Project topics and materials
Get Complete Project Material File(s) Now! »
From a classical formulation to the weak formulation.
We now assume that , u is a classical solution of the problem (1.12){(1.15), (1.18). (The classical solution is a solution which is continuous in together with all derivatives which appear in the equations (1.12), (1.13) and it satis es these equations at all points of . Moreover, the classical solution also admits the continuous extension of the expressions on the left{hand sides of (1.14) and (1.15) to the boundary of so that (1.14), (1.15), (1.18) hold.) Formally, the Helmholtz decomposition of u is u = v + r’ (2.1) where ’ is smooth and solves the Neumann problem ’ = div u in ; (2.2) @’ = 0 (2.3) @n @ and v := u r ’ is also smooth. We note that the requirement for v and ’ to be smooth is just formal (see De nition 2.1 below for the spaces in which we look for v and ’). This decomposition guarantees that div v = 0 in and v also satis es the boundary conditions (1.14), (1.15) and (1.18). If we use decomposition (2.1) of u in equations (1.12) and (1.13), we obtain @j (vj + @j’) (v + r’) + curl2v + r (2 + ) ’+P() + g in ; (2.4) = f in (2.5) div (v + r’) ) = 0 : This system is now completed by the boundary conditions v n @ =0; (2.6) curl v n @ = 0 (2.7) for function v, and by condition (2.3) for function ’. We moreover complete this system with the third scalar boundary condition: curl2v n @ = 0 (2.8).
This condition is actually satis ed for every \su ciently smooth » solution of system (2.3)-(2.7) (see Subsection 2.2.2 below).
Let us rst consider a test function from space D1;2( ) and let us multiply equation (2.4) by function and integrate on . Since v is in the domain of the operator (curl2)2 and is in the domain of curl2 (i.e. curl2 = curl by de nition), we can use the selfadjointness of operator curl2 and we can transform the integral of curl2v (which equals (curl2)2v ) to the integral of curl2v curl2 . Using also the integration by parts, we obtain the integral identity.
From the weak formulation to the classical formulation.
Now we assume that (v; ’; ) is a weak solution of the problem (2.3){(2.8) according to De nition 2.1. We show in this Subsection that if this solution is \su ciently smooth » then the pair u v + r’, is a classical solution of the problem (1.12){(1.15), (1.18),(1.19).
Equations (1.12) and (1.13).
Let us at rst consider the integral identities (2.9), (2.10) with the test functions and that have a compact support in . Integrating by parts in both the identities and summing them, we obtain Z @j (vj + @j’) (v + r’) + curl2v + r (2 + ) ’ + P ( ) ( + r ) dx Z = (2.12) f + g ( + r ) dx The set of all test functions , that have a compact support in , is dense in the space L2 ( ).
The set of all functions of the type r , where 2 is a test 2function with a compact support in , is dense in the orthogonal complement to L ( ) in L ( ). Thus, the sums + r are dense in L ( ). Consequently, (2.12) implies that the functions v, ’ and satisfy equation (2.4) a.e. in . According to (2.1), this is exactly the equation (1.12) for u. Considering test functions with a compact support in in the integral identity (2.11), we can show in the same way that the functions v, ’ and also satisfy the equation of continuity (2.5) everywhere in , which is exactly equation (1.13) for u.
Boundary conditions (1.14), (1.15) and (1.18).
The fact that function ’ satis es the Neumann boundary condition (2.3) is already involved in the de nition of the weak solution. The validity of the two conditions (2.6) and (2.7) directly follows from the condition v 2 D1;2( ). This implies that u satis es the boundary conditions (1.14) and (1.15). Thus, we still need to show that \su ciently smooth » weak solutions v + r’ (actually, conditions v 2 W 2;2( ) and r((2 + ) ’ P ( )) 2 L2( ) are needed) satisfy the boundary condition (2.8). We therefore return to the integral identities (2.9), (2.10) with general test functions , . (General in the sense that they satisfy all the requirements from De nition 2.1, but they do not need to have a compact support in .) Integrating again by parts, summing the identities, using (2.3), (2.6), (2.7) and nally, using also the already derived information that v, ’, satisfy equation (2.4), we arrive at Z curl v (n ) dS = 0: @ This equality can be rewritten in the form Z ( curl v) n dS = 0: (2.13) @ Since 2 D1;2( ), it coincides on @ with a gradient of some function 2 W 2;2( ) : = r on @ (See Bellout, Neustupa, Penel [2].) Using this form of in (2.13), we successively get
Table of contents :
I The compressible model
1 Introduction and notation of function spaces.
1.1 General considerations
1.2 Introduction to the studied model
1.3 Denition and basic properties of some function spaces
2 The studied model and the main results of the thesis.
2.1 From a classical formulation to the weak formulation
2.2 From the weak formulation to the classical formulation
2.2.1 Equations (1.12) and (1.13)
2.2.2 Boundary conditions (1.14), (1.15) and (1.18)
2.3 The main theorems
3 Mathematical preliminaries.
3.1 Useful tools from functional analysis and theory of partial dierential equations
3.2 The Leray{Schauder xed point theorem
II Approximate model I based on the Lions{Novotny{Straskraba approach.
4 Existence of approximations.
4.1 The boundary{value problem for approximations
4.2 Existence of approximations
4.2.1 The approximate density
4.2.2 The approximate velocity
5 The limit transitions for » ! 0 and ! 0.
5.1 Estimates independent of » and
5.2 Some limits for » ! 0
5.3 The equation of continuity: from D0( ) to D0(R3)
5.4 Fundamental lemmas to obtain renormalized equation of continuity
5.5 Strong convergence of »
5.6 The limit transition in the momentum equation and in the equation for eective pressure for » ! 0
5.7 -uniform estimates
5.8 Some limits for ! 0
5.9 Convergences when ! 0 for the continuity and the momentum equation in terms of WLf s , s 2 f ; 2; g
5.10 Strong convergence of density when ! 0
5.11 The limit transition in the momentum equation and in the equation for eective pressure for ! 0
6 Proof of Theorem 2.1.
6.1 Estimates independent of
6.2 Some limits for ! 0
6.3 Convergences when ! 0 for the continuity and the momentum equation in terms of WL e ; energy inequality
6.4 A new fundamental lemma
6.5 Strong convergence of density
6.6 The limit transition in the momentum equation and in the equation for eective pressure for ! 0
III Approximate model II, the setting with only one auxiliary small parameter.
7 Existence of approximations.
7.1 A formal justication of a new approach to system (2.3){(2.8)
7.2 Solution of the problem for density
7.3 Solution of the problem for the velocity
8 Proof of Theorem 2.2.
8.1 Estimates independent of »
8.2 Values of k for which the weak limit of » as » ! 0 is less or equal to k and « -uniform estimates
8.3 The limit process as » ! 0
8.3.1 Eective pressure
8.3.2 A bound for WLP(« ) »
8.3.3 The term WLP(« )
8.3.4 Strong convergence of the density
8.3.5 Better regularity for the divergence free part of the velocity
9 Final comments and remarks.
9.1 Originality of our approach due to the generalized impermeability boundary conditions
9.2 Energy inequalities
9.3 The non linear convective terms
9.4 The renormalization and the eective pressure
9.5 New results proved in the thesis
9.6 Conclusion
A Proofs of technical lemmas.
A.1 Technical Lemma A.1
A.2 Technical Lemma A.2
A.3 Technical Lemma A.3
A.4 Technical Lemma A.4
