On steady compressible Navier-Stokes equations in plane domains with corners. (English) Zbl 0844.35083
The following boundary value problem is considered in a bounded domain \(\Omega\subset \mathbb{R}^2\) with a corner point
\[
-\mu_1 \Delta \overline v- (\mu_1+ \mu_2)\nabla\text{ div } \overline v+ \nabla \rho= \rho\overline f- \rho(\overline v\cdot \nabla) \overline v,\quad x\in \Omega,
\]
\[ \text{div}(\rho\overline v)= 0,\quad x\in \Omega,\quad \overline v|_{\partial\Omega}= 0. \] Here \(\overline v= (v_1, v_2)\) is the unknown velocity vector, \(\rho> 0\) is an unknown density, \(\overline f\) is a given vector, \(\mu_1\), \(\mu_2\) are constants satisfying the condition \(\mu_1> 0\), \(\mu_2\geq - \mu_1\). The problem is investigated in weighted Sobolev spaces. It is proved that if \(\overline f\) is sufficiently small then the problem has a unique solution in a certain ball of a suitable function space. The authors use the method of decomposition proposed in [the second author and M. Padula, Arch. Ration. Mech. Anal. 126, No. 3, 243-297 (1994; Zbl 0809.76080)].
\[ \text{div}(\rho\overline v)= 0,\quad x\in \Omega,\quad \overline v|_{\partial\Omega}= 0. \] Here \(\overline v= (v_1, v_2)\) is the unknown velocity vector, \(\rho> 0\) is an unknown density, \(\overline f\) is a given vector, \(\mu_1\), \(\mu_2\) are constants satisfying the condition \(\mu_1> 0\), \(\mu_2\geq - \mu_1\). The problem is investigated in weighted Sobolev spaces. It is proved that if \(\overline f\) is sufficiently small then the problem has a unique solution in a certain ball of a suitable function space. The authors use the method of decomposition proposed in [the second author and M. Padula, Arch. Ration. Mech. Anal. 126, No. 3, 243-297 (1994; Zbl 0809.76080)].
Reviewer: I.Sh.Mogilevskij (Ferrara)
MSC:
| 35Q30 | Navier-Stokes equations |
| 76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |
| 35M10 | PDEs of mixed type |
Citations:
Zbl 0809.76080References:
| [1] | Beir?o da Veiga H.: OnL p -theory for then-dimensional, stationary comparissible Navier-Stokes equations and the incompressible limit for compressible fluids. The equilibrium solutions. Comm. Math. Phys.109 (1987), 229-248 · Zbl 0621.76074 · doi:10.1007/BF01215222 |
| [2] | Beir?o da Veiga H.: Boundary value problems for a class of first order partial differential equations in Sobolev spaces and applications to the Euler flow, Rend. Sem. Mat. Univ. Padova79 (1988), 247-273 · Zbl 0709.35082 |
| [3] | Borchers W., Pileckas K.: Existence, uniqueness and asymptotics of steady jets, Arch. Rat. Mech. Anal.120 (1992), 1-49 · Zbl 0772.76021 · doi:10.1007/BF00381276 |
| [4] | Dauge M.: Op?rateur de Stokes dans des espaces de Sobolev ? points sur des domains anguleux, Can. J. Math.34(4) (1982), 853-882 · doi:10.4153/CJM-1982-059-2 |
| [5] | Dauge M.: Stationary Stokes and Navier-Stokes systems on two and threedimensional domains with corners. Part I.: Linearized Equations, SIAM J. Math. Anal.20(1) (1989), 74-97 · Zbl 0681.35071 · doi:10.1137/0520006 |
| [6] | Dauge M.: Elliptic boundary value problems on corner domains, smoothness and asymptotics of solutions, Lecture Notes Math.1341 (1988) · Zbl 0668.35001 |
| [7] | Grisvard P.: Singularit?s des solutions du probl?me de Stokes dans un polygon, Universit? de Nice (preprint) (1979) |
| [8] | Grisvard P.: Elliptic problems in nonsmooth domains. Boston, London, Melbourne, Pitman Pub. Co. (1985) · Zbl 0695.35060 |
| [9] | Farwig R.: Stationary solutions of the Navier-Stokes equations with slip boundary conditions, Comm. Part. Diff. Eq.14(11) (1989), 1579-1606 · Zbl 0701.35127 · doi:10.1080/03605308908820667 |
