Existence and Uniqueness of Global Strong Solutions for One-Dimensional Compressible Navier–Stokes Equations (original) (raw)

2008, SIAM Journal on Mathematical Analysis

https://doi.org/10.1137/060658199

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Abstract

We consider the Navier-Stokes equations for compressible viscous flu- ids in one dimension. It is a well known fact that if the initial data are smooth and the initial density is bounded by below by a positive constant, then a strong solution exists for a small time. In this paper, we show that under the same hypothesis, the density remains

Regularity of 1D compressible isentropic Navier–Stokes equations with density-dependent viscosity

Journal of Differential Equations, 2008

In this paper, we consider one-dimensional compressible isentropic Navier-Stokes equations with the viscosity depending on density and with the free boundary. The viscosity coefficient μ is proportional to ρ θ with θ > 0, where ρ is the density. The existence, uniqueness, regularity of global weak solutions in H 1 ([0, 1]) have been established by Xin and Yao in [Z. Xin, Z. Yao, The existence, uniqueness and regularity for one-dimensional compressible Navier-Stokes equations, preprint]. Furthermore, under certain assumptions imposed on the initial data, we improve the regularity result obtained in [Z. Xin, Z. Yao, The existence, uniqueness and regularity for one-dimensional compressible Navier-Stokes equations, preprint] by driving some new a priori estimates.

Global solutions of the compressible navier-stokes equations with larger discontinuous initial data

Communications in Partial Differential Equations, 2000

We prove the global existence of weak solutions to the Navier-Stokes equations for compressible, heat-conducting flow in one space dimension with large, discontinuous initial data, and we obtain a-priori estimates for these solutions which are independent of time, sufficient to determine their asymptotic behavior. In particular, we show that, as time goes to infinity, the solution tends to a constant state determined by the initial mass and the initial energy, and that the magnitudes of singularities in the solution decay to zero.

1D and 2D GLOBAL STRONG SOLUTIONS OF NAVIER STOKES EXISTENCE AND UNIQUENESS

2022

Consider the Navier-Stokes equation for a one-dimensional and two-dimensional compressible viscous liquid. It is a well-known fact that there is a strong solution locally in time when the initial data is smooth and the initial density is limited down by a positive constant. In this article, under the same hypothesis, I show that the density remains uniformly limited in time from the bottom by a positive constant, and therefore a strong solution exists globally in time. In addition, most existing results are obtained with a positive viscosity factor, but current results are true even if the viscosity factor disappears with density. Finally, I prove that this solution is unique in a class of weak solutions that satisfy the usual entropy inequalities. The point of this work is the new entropy-like inequalities that Bresch and Desjardins introduced into the shallow water system of equations. This discrepancy gives the density additional regularity (assuming such regularity exists first)...

Global strong solutions to compressible Navier–Stokes system with degenerate heat conductivity and density-depending viscosity

Communications in Mathematical Sciences

We consider the compressible Navier-Stokes system where the viscosity depends on density and the heat conductivity is proportional to a positive power of the temperature under stress-free and thermally insulated boundary conditions. Under the same conditions on the initial data as those of the constant viscosity and heat conductivity case ([Kazhikhov-Shelukhin. J. Appl. Math. Mech. 41 (1977)], we obtain the existence and uniqueness of global strong solutions. Our result can be regarded as a natural generalization of the Kazhikhov's theory for the constant heat conductivity case to the degenerate and nonlinear one under stress-free and thermally insulated boundary conditions.

Blowup of solutions for the compressible Navier–Stokes equations with density-dependent viscosity coefficients

Nonlinear Analysis: Theory, Methods & Applications, 2013

We consider blowup of classical solutions to compressible Navier-Stokes equations with revised Maxwell's law which can be regarded as a relaxation to the classical Newtonian flow. For this new model, we show that for some special large initial data, the life span of any C 1 solution must be finite. This shows much difference with Newtonian flow where the global existence of C 1 solutions is still an open and famous problem in fluid dynamics. We exploit the property of finite propagation speed and the methods from Sideris (1985) to prove our results.

On the Global Existence of Weak Solutions for the Navier–Stokes Equations of Compressible Fluid Flows

SIAM Journal on Mathematical Analysis, 2006

We study radial solutions of a singularly perturbed nonlinear elliptic system of the FitzHugh-Nagumo type. In a particular parameter range, we find a large number of layered solutions. First we show the existence of solutions whose layers are well separated from each other and also separated from the origin and the boundary of the domain. Some of these solutions are local minimizers of a related functional while the others are critical points of saddle type. Although the local minimizers may be studied by the Γ-convergence method, the reduction procedure presented in this paper gives a more unified approach that shows the existence of both local minimizers and saddle points. Critical points of both types are all found in the reduced finite dimensional problem. The reduced finite dimensional problem is solved by a topological degree argument. Next we construct solutions with an odd number of layers that cluster near the boundary, again using the reduction method. In this case the reduced finite dimensional problem is solved by a maximization argument. * Supported in part by ARC. † Phone: 61 2 9351 4141. Fax: 61 2 9351 4534.

