Funkcialaj Ekvacioj, 60 (2017) 1–20 Solvability of p-Laplacian Parabolic Equations with Constraints Coupled with Navier-Stokes Equations in 3D Domains by Using Largeness of p By Takeshi Fukao*, Yutaka Tsuzuki and Tomomi Yokota† (Kyoto University of Education, Hiroshima Shudo University and Tokyo University of Science, Japan) Abstract. This paper is concerned with a system of p-Laplace heat equations with constraints and Navier-Stokes equations. The existence and uniqueness of solutions have been already proved for several types of the system in 2-dimensional domains. This paper gives the existence result in 3-dimensional domains, where the di¤usion term on heat equations is the p-Laplacian with pb3. This work provides a first insight towards the full case pb2. Key Words and Phrases. Heat equations, p-Laplacian, Time-dependent constraints, 3D Navier-Stokes equations, Subdi¤erential operators. 2010 Mathematics Subject Classification Numbers. Primary 35Q35, Secondary 47H05. 1. Introduction 1.1. Problem and related works Let 0>>>>>>>>>>>>>>>>>>>>>>>>:qdyyvði¼v0=;qv0(cid:4)tÞ¼;(cid:3)¼v0D¼yv0ðþ(cid:4)Þ;ðvvð(cid:4)0‘;Þ(cid:4)Þv¼¼vgððy(cid:4)ÞÞ(cid:3)‘p iiionnnnWQQS;;;:¼ð0;TÞ(cid:2)G; 0 0 * Supported by a Grant-in-Aid for Scientific Research (C) (No. 26400164), JSPS. † Supported by a Grant-in-Aid for Scientific Research (C) (No. 25400119), JSPS. 2 Takeshi Fukao, Yutaka Tsuzuki and Tomomi Yokota where y:Q!R, v¼ðv ;v ;v Þ:Q!R3 and p:Q!R represent the temper- 1 2 3 ature, the velocity and the pressure, respectively, and are unknown functions; c ;c :Q!R are given obstacle functions with c ac , and f :Q!R, 1 2 1 2 g¼ðg ;g ;g Þ:R!R3, y :W!R, v :W!R3, and pb3 are also given, 1 2 3 0 0 and D is the so-called p-Laplace operator defined as D y:¼divðj‘yjp(cid:3)2‘yÞ. p p The problem including the fifth equation is the so-called ‘‘Boussinesq system’’. From a viewpoint of physics, the problem (P) (practically in the case p¼2) represents the model describing temperature y¼yðt;xÞ and velocity v¼vðt;xÞ of incompressible fluid at each place x on the domain W at each time tA½0;T(cid:5). Artificially, we impose constraint on temperature. In more detail, the equation qy=qt(cid:3)D yþv(cid:4)‘y¼ f p holds when c 2 admits a unique solution in the classes (C2) and ðC1Þ yAH1ð0;T;L2ðWÞÞ\Lyð0;T;W1;pðWÞÞ with D yAL2ð0;T;L2ðWÞÞ: p 0 p There are two important parts to solve the problem (P) with pb2. One is to assert that (C1) and (C2) imply p ð1:1Þ v(cid:4)‘yAL2ð0;T;L2ðWÞÞ: Indeed, in the case p¼2 as in [9] and [22], noting that 4 Takeshi Fukao, Yutaka Tsuzuki and Tomomi Yokota ðC1Þ0 ‘yALyð0;T;L2ðWÞÞ\L2ð0;T;H1ðWÞÞ; 2 ðC2Þ0 vALyð0;T;L2ðWÞÞ\L2ð0;T;H1ðWÞÞ; s s we see from the Gagliardo-Nirenberg interpolation that vAL4ð0;T;L4ðWÞÞ and ‘yAL4ð0;T;L4ðWÞÞ, and hence (1.1) holds. On the other hand, in the case p>2 as in [24], the Gagliardo-Nirenberg interpolation with (C2)0 implies that vALpð0;T;Lp(cid:6)ðWÞÞ, where p(cid:6) is defined as 1 1 1 ð1:2Þ þ ¼ ; p p(cid:6) 2 moreover, it follows from (C1) that p ðC1Þ0 ‘yALyð0;T;LpðWÞÞ; p and hence (1.1) holds. The other of the two important parts is to introduce the two mappings, which are for example the two solution operators of the heat equation with fixed velocity and the Navier-Stokes equation with fixed temperature, and to construct solutions to (P) by combining the two mappings. Indeed in the both case p¼2 and p>2, they apply the Schauder fixed point theorem (as in [9]) or the contraction mapping principle (as in [22] and [24]) to the composition of the two mappings mentioned above. The purpose of this paper is to establish the existence of solutions to the problem (P) in 3-dimensional domains. We should pay more attention to verifying (1.1) because of loss of regularity for solutions to (P). Indeed, (C2)0 implies that vALqð0;T;LrðWÞÞ with 4=ð3qÞþ2=r¼1 for all qA½2;y(cid:5) and rA½2;6(cid:5). Therefore, to assert (1.1), it is required that ð1:1Þ0 ‘yALqq~ð0;T;Lrr~ðWÞÞ with 2=qq~þ3=~rra1 for some qq~A½2;y(cid:5) and ~rrA½3;y(cid:5). If we consider the case p¼2, then (C1)0 2 implies that ðC1Þ00 ‘yALqq~ð0;T;Lrr~ðWÞÞ with 4=ð3qq~Þþ2=~rr¼1 2 for all qq~A½2;y(cid:5) and ~rrA½2;6(cid:5). Since qq~ and ~rr in ðC1Þ00 satisfy 2 2=qq~þ3=~rr¼3=2 ð>1Þ; ðC1Þ00 does not imply ð1:1Þ0. Therefore, in this paper, we assume pb3 so 2 that ðC1Þ0 holds and it implies ð1:1Þ0. Since we do not know the uniqueness p of solutions to the Navier-Stokes equation in 3-dimensional domains, we can Systems of p-Laplacian Parabolic Equations and Navier-Stokes Equations 5 neither use the contraction mapping principle nor the Schauder fixed point theorem as in [9], [22] and [24]. Indeed, the application of the contraction mapping principle breaks down on a similar situation when we calculate to prove the uniqueness of solutions to the Navier-Stokes equation in 3-dimensional domains. The application of the Schauder fixed point theorem also breaks down because the solution operator mentioned above is multi- valued. Therefore, it is necessary to introduce another method to construct solutions to (P). In this paper, we use the method in [6] (see Section 3 in more detail). 