((˜A−˜B∞)uj+g′∞(uj))=limj→+∞(˜A−˜B∞)uj. | (3.16) |
But by (4-4′) of [17] there exists a real number α′>0 such that
||(˜A−˜B∞)u||≥α′||u||,∀u∈E. | (3.17) |
Hence by (3.17) we have
limj→+∞||(˜A−˜B∞)uj||=+∞, | (3.18) |
which contradicts (3.16). This proves (f1) in Theorem 3.1.
(H3) implies (f2) of Theorem 3.1 holds. Hence by Theorem 3.1, (1.1)-(1.2) has at least iL0√−1(˜B0)−iL0√−1(˜B∞) pairs of nontrivial solutions whenever iL0√−1(˜B0)−iL0√−1(˜B∞)>0. If iL0√−1(˜B0)−iL0√−1(˜B∞)>0, we replace f by −f and let E+=M−(˜A−˜B∞) and E−=M+(˜A−˜B0). By almost the same proof we can show that (f1)-(f3) of Theorem 3.1 hold. And by Theorems 1.2 and 3.1 (1.1)-(1.2) has at least iL0√−1(˜B∞)−iL0√−1(˜B0) pairs of nontrivial brake orbit solution. The proof of Theorem 1.3 is completed.
Similarly to Theorems 1.4 and 1.5 of [5] or [1]), we have
Remark 3.1. If νL0√−1(˜B∞)>0, we can prove (f1) of Theorem 3.1 under other additional conditions while we can prove (f2) and (f3) are satisfied under (H1)-(H3) by the same proof of Theorem 1.3.
Suppose the following condition:
(H4) V′∞(t,x) is bounded and V(t,x)→+∞ as |x|→+∞, uniformly in t.
By the proof of Theorem 5.2 of [17] and Theorem 4.1 of [18] (f1) holds.
Suppose the following conditions:
(H5) There is r>0 and p∈(1,2) such that
pV∞(t,x)≥(z,V′∞(t,x))>0for|z|≥r,t∈R. |
(H6) ¯lim|x|→∞|x|−1|V′∞(t,x)|≤c<12.
(H7) There are constant a1>0 and a2>0 such that V∞(t,x)≥a|z|p−a2.
By the proof of Theorem 4.11 of [18] (f1) holds.
Then under either additional condition (H4) or (H5)-(H7), (1.1)-(1.2) has at least iL0√−1(˜B0)−iL0√−1(˜B∞)−νL0√−1(˜B∞) pairs of nontrivial solutions whenever iL0√−1(˜B0)−iL0√−1(˜B∞)−νL0√−1(˜B∞)>0.
This work is partially supported by the NSFC Grants 11790271 and 11171341, National Key R & D Program of China 2020YFA0713301, and LPMC of Nankai University. The authors sincerely thanks the referees for their careful reading and valuable comments and suggestions.
The authors declare there is no conflict of interest.
[1] | R. A. Adams, J. J. Fournier, Sobolev Spaces, Academic Press, 2003. |
[2] |
S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Commun. Pur. Appl. Math., 12 (1959), 623-727. doi: 10.1002/cpa.3160120405
![]() |
[3] |
S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II, Commun. Pur. Appl. Math., 17 (1964), 35-92. doi: 10.1002/cpa.3160170104
![]() |
[4] |
M. Amara, D. C. Papaghiuc, E. Chaćon-Vera, et al. Vorticity-velocity-pressure formulation for Navier-Stokes equations, Comput. Visual. Sci., 6 (2004), 47-52. doi: 10.1007/s00791-003-0107-y
![]() |
[5] |
C. Amrouche, P. Penel, N. Seloula, Some remarks on the boundary conditions in the theory of Navier-Stokes equations, Anna. Math. Blais. Pasc., 20 (2013), 37-73. doi: 10.5802/ambp.321
![]() |
[6] |
I. Babuška, Finite element method for domains with corners, Computing, 6 (1970), 264-273. doi: 10.1007/BF02238811
![]() |
[7] | G. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, 1967. |
[8] |
J. M. Bernard, Time-dependent Stokes and Navier-Stokes problems with boundary conditions involving pressure, existence and regularity, Nonlinear Anal. Real., 4 (2003), 805-839. doi: 10.1016/S1468-1218(03)00016-6
![]() |
[9] |
L. Bers, Survey of local properties of solutions of elliptic partial differential equations, Commun. Pur. Appl. Math., 9 (1956), 339-350. doi: 10.1002/cpa.3160090306
![]() |
[10] |
H. Blum, M. Dobrowolski, On finite element methods for elliptic equations on domains with corners, Computing, 28 (1982), 53-63. doi: 10.1007/BF02237995
![]() |
[11] |
G. A. Brés, T. Colonius, Three-dimensional instabilities in compressible flow over open cavities, J. Fluid Mech., 599 (2008), 309-339. doi: 10.1017/S0022112007009925
![]() |
