Existence and uniqueness for some equation of a mixed elliptic-hyperbolic type

R E S E A R C H

Open Access

Existence and uniqueness for some equation

of a mixed elliptic-hyperbolic type

Zhang Haixia

_{and Xiao Wei}

*_{Correspondence:} [email protected]

2_{School of Sciences, Chang’an} University, Xi’an, P.R. China Full list of author information is available at the end of the article

Abstract

In the paper, we prove results on existence and uniqueness of weak solutions for closed Dirichlet problem for a transonic flow model, and interior regularity results are also given in the important special case of the Tricomi equation. The method employed consists in variants of the a-b-c integral method of Friedrichs in Sobolev spaces with suitable weights. Particular attention is paid to the problem of attaining results with a minimum of restrictions on the boundary geometry and the form of the type change function.

Keywords: Mixed elliptic-hyperbolic type equation; Tomotika–Tamada model; a-b-c integral method; Star-shaped

1 Introduction and results

In this paper, we consider the following initial boundary value problem:

⎧ ⎨ ⎩

Lu≡A(1 –e–2By_)u

xx(x,y) +uyy(x,y) =f(x,y) inΩ,

u= 0 on∂Ω

(1)

withf(x,y) is a known function, andA,Bare constants,A> 0,B> 0,u(x,y) is an unknown function. LetK(y) =A(1 –e–2By_{),Ωis a bounded, open, and connected subset ofR}2_with

piecewiseC1boundary. We assume throughout that

Ω±:=Ω∩R2_±=Ø (2)

so that (1) is of mixed elliptic-hyperbolic type. Such an equation is of Tricomi type and it is important in the problem of transonic fluid flow (see [18]).

Such a boundary value problem will be called closed in the sense that the boundary condition (2) is imposed on the entire boundary as opposed to an open problem, in which (2) is imposed on a proper subsetΓ ⊂∂Ω. Both kinds of problems are interesting for transonic flow. Much more is known about open problems, beginning with the work of Tricomi [19].

On the other hand, much less is known about closed problems. Under mild assumptions on the functionKand the geometry of the boundary, one has a uniqueness theorem for regular solutions to the Tricomi problem. Such uniqueness theorems have been proven

by a variety of methods, including energy integrals as in [16] and maximum principles as in [2,10]. In order to prove well-posedness, one must choose some reasonable function space that admits a singularity strong enough to allow for existence.

For the interest in closed problems for mixed-type equations, the literature essentially contains only two results on well-posedness. The first, due to Morawetz [11], concerns the Dirichlet problem for the Tricomi equation, and the second due to Pilant [15] con-cerns the natural analogue of the Neumann problem (conormal boundary conditions) for the Lavrentiev–Bitsadze equation. Inspired by [5–8,13,17,20], we consider the closed Dirichlet problem in the Tomotika–Tamada model (1).

The existence and uniqueness results of Sect.2for the Dirichlet problem do not require some smoothness assumptions on the boundary, no geometric assumptions on the ge-ometry of the elliptic boundary an arbitrarily small distance away from the parabolic line need to be made. Similar considerations have been exploited for the Tricomi problem and will hold for other problems with an open boundary condition, such as the Frankl prob-lem. We also note that the arguments used in obtaining the interior regularity results also apply for problems with open boundary conditions as well. The paper is organized as fol-lows. In Sect.1, we investigate the introduction, notions, and results. In Sect.2, we show the existence and uniqueness of a weak solution to a problem with Dirichlet conditions. In Sect.3, we show the regularity of the weak solution.