1.3. Formulation of the problem and main results We use the following notation: H :¼L2ðWÞ; V :¼W1;pðWÞ and H :¼L2ðWÞ; V :¼H1ðWÞ: p 0 s s Here H and V are equipped with standard inner product and k(cid:4)k :¼ p Vp k‘(cid:4)k , respectively. Also, L2ðWÞ and H1ðWÞ are the closures of D ðWÞ LpðWÞ s s s :¼fvACyðWÞjdivv¼0 in Wg in L2ðWÞ and H1ðWÞ, respectively. Then we 0 have the dense and compact imbeddings V ,!H and V ,!H ,!V(cid:6). Under p these spaces, we define the closed and convex set KðtÞ as KðtÞ:¼fzAHjc ðtÞazac ðtÞ a:e: on Wg; tA½0;T(cid:5); 1 2 and we define að(cid:4);(cid:4)Þ:V (cid:2)V !R and bð(cid:4);(cid:4);(cid:4)Þ:V (cid:2)V (cid:2)V !R respectively as aðu;zÞ:¼ X3 ð quj qzj dx; u;zAV; qx qx i;j¼1 W i i bðu;v;zÞ:¼ X3 ð u qvjz dx; u;v;zAV; iqx j i;j¼1 W i and we also define A:V !V(cid:6) and B:V (cid:2)V !V(cid:6) respectively as hAu;zi :¼aðu;zÞ; u;zAV; V(cid:6);V hBðu;vÞ;zi :¼bðu;v;zÞ; u;v;zAV; V(cid:6);V where A is the so-called Stokes operator and represented as (cid:3)PD, where P:L2ðWÞ!H is the Helmholtz projection. Under these settings we define solutions to the problem (P). Definition 1.1. A pair ðy;vÞAL2ðQÞ(cid:2)L2ðQÞ is called a solution to the problem (P) if the following conditions are satisfied: 6 Takeshi Fukao, Yutaka Tsuzuki and Tomomi Yokota (D1) y and v belong to the following classes, respectively: ðD1aÞ yAH1ð0;T;HÞ\Lyð0;T;V Þ; D yAL2ð0;T;HÞ; p p ðD1bÞ yðtÞAKðtÞ for all tA½0;T(cid:5); ðD1cÞ vAW1;4=3ð0;T;V(cid:6)Þ\Lyð0;T;HÞ\L2ð0;T;VÞ; (D2) y and v satisfy (D2a) for a.a. tAð0;TÞ and all zAKðtÞ and (D2b) for all zAV respectively: ðD2aÞ ððdy=dtÞðtÞ(cid:3)D yðtÞþvðtÞ(cid:4)‘yðtÞ(cid:3) fðtÞ;yðtÞ(cid:3)zÞ a0; p H ðD2bÞ hdv=dt;zi þaðv;zÞþbðv;v;zÞ¼ðPgðyÞ;zÞ a:e: on ð0;TÞ V(cid:6);V H ð,dv=dtþAvþBðv;vÞ¼PgðyÞ in V(cid:6) a:e: on ð0;TÞÞ; (D3) ðyð0Þ;vð0ÞÞ¼ðy ;v Þ in H(cid:2)H. 0 0 In this paper, especially in the two main theorems, we assume the following conditions: (A1) c ;c AH1ð0;T;HÞ, D c AL2ð0;T;HÞ, cðtÞAW1;pðWÞ a.a. tA 1 2 p i i ð0;TÞ for i¼1;2, c ðtÞac ðtÞ a.e. on W, c ðtÞa0ac ðtÞ a.e. on G for 1 2 1 2 all tA½0;T(cid:5); (A2) f AL2ð0;T;HÞ and g:R!R3 is Lipschitz continuous; (A3) ðy ;v ÞAðV \Kð0ÞÞ(cid:2)H. 0 0 p Now we are in a position to state the main results. Theorem 1.1. Assume (A1), (A2), (A3) and p>4: Then there exists at least one solution ðy;vÞ to the problem (P). Theorem 1.2. Let pb3. Assume (A1), (A2), (A3) and c ;c ALyð0;T;W1;pðWÞÞ: 1 2 Then there exists