[12] |
Z. Cai, S. Kim, B. C. Shin, Solution methods for the Poisson equation with corner singularities: Numerical results, SIAM J. Sci. Comput., 23 (2001), 672-682. doi: 10.1137/S1064827500372778
![]() |
[13] | G. F. Carrier, M. Krook, C. E. Pearson, Functions of a Complex Variable: Theory and Technique, Cambridge University Press, 2005. |
[14] |
S. E. Chen, R. B. Kellogg, An interior discontinuity of a nonlinear elliptic-hyperbolic system, SIAM J. Math. Anal., 22 (1991), 602-622. doi: 10.1137/0522038
![]() |
[15] |
X. F. Chen, W. Q. Xie, Discontinuous solutions of steady state, viscous compressible NavierStokes equations, J. Differ. Equations, 115 (1995), 99-119. doi: 10.1006/jdeq.1995.1006
![]() |
[16] |
Y. Chen, T. Jiang, The pressure boundary conditions for the incompressible navier-stokes equations computation, Commun. Nonlinear. Sci., 1 (1996), 70-72. doi: 10.1016/S1007-5704(96)90042-8
![]() |
[17] | Y. Z. Chen, L. C. Wu, Second Order Elliptic Equations and Elliptic Systems, American Mathematical Society, 2004. |
[18] |
H. J. Choi, J. R. Kweon, For the stationary compressible viscous Navier-Stokes equations with no-slip condition on a convex polygon, J. Differ. Equation, 250 (2011), 2440-2461. doi: 10.1016/j.jde.2010.12.018
![]() |
[19] |
H. J. Choi, J. R. Kweon, The stationary Navier-Stokes system with no-slip boundary condition on polygons: Corner singularity and regularity, Commun. Part. Diff. Eq., 38 (2013), 1235-1255. doi: 10.1080/03605302.2012.752386
![]() |
[20] |
H. J. Choi, J. R. Kweon, A finite element method for singular solutions of the Navier-Stokes equations on a non-convex polygon, J. Comput. Appl. Math., 292 (2016), 342-362. doi: 10.1016/j.cam.2015.07.006
![]() |
[21] | N. Chorfi, Geometric singularities of the Stokes problem, Abstr. Appl. Anal., 2014 (2014), 1-8. |
[22] | P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Society for Industrial and Applied Mathematics, 2002. |
[23] |
M. G. Crandall, P. H. Rabinowitz, L. Tartar, On a dirichlet problem with a singular nonlinearity, Commun. Part. Diff. Eq., 2 (1977), 193-222. doi: 10.1080/03605307708820029
![]() |
[24] | D. G. Crowdy, S. J. Brzezicki, Analytical solutions for two-dimensional Stokes flow singularities in a no-slip wedge of arbitrary angle, Proc. R. Soc. A., 473 (2017), 20170134. |
[25] | D. G. Crowdy, A. M. J. Davis, Stokes flow singularities in a two-dimensional channel: A novel transform approach with application to microswimming, Proc. R. Soc. A., 469 (2013), 20130198. |
[26] | M. Dauge, Singularities along the edges, In: Elliptic Boundary Value Problems on Corner Domains, Berlin: Springer, 1988, 128-152. |
[27] |
M. Dauge, Stationary Stokes and Navier-Stokes systems on two or three-dimensional domains with corners. Part I. Linearized equations, SIAM J. Math. Anal., 20 (1989), 74-97. doi: 10.1137/0520006
![]() |
[28] |
M. Dauge, Singularities of corner problems and problems of corner singularities, ESAIM: Proc., 6 (1999), 19-40. doi: 10.1051/proc:1999044
![]() |
[29] | M. Dauge, Elliptic boundary value problems on corner domains: Smoothness and asymptotics of solutions, Springer, 2006. |
[30] |
W. R. Dean, P. E. Montagnon, On the steady motion of viscous liquid in a corner, Math. Proc. Cambridge, 45 (1949), 389-394. doi: 10.1017/S0305004100025019
![]() |
[31] |
F. Dubois, Vorticity-velocity-pressure formulation for the Stokes problem, Math. Meth. Appl. Sci., 25 (2002), 1091-1119. doi: 10.1002/mma.328
![]() |
[32] |
F. Dubois, M. Saläun, S. Salmon, Vorticity-velocity-pressure and stream function-vorticity formulations for the Stokes problem, J. Math. Pure. Appl., 82 (2003), 1395-1451. doi: 10.1016/j.matpur.2003.09.002
![]() |