We defineL2_(Ω;|K|–1_{) for the givenC}1_(R2_{) functionK(x,y) in the same way as (see [5])}

L2Ω;|K|–1:=f ∈L2(Ω) :|K|–1/2f ∈L2(Ω)

equipped with its norm

fL2_(Ω;|K|–1₎= Ω

|K|–1f2dx dy

1/2

(3)

which is the dual space to the weighted spaceL2_{(Ω;|K|) defined as the equivalence classes}

of square-integrable functions with respect to the measure|K|dx dy; that is, with finite norm

fL2_(Ω;|K|)= Ω

|K|f2dx dy

1/2

. (4)

One has the obvious chain of inclusions

L2Ω;|K|–1⊂L2(Ω)⊂L2Ω;|K|, (5)

where the inclusion maps are continuous and injective. We defineH1

0(Ω;K) as the closure ofC∞0 (Ω) (smooth functions with compact support

inΩwith respect to the weighted Sobolev norm):

u_H1_(Ω;|K|):= Ω

|K|u2_x+u2_y+u2dx dy

1/2

Sinceu∈H1_{(Ω;|K|) vanishes weakly on the entire boundary, one has a Poincaré}

inequal-ity: there existsCP=CP(Ω,K)

u2_L2(Ω)≤CP

Ω

|K|u2_x+u2_ydx dy; u∈H1Ω;|K|. (7)

An equivalent norm onH1

0(Ω;K) is thus given by

u_H1

0(Ω;K):= _Ω

|K|u2_x+u2_ydx dy

1/2

. (8)

We denote byH–1_{(Ω;K) the dual space toH}1

0(Ω;K) equipped with its norm [1]

uH–1_(Ω;K):= sup 0=φ∈C∞0 (Ω)

|u,φ|

φ_H1 0(Ω;K)

, (9)

where·,·is the duality bracket and one has the generalized Schwarz inequality

u,φ≤ uH–1_(Ω;K)φ_H1

0(Ω;K); u∈H

–1_{(Ω;K),φ∈H}1

0(Ω;K). (10)

One clearly has a rigged triple of Hilbert spaces

H₀1(Ω;K)⊂L2(Ω)⊂H–1(Ω;K), (11)

where the scalar product (onL2_{, for example) will be denoted by (·,·)} L2_(Ω).

It is routine to check that the second-order operator Lin (1) is formally self-adjoint when acting on distributionsD(Ω) and gives rise to a unique continuous and self-adjoint extension

L:H₀1(Ω;K)−→H–1(Ω;K). (12)

Now, we give the definition and the main theorem in the paper.

Definition 1 We sayΩis star-shaped with respect to the vector field of a given (Lipschitz) continuous vector fieldV= (V1(x,y),V2(x,y)); that is, for every (x0,y0)∈Ω, (the closure

ofΩ) one hasFt(x0,y0)∈Ω for eacht∈[0, +∞] whereFt(x0,y0) represents the time-t

flow of (x0,y0) in the direction ofV.

IfΩis star-shaped with respect to the flow ofV, thenΩ is simply connected and will have aV-starlike boundary in the sense thatV(x,y)·ν≥0, whereνis the unit exterior normal (see Lemma 2.2 of [6]).

Definition 2 We say thatu∈H₀1(Ω;K) is a weak solution of the Dirichlet problem (1) withK(y) =A(1 –e–2By_{) if there exists a sequenceu}

n∈C0∞(Ω) such that

or, equivalently,

Lu,ϕ= –

Ω

(Kuxϕx+uyϕy)dx dy=f,ϕ, ϕ∈H01(Ω,K), (14)

where·,·is the duality pairing betweenH–1_{(Ω;K) andH}1

0(Ω;K),Lis the continuous

extension defined in (12), and the relevant norms are defined in (8) and (9).

Theorem 1 LetΩbe a bounded mixed domain with piecewise C1_{boundary and parabolic}

segment CO with O= 0and2sup_Ω+e2By< 3 + 2infΩ+e2By.Assume thatΩ is star-shaped

with respect to the vector field V = (b0x, –e 2By_–1

B )with

b0= ⎧ ⎨ ⎩

–1₂++ 2sup_Ω+e2By Ω+, 5

2++ 2supΩ–e2By Ω–

(15)

for some . Then, for each f ∈L2_(Ω;|K|–1_{), there exists a unique weak solution u∈}

H1

0(Ω;K)in the sense of Definition1to the Dirichlet problem(1).

Theorem 2 LetΩbe a mixed domain and f ∈L2_{(Ω).If u∈H}1_{(Ω,K)is a weak solution}

to the equation Lu=f,then u∈H1 loc(Ω).

