On the stationary transport equations. (English) Zbl 1139.35327
Summary: We find solutions in \(L^{1}\) and \(L^\infty \) for the general domains for certain transport equations arising in the theory of the compressible Navier-Stokes equations. Some a priori estimates in \(L^{1}\) and \(L^\infty \) have been found. Then, the asymptotic behavior of \(L^\infty \) solutions is found. We also consider solutions in \(\mathcal H^1\) and BMO. We show the uniqueness of the solutions in these four function spaces.
MSC:
| 35D05 | Existence of generalized solutions of PDE (MSC2000) |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35L60 | First-order nonlinear hyperbolic equations |
| 35Q30 | Navier-Stokes equations |
| 35B45 | A priori estimates in context of PDEs |
| 35D10 | Regularity of generalized solutions of PDE (MSC2000) |
References:
| [1] | Beirão da Veiga, H., Existence results in Sobolev spaces for a stationary transport equation, Ricerche Mat., 36, suppl., 173-184 (1987) · Zbl 0691.35087 |
| [2] | Beirão da Veiga, H., An \(L^p\)-theory for the \(n\)-dimensional, stationary compressible, Navier-Stokes equations, and the incompressible limit for compressible fluids. The equilibrium solutions, Comm. Math. Phys., 109, 229-248 (1987) · Zbl 0621.76074 |
| [3] | DiPerna, R. J.; Lions, P. L., Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98, 511-547 (1989) · Zbl 0696.34049 |
| [4] | Evans, L. C., (Partial Differential Equations. Partial Differential Equations, Graduate Studies in Mathematics, vol. 19 (1998), American Math. Soc.) · Zbl 0902.35002 |
| [5] | Farwig, R., Stationary solutions of the Navier-Stokes equations with slip boundary conditions, Comm. Partial Differential Equations, 14, 1579-1606 (1989) · Zbl 0701.35127 |
| [6] | Fichera, G., Sulle equazioni diff. lineari ellittico-paraboliche del secondo ordine, Acad. Naz. Lincei, Mem. Cl. Sc. Fis. Mat. Nat., Sez. I, 5, 1-30 (1991) · Zbl 0075.28102 |
| [7] | Friedrichs, K. O., Symmetric positive linear differential equations, Comm. Pure Appl. Math., 11, 333-418 (1958) · Zbl 0083.31802 |
| [8] | Kohn, J. J.; Nirenberg, L., Degenerate elliptic-parabolic equations of second order, Comm. Pure Appl. Math., 20, 797-872 (1967) · Zbl 0153.14503 |
| [9] | Lax, P. D.; Phillips, R. S., Local boundary conditions for dissipative symmetric linear differential operators, Comm. Pure Appl. Math., 13, 427-455 (1960) · Zbl 0094.07502 |
| [10] | Novotný, A., About steady transport equation \(I — L^p\)-approach in domains with smooth boundaries, Comment. Math. Univ. Carolin., 37, 1, 43-89 (1996) · Zbl 0852.35115 |
| [11] | Novotný, A.; Padula, M., \(L^p\)-approach to steady flows of viscous compressible fluid in exterior domains, Arch. Ration. Mech. Anal., 126, 243-297 (1994) · Zbl 0809.76080 |
| [12] | Oleinik, O. A.; Radkevic, E. V., Second order equations with nonnegative characteristic form, Amer. Math. Soc. English Trans. (1973) |
| [13] | Stein, E. M., Harmonic analysis: Real-variable methods, orthogonality and oscillatory integrals, (Princeton Mathematical Series, vol. 43 (1993)) · Zbl 0821.42001 |
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