[33] | M. Durand, Singularities in elliptic problems, In: Singularities and Constructive Methods for Their Treatment, Berlin: Springer, 1985, 104-112. |
[34] |
M. Elliotis, G. Georgiou, C. Xenophontos, The solution of Laplacian problems over L-shaped domains with a singular function boundary integral method, Commun. Num. Meth. Eng., 18 (2002), 213-222. doi: 10.1002/cnm.489
![]() |
[35] | L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 2010. |
[36] | M. Feistauer, Mathematical Methods in Fluid Dynamics, Chapman and Hall/CRC, 1993. |
[37] |
G. Georgiou, A. Boudouvis, A. Poullikkas, Comparison of two methods for the computation of singular solutions in elliptic problems, J. Comput. Appl. Maths., 79 (1997), 277-287. doi: 10.1016/S0377-0427(96)00173-2
![]() |
[38] | D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2015. |
[39] | V. Girault, P. A. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Springer Science and Business Media, 2012. |
[40] |
S. Gontara, H. Mâagli, S. Masmoudi, et al. Asymptotic behavior of positive solutions of a singular nonlinear Dirichlet problem, J. Math. Anal. Appl., 369 (2010), 719-729. doi: 10.1016/j.jmaa.2010.04.008
![]() |
[41] |
P. Grisvard, Edge behavior of the solution of an elliptic problem, Math. Nachr., 132 (1987), 281-299. doi: 10.1002/mana.19871320119
![]() |
[42] | P. Grisvard, Singularities in Boundary Value Problems, Springer, 1992. |
[43] | P. Grisvard, Singular behavior of elliptic problems in non-Hilbertian Sobolev spaces, J. Math. Pure. Appl., 74 (1995), 3-33. |
[44] | P. Grisvard, Behavior of the solutions of an elliptic boundary value problem in a polygonal or polyhedral domain, In: Numerical solution of partial differential equations-III, Elsevier, 1976, 207-274. |
[45] | P. Grisvard, Elliptic problems in nonsmooth domains, Pitman Advanced Pub, Program, Boston, 2 (1985), 2-2. |
[46] | W. Hackbusch, Elliptic Differential Equations Theory and Numerical Treatment: The Poisson Equation, Springer, 2010. |
[47] |
J. H. Han, J. R. Kweon, M. Park, Interior discontinuity for a stationary compressible Stokes system with inflow datum, Comput. Math. Appl., 74 (2017), 2321-2329. doi: 10.1016/j.camwa.2017.07.002
![]() |
[48] | J. Hernández, F. J. Mancebo, J. M. Vega, On the linearization of some singular, nonlinear elliptic problems and applications, Ann. I. H. Poincaré-AN, 19 (2002), 777-813. |
[49] | D. Hoff, Construction of solutions for compressible, isentropic Navier-Stokes equations in one space dimension with non-smooth initial data, P. Roy. Soc. Edinb. A., 103 (1986), 301-315. |
[50] | D. Hoff, Dynamics of singularity surfaces for compressible, viscous flows in two space dimensions, Commun. Pur. Appl. Math., 55 (2002), 1365-1407. |
[51] |
S. Itoh, N. Tanaka, A. Tani, On some boundary value problem for the stokes equations in an infinite sector, Anal. Appl., 4 (2006), 357-375. doi: 10.1142/S0219530506000826
![]() |
[52] | C. Johnson, Streamline diffusion finite element methods for incompressible and compressible fluid flow, In: Computational Fluid Dynamics and Reacting Gas Flows, Springer, 1988, 87-106. |
[53] | C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Dover Publications, 2009. |
[54] |
T. Jonsson, M. G. Larson, K. Larsson, Graded parametric CutFEM and CutIGA for elliptic boundary value problems in domains with corners, Comput. Meth. Appl. Mech. Eng., 354 (2019), 331-350. doi: 10.1016/j.cma.2019.05.024
![]() |
[55] | V. V. Katrakhov, S. V. Kiselevskaya, A singular elliptic boundary value problem in domains with corner points. I. Function spaces, Diff. Equat., 42 (2006), 395-403. |
[56] |