In the following sections, we will prove the results on existence, uniqueness, and interior regularity by the a-b-c integral method in Sobolev spaces with suitable weights.

2 Existence and uniqueness of weak solution

Lemma 1 Under the hypotheses of Theorem1,one has the a priori estimate.ThenΩ is admissible,that is,there exists a positive constant C1=C1(Ω,K)such that

uL2_(Ω;|K|)≤C₁Lu_H–1_(Ω;K); u∈C₀∞(Ω). (16)

Proof DefineMv=av+bvx+cvy, where (a,b,c) = (–1₄,b0x,e 2By_–1

B ). We claim that for every

u∈C∞₀ (Ω), there existsv∈C∞(Ω)∩C0_(Ω¯_{\{0, 0}) solving} ⎧

⎨ ⎩

Mv=u inΩ,

v= 0 on∂Ω\{0, 0}; (17)

furthermore,_Ω(|K|v2

x+v2y)dx dy< +∞. In fact, parameterizing the integral curve of (b,c)

byγ(t) = (x(t),y(t)) = (x0eb0t, –ln[1–e

2t_(1–e–2By0_)]

2B ) for (x0,y0)∈∂Ωandt∈(–∞, 0] and by the

assumption thatΩis star-shaped, the method of characteristic gives the uniqueC∞(Ω)∩ C0_(Ω¯_{\{0, 0}) solution of (17)}

vx(t),y(t)=e–at

t

0

easux(s),y(s)ds (18)

with centerO, thenBε(O)∩suppu=∅. For each (x,y)∈Bε(O)∩Ω, we can re-initialize the

Cauchy problem at timet=T< 0 by starting from points (xT,yT) on∂Bε∩Ω, thenv(x,y) =

v(xT,yT)e 1

4(t–T)_{. Sincev}∈_C1_{onΩ, the variation inv(xT,yT) is bounded for (xT,yT) on the}

initial data surface,vsatisfies (18).

We proceed to estimate the expression (v,Lu)L2_(Ω).

(v,Lu)_L2_(Ω)=

Ω

divv(Kux,uy)

–∇v·(Kux,uy)dx dy

= –

Ω

vxK(av+bvx+cvy)x+vy(av+bvx+cvy)y+

∇Ω

v(Kux,uy)· n ds

=1 2

Ω

–2aK–bxK+bKx+ (Kc)y

v2_x+ 2(–Kcx–by)vxvy

+ (–2a+bx–cy)v2y+v2La dx dy +1 2 ∂Ω

2v(Kux,uy) –

Kv2_x+v2_y(b,c) –v2(Kax,ay)

n ds, (19)

wherenis the unit exterior normal vector whiledsis the arc length element.

Using the smoothness and compact support ofu, the continuity ofv,Ωis star-shaped withV, the boundary integrals vanish. Notice the value of (a,b,c).

–2aK–bxK+bKx+ (Kc)y=

K

2 –b0K+ 2K+ 2e

2By_K= 5

2–b0+ 2e

2By

K, (20)

–Kcx–by= 0, (21)

–2a+bx–cy=

1

2+b0– 2e

2By_{. (22)}

On the elliptic regionΩ+,K> 0, and

5

2–b0+ 2e

2By₌5

2– – 1

2++ 2supΩ+

e2By

+ 2e2By= 3 –– 2sup

Ω+

e2By+ 2e2By

= 3 ––3 + 2inf

Ω+

e2By–¯

+ 2e2By>¯–+ 2e2By– 2inf

Ω+

e2By, (23)

where¯= 3 + 2infΩ+e2By– 2sup_Ω+e2By

1

2+b0– 2e

2By₌1

2– 1

2++ 2supΩ+e

2By_{– 2e}2By_{>. (24)}

On the hyperbolic regionΩ–,K< 0 and

–5

2+b0– 2e

2By_{= –}5

2+ 5

2++ 2supΩ–

e2By– 2e2By>, (25)