| [10] | Kellog R.B., Osborn J.E.: A regularity result for the Stokes problem in a convex polygon, J. Funct. Anal.21 (1976), 397-431 · Zbl 0317.35037 · doi:10.1016/0022-1236(76)90035-5 |
| [11] | Kondrat’ev V.: Boundary value problems for elliptic equations in domains with conical or angular points. Trudy Moskov. Mat. Obsc.16 (1967), 209-292 (in Russian); English translation in Trans. Moscow Math. Soc.16 (1967), 227-313 |
| [12] | Kondrat’ev V.: Asymptotic of solution of the Navier-Stokes equations near the angular point of the boundary, J. Appl. Math. Mech.31 (1967), 125-129 · doi:10.1016/0021-8928(67)90073-1 |
| [13] | Kozlov V.A., Maz’ya V.G.: On quasihomogeneous solutions to the Dirichlet problem for the Stokes system in a dihedral angle, Univ. Link?ping LiTH-MAT-R-91-15 (preprint), 1991 |
| [14] | Kozlov V.A., Maz’ya V.G., Schwab C.: On singularities of solutions to the Dirichlet problem of hydrodynamics near the vertex of a 3-D cone, Univ. Link?ping LiTH-MAT-R-90-23 (preprint) (1990) |
| [15] | Kozlov V.A., Maz’ya V.G., Schwab C.: On singularities of solutions to the boundary value problems near the vertex of a rotational cone, Univ. Link?ping LiTH-MAT-R-91-24 (preprint) (1991) |
| [16] | Matsumura A., Nishida T.: Exterior stationary problems for the equations of motion of compressible viscous and heat conductive fluids. Proc. EQUADIFF 89. Ed. Dafermos, Ladas, Papanicolau, M. Dekker Inc. (1989), 473-479 · Zbl 0679.76076 |
| [17] | Maz’ya V.G., Plamenevskii B.A.: About asymptotics of solutions of Navier-Stokes equations near edges, Dokl. Acad. Nauk SSSR210(4) (1973), 803-806 |
| [18] | Maz’ya V.G., Plamenevskii B.A.: Problem of the motion of the fluid with a free surface in a vessel with edges, Problemy Mat. Analiza7, LGU (1979), 100-145 (in Russian) |
| [19] | Maz’ya V.G., Plamenevskii B.A.: On properties of solutions of threedimensional problems in the elasticity and hydrodynamics in domains with isolated singular points, Din. Sploschnoj Sredy50, Novosibirsk (1981), 99-121 (in Russian); English translation in Amer. Math. Soc. Transl.123 (1984), 109-123 |
| [20] | Maz’ya V.G., Plamenevskii B.A.: First boundary value problem for the equations of hydrodynamics, Zap. Nauchn. Sem. LOMI96 (1980), 179-196 (in Russian); English translation in J. Sov. Math.21(5) (1983), 777-783 |
| [21] | Maz’ya V.G., Plamenevskii B.A.: On the coefficients in the asymptotics of solutions of elliptic boundary value problems in domains with conical points, Math. Nachr.76 (1977), 29-60 (in Russian); English translation in Amer. Math. Soc. Transl.123(2) (1984), 57-88 · Zbl 0359.35024 · doi:10.1002/mana.19770760103 |
| [22] | Maz’ya V.G., Plamenevskii B.A.: Estimates inL p and H?lder classes and the Miranda-Agmon maximum principle for solutions of elliptic boundary value problems. Math. Nachr.81 (1978), 25-82 (in Russian); English translation in Amer. Math. Soc. Transl.123(2) (1984), 1-56 · Zbl 0371.35018 · doi:10.1002/mana.19780810103 |
| [23] | Maz’ya V.G., Plamenevskii B.A., Stupelis L.I.: The three-dimensional problem about the motion of fluid with the free boundary, Differential equations and their application, Vilnius,23 (1979), 29-153 (in Russian); English translation in Amer. Math. Soc. Transl.123(2) (1984), 171-268 |
| [24] | Mizohata S.: The theory of partial differential equations, Cambridge Univ. Press (1973) · Zbl 0263.35001 |
| [25] | Nazarov S.A., Plamenevskii B.A.: Elliptic boundary value problems in domains with picewise smooth boundary, Valter de Gruyter and Co, Berlin (1993) |
| [26] | Novotn? A.: Steady flows of viscous compressible fluids in exterior domains under small perturbations of great potential forces, Math. Meth. Model. Appl. Sci.3(6),(1993), 725-757 · Zbl 0803.76072 · doi:10.1142/S0218202593000370 |