B. Kellogg, Some simple boundary value problems with corner singularities and boundary layers, Comput. Math. Appl., 51 (2006), 783-792. doi: 10.1016/j.camwa.2006.03.010
![]() |
[57] |
R. B. Kellogg, J. E. Osborn, A regularity result for the Stokes problem in a convex polygon, J. Funct. Anal., 21 (1976), 397-431. doi: 10.1016/0022-1236(76)90035-5
![]() |
[58] | R. B. Kellogg, Discontinuous solutions of the linearized, steady state, compressible, viscous, Navier-Stokes equations, SIAM J. Math. Anal., 19 (1988), 567-579. |
[59] | R. B. Kellogg, Corner singularities and singular perturbations, Ann. Univ. Ferrara, 47 (2001), 177-206. |
[60] | V. A. Kondratíev, Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Mos. Matem. Obsh., 16 (1967), 209-292. |
[61] | V. A. Kozlov, V. G. Maz'ya, J. Rossmann, Elliptic Boundary Value Problems in Domains with Point Singularities, American Mathematical Society, 1997. |
[62] | V. A. Kozlov, V. G. Maz'ya, J. Rossmann, Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations, American Mathematical Society, 2001. |
[63] | M. Kumar, G. Mishra, A review on nonlinear elliptic partial differential equations and approaches for solution, Int. J. Nonlin. Sci., 13 (2012), 401-418. |
[64] |
J. R. Kweon, A regularity result of solution to the compressible Stokes equations on a convex polygon, Z. Angew. Math. Phys., 55 (2004), 435-450. doi: 10.1007/s00033-003-2042-7
![]() |
[65] | J. R. Kweon, Singularities of a compressible Stokes system in a domain with concave edge in R3, J. Differ. Equations, 229 (2006), 24-48. |
[66] |
J. R. Kweon, Regularity of solutions for the Navier-Stokes system of incompressible flows on a polygon, J. Differ. Equations, 235 (2007), 166-198. doi: 10.1016/j.jde.2006.12.008
![]() |
[67] | J. R. Kweon, Edge singular behavior for the heat equation on polyhedral cylinders in R3, Potential Anal., 38 (2013), 589-610. |
[68] |
J. R. Kweon, Corner singularity dynamics and regularity of compressible viscous Navier-Stokes flows, SIAM J. Math. Anal., 44 (2012), 3127-3161. doi: 10.1137/120867937
![]() |
[69] |
J. R. Kweon, A jump discontinuity of compressible viscous flows grazing a nonconvex corner, J. Math. Pure. Appl., 100 (2013), 410-432. doi: 10.1016/j.matpur.2013.01.007
![]() |
[70] |
J. R. Kweon, Jump dynamics due to jump datum of compressible viscous NavierStokes flows in a bounded plane domain, J. Differ. Equations, 261 (2016), 3463-3492. doi: 10.1016/j.jde.2016.05.031
![]() |
[71] |
J. R. Kweon, The compressible Stokes flows with no-slip boundary condition on non-convex polygons, J. Math. Fluid Mech., 19 (2017), 47-57. doi: 10.1007/s00021-016-0264-7
![]() |
[72] |
J. R. Kweon, R. B. Kellogg, Compressible Navier-Stokes equations in a bounded domain with inflow boundary condition, SIAM J. Math. Anal., 28 (1997), 94-108. doi: 10.1137/S0036141095284254
![]() |
[73] |
J. R. Kweon, R. B. Kellogg, Compressible Stokes problem on nonconvex polygonal domains, J. Differ. Equations, 176 (2001), 290-314. doi: 10.1006/jdeq.2000.3964
![]() |
[74] |
J. R. Kweon, R. B. Kellogg, Regularity of solutions to the Navier-Stokes equations for compressible barotropic flows on a polygon, Arch. Ration. Mech. Anal., 163 (2002), 35-64. doi: 10.1007/s002050200191
![]() |
[75] |
J. R. Kweon, R. B. Kellogg, The pressure singularity for compressible Stokes flows in a concave polygon, J. Math. Fluid Mech., 11 (2009), 1-21. doi: 10.1007/s00021-007-0245-y
![]() |
[76] |
J. R. Kweon, M. Song, A discontinuous solution for an evolution compressible Stokes system in a bounded domain, J. Differ. Equations, 219 (2005), 202-220. doi: 10.1016/j.jde.2004.10.001
![]() |
[77] |
O. S. Kwon, J. R. Kweon, For the vorticity-velocity-pressure form of the Navier-Stokes equations on a bounded plane domain with corners, Nonlinear. Anal. Theor., 75 (2012), 2936-2956. doi: 10.1016/j.na.2011.11.037