1

2+b0– 2e

2By₌1

2+ 5

2++ 2supΩ–e

2By_{– 2e}2By_{> 3 +. (26)}

Choose0> 0 such that¯–+ 2e2By– 2infΩ+e2By>0. So that

(v,Lu)L2_(Ω)≥₀

Ω

On the other hand, by the generalized Schwarz inequality,

(v,Lu)_L2_(Ω)≤ v_H1

0(Ω;K)LuH–1(Ω;K). (28)

Using the Cauchy–Schwarz inequality and Poincaré inequality, we have

u2_L2_(Ω;|K|)=Mv2_L2_(Ω;|K|)=

Ω

|K|(av+bvx+cvy)2dx dy

≤

Ω

|K|v2_x+v_y2dx dy=Cv_H1

0(Ω;K). (29)

Combining (27)–(29) gives the desired estimate

uL2_(Ω;|K|)≤CLu_H–1_(Ω;K). (30)

This competes the proof of this lemma.

Proof of Theorem1 Define a linear functionalJf forϕ∈C0∞(Ω) by the formulaJf(Lϕ) =

(f,ϕ)_L2_(Ω), and the estimate together with the Cauchy–Schwarz inequality yields

Jf(Lϕ)≤ fL2_(Ω;|K|–1₎ϕ_L2_(Ω;|K|)≤C₁f_L2_(Ω;|K|–1₎Lϕ_H–1_(Ω;K). (31)

HenceJf is bounded on the subspaceV ofH–1(Ω;K) of elements of the formLϕwith ϕ∈C₀∞(Ω). By the Hahn–Banach theorem,Jf extends to the closure ofVinH–1(Ω;K) in

a bounded way. Extension by zero on the orthogonal complement ofV¯ gives a bounded linear functional on all ofH–1(Ω;K), and so, by the Riesz representation theorem, there existsu∈H₀1(Ω;K) such that

u,Lϕ= (f,ϕ)_L2_(Ω), ϕ∈H₀1(Ω;K);Lϕ∈H–1(Ω;K), (32)

whereLis the self-adjoint extension defined in (12). This distributional solution is a weak solution in the sense of Definition2. In fact, given a sequenceun∈C0∞(Ω) thatun→uin

H1

0(Ω;K), the continuity property (12) shows thatfn:=Lun H–1(Ω,K)

−→ ˜f. One also has

un,Lϕ= (fn,ϕ)L2_(Ω). (33)

Taking the difference between (32) and (33) and passing to the limit shows thatf˜=f, and hence (13) holds.

For the uniqueness, we use estimate (16). In fact, for fixedf, letu,v∈H₀1(Ω;K) be two weak solutions which approximate sequences{un}and{vn}satisfying (13). From the

lin-earity ofLand (16), one has thatun–vn L2(Ω,K)

−→ 0 by the injectivity of the first inclusion in (5) and the Poincaré inequality (7). Thusu=vinH₀1(Ω;K).

3 Regularity of the weak solution

there exists a constant C=C(K,G)such that

G

u2_xdx dy≤C(E+F). (34)

Proof Pick a smooth cutoff functionζ∈C₀∞(Ω) such that 0≤ζ≤1 andζ ≡1 onG. One merely computes by a sequence of integrations by parts, which will need to be justified later. Starting from

AB

Ω

e–2Byζu2_xdx dy=

Ω

K(y)ζu2_xdx dy=

Ω

(Kζ)yu2xdx dy–

Ω

Kζyu2xdx dy, (35)

we find there exists a positive number0such that

2AB

Ω

e–2Byζu2_xdx dy≥0

G

u2_xdx dy. (36)

On the other hand,

Ω

(Kζ)yu2xdx dy–

Ω

Kζyu2xdx dy

= –

Ω

Kζyu2xdx dy–

Ω

2Kζuxuxydx dy

≤C1E+ 2

Ω

(Kζux)xuydx dy≤C2E+ 2

Ω

Kζuxxuydx dy

=C2E+ 2

Ω

ζ(f–uyy)uydx dy≤C3(E+F) – 2

Ω

ζuyyuydx dy

=C3(E+F) –

Ω

ζu2_yydx dy

≤C4(E+F). (37)

Thus

G

u2_xdx dy≤C(E+F). (38)

Mollification η∈C₀∞(R) such that 0≤η≤1, _Rη(t)dt= 1 for > 0, defineη(t) =

–1η(t) so that η ∈C₀∞(B(0;)) and _Rη(t)dt= 1. Given anyu∈L1_loc(Ω), measurable and locally integrable with respect to Lebesgue measure, define

u(x,y) :=

R

η(x–t)u(t,y)dt=

B(0;)

η(t)u(x–t,y)dt, (39)

where we extenduby 0 outside ofΩ.