| [27] | Novotn? A.: steady flows of viscous compressible fluids.L 2 approach, Proc. of EQUAM 92, Salvi R., Straskraba I., Eds. Math. Meth. Model. Appl. Sci. (in press) |
| [28] | Novotn? A.: On the steady transport equation. I.L p approach in domains with sufficiently smooth boundary Comm. Math. Univ. Carolinae (in press) |
| [29] | Novotn? A., Padula M.:L p -approach to steady flows of viscous compressible fluids in exterior domains, Arch. Rat. Mech. Anal.126 (1994), 243-297 · Zbl 0809.76080 · doi:10.1007/BF00375644 |
| [30] | Novotn? A., Padula M.: Existence and uniqueness of stationary solutions for viscous compressible heat conductive fluid with large potential and small nonpotential external forces, Sib. Math. J.34(5) (1993), 120-146 · Zbl 0808.76083 |
| [31] | Novotn? A., Penel P.: OnL p -approach for steady flows of viscous compressible heat conductive gas. Math. Meth. Model. Appl. Sci. (in press) |
| [32] | Osborn V.E.: Regularity of solutions of the Stokes problem in a polygonal domain, Num. Sol. of Part. Diff. Eq. III. Synspade 1975, Academic Press (1976), 393-421 |
| [33] | Padula M.: A representation formula for steady solutions of a compressible m fluid moving at low speed, Transp. Th. Stat. Phys.21 (1992), 593-614 · Zbl 0793.76071 · doi:10.1080/00411459208203800 |
| [34] | Padula M.: Existence and uniqueness for viscous steady compressible motions, Arch. Rat. Mech. Anal.77(2) (1987), 89-102 · Zbl 0644.76086 |
| [35] | Padula M.: Existence and uniqueness for viscous steady compressible motions, Proc. Sem. Fis. Mat. Trieste, Dinamica dei fluidi e dei gaz ionizzati (1982) · Zbl 0644.76086 |
| [36] | Pileckas K., Zajaczkowski W.: On the free boundary problem for stationary compressible Navier-Stokes equations, Comm. Math. Phys.129 (1990), 169-204 · Zbl 0715.35088 · doi:10.1007/BF02096785 |
| [37] | Solonnikov V.A.: Solvability of the problem of plane motion of a heavy viscous incompressible capillary liquid, partially filling a container, Izv. Akad. Nauk SSSR43 (1977), 203-236 (in Russian); English translation in Math. USSR-Izv.14 (1980), 193-221 |
| [38] | Solonnikov V.A.: On a free boundary problem for the system of Navier-Stokes equations, Trudy Sem. S.L. Sobolev2 (1978), 127-140 (in Russian) |
| [39] | Solonnikov V.A.: On the Stokes equation in domains with nonsmooth boundaries and on a viscous incompressible flow with a free surface, Nonlinear Part. Diff. Equations.3, College de France Seminar (1980/1981), 340-423 |
| [40] | Solonnikov V.A.: Solvability of the three-dimensional boundary value problem with a free surface for the stationary Navier-Stokes system, Zap. Nauchn. Sem. LOMI84 (1976), 252-285 (in Russian); English translation in J. Sov. Math.21 (3) (1983), 427-450 · Zbl 0414.35062 |
| [41] | Solonnikov V.A.: Solvability of three-dimensional boundary value problem with a free surface for the stationary Navier-Stokes system, Partial Differential Equations. Banach Center Publ.10 (1983), 361-403 · Zbl 0575.35071 |
| [42] | Solonnikov V.A., Zajaczkowski W.: Neumann problem for second order elliptic equations in domains with edges on the boundary, Zap. Nauchn. Sem. LOMI127 (1983), 7-48 (in Russian); English translation in J. Sov. Math.27 (2) (1984), 2561-2586 · Zbl 0518.35030 |
| [43] | Valli A.: On the existence of stationary solutions to compressible Navier-Stokes equations, Ann. Inst. H. Poincare,4(1) (1987), 99-113 · Zbl 0627.76080 |
| [44] | Valli A.: Periodic and stationary solutions for compressible Navier-Stokes equations via stability method, Ann. Sc. Norm. Sup. Pisa4 (1983), 607-647 · Zbl 0542.35062 |
| [45] | Zeidler E.: Vorlesungen ?ber nichtlinear? functional Analysis I (Fixpunkts?te), Leipzig, (1976) · Zbl 0326.47053 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