![]() |
[78] |
O. S. Kwon, J. R. Kweon, Interior jump and regularity of compressible viscous Navier-Stokes flows through a cut, SIAM J. Math. Anal., 49 (2017), 1982-2008. doi: 10.1137/15M1042826
![]() |
[79] |
O. S. Kwon, J. R. Kweon, Compressible Navier-Stokes equations in a polyhedral cylinder with inflow boundary condition, J. Math. Fluid Mech., 20 (2018), 581-601. doi: 10.1007/s00021-017-0336-3
![]() |
[80] |
L. Larchevêque, P. Sagaut, I. Mary, et al. Large-eddy simulation of a compressible flow past a deep cavity, Phys. Fluids, 15 (2003), 193-210. doi: 10.1063/1.1522379
![]() |
[81] |
Z. C. Li, Y. L. Chan, G. C. Georgiou, et al. Special boundary approximation methods for laplace equation problems with boundary singularities-applications to the motz problem, Comput. Math. Appl., 51 (2006), 115-142. doi: 10.1016/j.camwa.2005.01.030
![]() |
[82] |
Z. C. Li, T. T. Lu, Singularities and treatments of elliptic boundary value problems, Math. Comput. Model., 31 (2000), 97-145. doi: 10.1016/S0895-7177(00)00062-5
![]() |
[83] |
Z. C. Li, The method of fundamental solutions for annular shaped domains, J. Comput. Appl. Math., 228 (2009), 355-372. doi: 10.1016/j.cam.2008.09.027
![]() |
[84] | Z. C. Li, Combined Methods for Elliptic Equations with Singularities, Interfaces and Infinities, Springer, 2011. |
[85] |
V. Maz'ya, J. Rossmann, Mixed boundary value problems for the stationary Navier-Stokes system in polyhedral domains, Arch. Rati. Mech. Anal., 194 (2009), 669-712. doi: 10.1007/s00205-008-0171-z
![]() |
[86] | W. McLean, Corner singularities and boundary integral equations, In: Contributions of Mathematical Analysis to the Numerical Solution of Partial Differential Equations, Centre Math. Anal. Austral. Nat. Univ., 7 (1984), 197-213. |
[87] |
S. A. Nazarov, A. Novotny, K. Pileckas, On steady compressible Navier-Stokes equations in plane domains with corners, Math. Annal., 304 (1996), 121-150. doi: 10.1007/BF01446288
![]() |
[88] |
B. Nkemzi, S. Tanekou, Predictor-corrector p-and hp-versions of the finite element method for Poisson's equation in polygonal domains, Comput. Meth. Appl. Mech. Eng., 333 (2018), 74-93. doi: 10.1016/j.cma.2018.01.027
![]() |
[89] | B. Nkemzi, M. Jung. Flux intensity functions for the Laplacian at polyhedral edges, Int. J. Fracture, 175 (2012), 167-185. |
[90] |
B. Nkemzi, M. Jung, Flux intensity functions for the Laplacian at axisymmetric edges, Math. Meth. Appl. Sci., 36 (2013), 154-168. doi: 10.1002/mma.2578
![]() |
[91] |
M. Renardy, Corner singularities between free surfaces and open boundaries, Z. Angew. Math. Phys., 41(1990), 419-425. doi: 10.1007/BF00959988
![]() |
[92] |
P. N. Shankar, M. D. Deshpande, Fluid mechanics in the driven cavity, Annu. Rev. Fluid Mech., 32 (2000), 93-136. doi: 10.1146/annurev.fluid.32.1.93
![]() |
[93] | R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland: Elsevier, 1979. |
[94] |
H. B. D. Veiga, An Lp-theory for three-dimensional, stationary, compressible Navier-Stokes equations, and the incompressible limit for compressible fluids. The equilibrium solutions, Commun. Math. Phys., 109 (1987), 229-248. doi: 10.1007/BF01215222
![]() |
[95] |
J. R. Whiteman, N. Papamichael, Treatment of harmonic mixed boundary problems by conformal transformation methods, Z. Angew. Math. Phys., 23 (1972), 655-664. doi: 10.1007/BF01593987
![]() |
[96] |
Z. Zhang, The existence and asymptotic behaviour of the unique solution near the boundary to a singular Dirichlet problem with a convection term, P. Roy. Soc. Edinb. A., 136 (2006), 209-222. doi: 10.1017/S0308210500004522
![]() |