DefineI(y) ={x∈R|(x,y)∈Ω},I(y) ={x∈I(y)|x±∈I(y)}. Lemma 3(see [4]) Let u∈L1

loc(Ω),then one has,for almost every y∈π2(Ω) :={y:∃x∈

(i) u(·,y)∈C∞(I(y))and for eachk∈N,Dk_xu(x,y) =

I(y)Dkη(x–t)u(x,t)dtfor

∀x∈I(y);

(ii) u(·,y)−→u(·,y)almost everywhere onI(y)as−→0;

(iii) Ifu(·,y)∈C0_(I(y)),thenu

(·,y)−→u(·,y)uniformly on compact subsets ofI(y);

(iv) If1≤p<∞andu∈Lp_loc(I(y)),thenu(·,y)−→u(·,y)inL p loc(I(y)).

Moreover,let u∈L2(Ω).Then one has: (v) u∈L2(Ω)andu−→uinL2(Ω);

(vi) Dk_xJ:L2(I(y))−→L2(I(y))is bounded for everyk∈N,whereJu:=u;

(vii) Dk

xJ:L2(Ω)−→L2(Ω)is bounded for everyk∈N;

(viii) For everyg∈L2(Ω),ϕ∈C₀∞(Ω),one has

e–2Byg=e–2Byg, (40)

Dx(ϕ) = (Dxϕ), Dy(ϕ) = (Dyϕ), (41)

Ω

gϕdx dy=

Ω

gϕdx dy. (42)

Lemma 4 Let u∈H1_{(Ω;K)be a weak solution to(1)with f∈L}2_{(Ω).Then u}

∈H2(Ω)is

a weak solution to the equation

Lu=f, (43)

where f=Jf is the mollification in x of f.

Proof Ifu∈H1_{(Ω;K), thenu,|K|}1/2_u

xanduy∈L2(Ω), and by Lemma3, we haveu ∈

L2_{(Ω) andD}k

xu∈L2(Ω) for eachk≥1. As forDyu, since

Ω

uDyϕdx dy= –

Ω

(u)yϕdx dy= –

Ω

(uy)ϕdx dy ∀ϕ∈C₀∞(Ω) (44)

and (uy)∈L2(Ω), therefore (u)y= (uy)∈L2(Ω).

Next we claim that ifu∈L2_{(Ω) solves (1) in the sense of distributions, thenu} solves

(43) in the sense of distributions. Indeed, starting from

Ω

u(Kϕx)x+ϕyy

dx dy=

Ω

f dx dy, ϕ∈C₀∞(Ω), (45)

take > 0 small enough so that the support ofϕ inside Ω, apply (44) toϕ, and use Lemma3to find

Ω

fϕdx dy=

Ω

u(Kϕx)x+ (ϕ)yydx dy=

Ω

u(Kϕxx+ϕyy)dx dy, (46)

which gives the claim. Rewriting (46) in the form

Ω

uϕyydx dy=

Ω

f–K(u)xx

and noticingf –K(u)x, we know that the weak derivativeD2yu exists and belongs to

L2_{(Ω). SinceD}

xDyu=Dx(Dyu) andDyu∈L2(Ω), then by Lemma3, we knowDxDyu∈

L2_{(Ω). Hence we haveu∈H}2_{(Ω), which completes the proof.}

Proof of Theorem2 By Lemma3, we have

Ω

ζ(u)2xdx dy≤C

Ω

|K|(u)2x+ (u)2y+f2

dx dy, (48)

whereu∈L2_{(Ω), then Lemma3and applying Fatou’s lemma gives the needed estimate}

Ω

ζu2_xdx dy≤C

Ω

|K|u2_x+u2_y+f2dx dy, (49)

and the theorem follows.

We conclude the results on existence, uniqueness, and regularity of weak solutions for closed Dirichlet problem. One can generalize Theorems1and2to include more general type change functionsK(y) provided thatKis sufficiently smooth.

Acknowledgements

The authors would like to thank the anonymous reviewers for their careful work and comments that helped to improve the clarity of the paper.

Funding

The first author is supported by the Natural Science Foundation of Shaanxi Province of China 2019GY-202. The second author is supported by NSFC 11401043, Natural Science Foundation of Shaanxi Province of China 2018JQ1084, and the Fundamental Research Funds for the Central Universities 310812171005.

Availability of data and materials Not applicable.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors have contributed equally to the paper. All authors read and approved the final manuscript.

Author details

1_{School of Highway Engineering, Chang’an University, Xi’an, P.R. China.}2_{School of Sciences, Chang’an University, Xi’an,}

P.R. China.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 7 March 2019 Accepted: 5 July 2019 References

1. Adams, R.A., Fournier, J.J.: Sobolev Spaces. Elsevier, New York (2003)

2. Agmon, S., Nirenberg, L., Protter, M.H.: A maximum principle for a class of hyperbolic equations and applications to equations of mixed elliptic-hyperbolic type. Commun. Pure Appl. Math.6, 455–470 (1953)

3. Bers, L.: Mathematical Aspects of Subsonic and Transonic Gas Dynamics. Dover, New York (2016) 4. Evans, L.C.: Partial Differential Equations. Am. Math. Soc., Providence (2010)

5. Lupo, D., Morawetz, C.S., Payne, K.R.: On closed boundary value problems for equations of mixed elliptic-hyperbolic type. Commun. Pure Appl. Math.60, 1319–1348 (2007)

6. Lupo, D., Payne, K.R.: Critical exponents for semilinear equations of mixed elliptic-hyperbolic and degenerate types. Commun. Pure Appl. Math.56, 403–424 (2003)

7. Marin, M.: Cesaro means in thermoelasticity of dipolar bodies. Acta Mech.122, 155–168 (1997)

8. Marin, M., Öchsner, A.: The effect of a dipolar structure on the Hölder stability in Green–Naghdi thermoelasticity. Contin. Mech. Thermodyn.29, 1365–1374 (2017)

10. Morawetz, C.S.: Note on a maximum principle and a uniqueness theorem for an elliptic-hyperbolic equation. Proc. R. Soc. Lond.236, 141–144 (1956)

11. Morawetz, C.S.: The Dirichlet problem for the Tricomi equation. Commun. Pure Appl. Math.23, 587–601 (1970) 12. Morawetz, C.S.: Mixed equations and transonic flow. J. Hyperbolic Differ. Equ.1, 1–26 (2004)

13. Otway, T.H.: Unique solutions to boundary value problems in the cold plasma model. SIAM J. Math. Anal.42, 3045–3053 (2010)

14. Picone, M.: Sulle equazioni alle derivate parziali del secondordine del tipo iperbolico in due variabili indipendenti. Rend. Circ. Mat. Palermo30, 349–376 (1910)

15. Pilant, M.: The Neumann problem for an equation of Lavrent’ev–Bitsadze type. J. Math. Anal. Appl.106, 321–359 (1985)

16. Protter, M.H.: Uniqueness theorems for the Tricomi problem. II. J. Ration. Mech. Anal.4, 721–732 (1955) 17. Ruziev, M.K.: A problem with conditions given on inner characteristics and on the line of degeneracy for a

mixed-type equation with singular coefficients. Bound. Value Probl.2013, 210 (2013)

18. Tomotika, S., Tamada, K.: Studies on two-dimensional transonic flows of compressible fluid. Q. Appl. Math.7, 381–397 (1950)

19. Tricomi, F.: Sulle equazioni lineari alle derivate parziali di tipo miste. Rend. Atti Accad. Naz. Lincei14, 134–147 (1923) 20. Yuan, C., Jiang, D., Zhang, Y.: Existence and uniqueness of solutions for singular higher order continuous and discrete