at least one solution ðy;vÞ to the problem (P). Remark 1.1. The assumption pb3 assures that V ,!Lp(cid:6)ðWÞ with p(cid:6) defined as (1.2) and kzkLp(cid:6)ðWÞackzkV for some positive constant c. Therefore, yALyð0;T;V Þ and vAL2ð0;T;VÞ imply that p v(cid:4)‘yAL2ð0;T;HÞ with estimate ð1:3Þ kv(cid:4)‘yk ackvk kyk : L2ð0;T;HÞ L2ð0;T;VÞ Lyð0;T;VpÞ This fact is fundamental in dealing with the heat equation in L2ð0;T;HÞ. Systems of p-Laplacian Parabolic Equations and Navier-Stokes Equations 7 2. Preliminaries 2.1. Solvability of the approximate heat equation To construct solutions to the heat equation, we regard the variational inequality (D2a) as the following evolution equation in H dy ðD2aÞ0 ðtÞþqjðyðtÞÞþqI ðyðtÞÞC fðtÞ(cid:3)vðtÞ(cid:4)‘yðtÞ a:a: tAð0;TÞ dt KðtÞ by using subdi¤erential operators, and we also introduce the Yosida approximation and the Moreau-Yosida approximation. We define two proper, lower semi-continuous and convex functions as follows: 81ð >< j‘yjpdx; yAV ; ð2:1Þ jðyÞ:¼ p p W >:y; yAHnV ; p (cid:1)0; yAKðtÞ; ð2:2Þ I ðyÞ:¼ tA½0;T(cid:5): KðtÞ y; yAHnKðtÞ; We introduce the subdi¤erential operator qf:DðqfÞ(cid:1)H !2H, for a proper, lower semi-continuous and convex function f:H !R[fyg with DðfÞ:¼ fuAHjfðuÞ><ðdyy~=dtÞðtÞþqjðyy~ðtÞÞþqIKnðtÞðyy~ðtÞÞ ðNNgHHÞn ¼ fðtÞ(cid:3)vðtÞ(cid:4)‘yðtÞ a:a: tAð0;TÞ; >>:yy~ð0Þ¼y : 0 Systems of p-Laplacian Parabolic Equations and Navier-Stokes Equations 9 Then for all tA½0;T(cid:5), ð2:6Þ kdyy~=dtk2 þjðyy~ðtÞÞþIn ðyy~ðtÞÞ L2ð0;T;HÞ KðtÞ þkqjðyy~Þk2 þkqIn ðyy~Þk2 L2ð0;T;HÞ Kð(cid:4)Þ L2ð0;T;HÞ aC ðkvk2 kyk2 þIn ðy Þþ1Þ; 1 L2ð0;T;VÞ Lyð0;T;VpÞ Kð0Þ 0 and moreover, if y¼yy~¼:y(cid:6) and y AKð0Þ, then for all tA½0;T(cid:5), 0 ð2:7Þ kdy(cid:6)=dtk2 þjðy(cid:6)ðtÞÞþIn ðy(cid:6)ðtÞÞ L2ð0;T;HÞ KðtÞ þkqjðy(cid:6)Þk2 þkqIn ðy(cid:6)Þk2 L2ð0;T;HÞ Kð(cid:4)Þ L2ð0;T;HÞ aC0ðkvkp(cid:6) þ1Þ: 1 L2ð0;T;VÞ Here C and C0 are positive constants depending only on T, jðy Þ, f, c and c 1 1 0 1 2 (independent of n) and p(cid:6) is the constant defined as (1.2). Proof. We multiply the equation in ðNNgHHÞ by qjðyy~ðtÞÞþqIn ðyy~ðtÞÞ. n KðtÞ Using Lemmas 2.1 and 2.2, we see that for a.a. tAð0;TÞ, d ½jðyy~ðtÞÞþIn ðyy~ðtÞÞ(cid:5)þkqjðyy~ðtÞÞk2 þkqIn ðyy~ðtÞÞk2 dt KðtÞ H KðtÞ H aðC(cid:6)ðtÞþ2CDpðtÞÞkqIn ðyy~ðtÞÞk KðtÞ H þðkfðtÞk þkvðtÞ(cid:4)‘yðtÞk Þðkqjðyy~ðtÞÞk þkqIn ðyy~ðtÞÞk Þ H H H KðtÞ H aðC(cid:6)ðtÞþ2CDpðtÞþkfðtÞk þckvðtÞk kyðtÞk Þ H V Vp (cid:2)ðkqjðyy~ðtÞÞk þkqIn ðyy~ðtÞÞk Þ H KðtÞ H a4ðjC(cid:6)ðtÞj2þ4jCDpðtÞj2þkfðtÞk2 þc2kvðtÞk2kyðtÞk2 Þ H V Vp 1 þ ðkqjðyy~ðtÞÞk2 þkqIn ðyy~ðtÞÞk2Þ: 2 H KðtÞ H Integrate it over ½0;t0(cid:5) for all t0 A½0;T(cid:5) (denoted by t again). Then we see that for all tA½0;T(cid:5), 2jðyy~ðtÞÞþ2In ðyy~ðtÞÞþkqjðyy~Þk2 þkqIn ðyy~Þk2 KðtÞ L2ð0;t;HÞ Kð(cid:4)Þ L2ð0;t;HÞ a2jðy Þþ2In ðy Þþ8ðkC(cid:6)k2 þ4kCDpk2 þkfk2 0 Kð0Þ 0 L2ð0;TÞ L2ð0;TÞ L2ð0;T;HÞ þc2kvk2 kyk2 Þ: L2ð0;T;VÞ Lyð0;T;VpÞ 10 Takeshi Fukao, Yutaka Tsuzuki and Tomomi Yokota This inequality together with the equation in ðNNgHHÞ gives the estimate for n kdyy~=dtk2 and hence we obtain (2.6). Furthermore, if y¼yy~¼:y(cid:6) and L2ð0;T;HÞ y AKð0Þ, then (2.6) implies (2.7). To see it, note In ðy Þ¼0 and use the 0 Kð0Þ 0 Young inequality with (1.2). r Now we state the solvability of the approximate heat equation in ðNNgHHÞ n with y¼yy~¼:y(cid:6) for given velocity. n Proposition 2.4. Assume (A1), (A2) and y AV . Let nAN. Then for all 0 p vAL2ð0;T;VÞ, there exists a unique solution y(cid:6) belonging to the class (D1a) n such that 8ðdy(cid:6)=dtÞðtÞþqjðy(cid:6)ðtÞÞþqIn ðy(cid:6)ðtÞÞ >< n n KðtÞ n ðNHÞ ¼ fðtÞ(cid:3)vðtÞ(cid:4)‘y(cid:6)ðtÞ a:a: tAð0;TÞ; n n >:y(cid:6)ð0Þ¼y : n 0 Proof. Since j is proper, lower-semi continuous and convex and qIn is KðtÞ Lipschitz continuous, we see that for all yALyð0;T;V Þ there exists a unique p solution yy~belonging to the class (D1a) to the problem ðNNgHHÞ as in Lemma 2.3, n and hence we obtain the estimate (2.6). Here we introduce the mapping S :yAX !yy~AX n M M with the compact and convex set X defined as M X :¼fyACð½0;T(cid:5);HÞjkdy=dtk2 þkykp aMg M L2ð0;T;HÞ Lyð0;T;VpÞ with the topology on Cð½0;T(cid:5);HÞ, where M >0 is large enough. Using the Schauder fixed point theorem, we obtain a solution y(cid:6) to (NH) in the same n n way as in [24, Proof of Proposition 3.2]. The uniqueness is obtained by the accretivity of qj and qIn and the divergence freeness of v. r KðtÞ 2.2. Further estimate toward Theorem 1.2 In this section we suppose c ;c ALyð0;T;W1;pðWÞÞ, which is the 1 2 assumption of Theorem 1.2. Lemma 2.5. Let nAN, vAV and zAV . Assume that c , c satisfy (A1) p 1 2 and c ALyð0;T;W1;pðWÞÞ for i ¼1;2. Then the following estimate holds: i ðv(cid:4)‘z;qIn ðzÞÞ b(cid:3)C‘ðtÞkvk kqIn ðzÞk a:a: tAð0;TÞ; KðtÞ H V KðtÞ H where C‘ ALyð0;TÞ is defined as C‘ðtÞ:¼cðk‘c ðtÞk2 þk‘c ðtÞk2 Þ1=2; tA½0;T(cid:5): 1 LpðWÞ 2 LpðWÞ

Solvability of p-Laplacian Parabolic Equations with Constraints Coupled with Navier-Stokes ...

Similar Solvability of p-Laplacian Parabolic Equations with Constraints Coupled with Navier-Stokes ...