On the isentropic compressible Navier-Stokes equation

2005

We consider the compressible Navier-Stokes equation with density dependent viscosity coefficients, focusing on the case where those coefficients vanish on vacuum. We prove the stability of weak solutions both in the torus and in the whole space in dimension 2 and 3. The pressure is given by p=rho^gamma, and our result holds for any gamma>1. In particular, we obtain the

Compact attractors for the Navier-Stokes equations of one-dimensional, compressible flow

Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 1999

We prove the existence of a compact attractor for the Navier-Stokes equations of compressible fluid flow in one space dimension. We also show that the large-time behavior of a given solution is entirely determined by its values for all time at a finite number of points, given in terms of a certain dimensionless quantity associated with a canonical scaling of the system. Our results are based on a well-posedness theory for these equations which goes beyond previously known results. In particular, we establish the global existence and regularity of solutions with large external forces and large, nonsmooth initial data, with regularity estimates independent of time. 0 AcadCmie des Sciences/Elsevier, Paris Attructeurs compacts pour les bquations de Navier-Stokes d 'un jluide compressible monodimensionel R&urn& Dans cette Note nous &tab&sons l'existence d'un attracteur compact pour les iquations de Navier-Stokes gouvernant unJuide compressible en une dimension de l'espace. Nous montrons aussi que le comportement ir l'infini des solutions est entiPrement dPterminP par la connaissance de la solution en tout temps sur un ensemble$ni de points, dont le nombre est major& par une constante associPe ir une renormalisation canonique du systkme. Nos r&.dtats sent basis sur une tht?orie d'existence qui va au de& des r&ultats connus. En particulier, nous montrons 1 'existence globale de solutions borne'es pour de grandes forces et don&es initiales peu r&ditre.s. 0 AcadCmie des ScienceslElsevier, Paris Version frangaise abr6g6e Dans cette Note nous Ctablissons l'existence d'un attracteur compact pour les equations de Navier-Stokes gouvemant un fluide compressible en une dimension de l'espace. Nous montrons aussi que le comportement B l'infini des solutions est entikrement determine par la connaissance de la solution en tout temps sur un ensemble fini de points, dont le nombre est majort par une constante associCe Note prCsentCe par Pierre-Louis LIONS. 0764-4442/99/03280239 0 AcadCmie des ScienceslElsevier, Paris 239 D. Hoff, M. Ziane a une renormalisation canonique du systeme (voir 131, [5]). Nos resultats sont bases sur une theorie d'existence qui va au dela des resultats connus. En particulier, nous montrons l'existence et la regularit& globale de solutions pour de grandes forces et donnees initiales peu regulieres. De plus,

Unique solvability of the initial boundary value problems for compressible viscous fluids

Journal de Mathématiques Pures et Appliquées, 2004

We study the Navier-Stokes equations for compressible barotropic fluids in a domain Ω ⊂ R 3. We first prove the local existence of the unique strong solution, provided the initial data satisfy a natural compatibility condition. The initial density needs not be bounded away from zero; it may vanish in an open subset (vacuum) of Ω or decay at infinity when Ω is unbounded. We also prove a blow-up criterion for the local strong solution, which is new even for the case of positive initial densities. Finally, we prove that if the initial vacuum is not so irregular, then the compatibility condition of the initial data is necessary and sufficient to guarantee the existence of a unique strong solution.

Well-posedness of three-dimensional isentropic compressible Navier-Stokes equations with degenerate viscosities and far field vacuum

2018

In this paper, the Cauchy problem for the three-dimensional (3-D) isentropic compressible Navier-Stokes equations is considered. When viscosity coefficients are given as a constant multiple of the density's power (ρ^δ with 0<δ<1), based on some analysis of the nonlinear structure of this system, we identify the class of initial data admitting a local regular solution with far field vacuum and finite energy in some inhomogeneous Sobolev spaces by introducing some new variables and initial compatibility conditions, which solves an open problem of degenerate viscous flow partially mentioned by Bresh-Desjardins-Metivier [3], Jiu-Wang-Xin [11] and so on. Moreover, in contrast to the classical theory in the case of the constant viscosity, we show that one can not obtain any global regular solution whose L^∞ norm of u decays to zero as time t goes to infinity.

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Journal of Mathematical Fluid Mechanics, 2011

We consider the Cauchy problem for the equations of spherically symmetric motions in R 3 , of a selfgravitating barotropic gas, with possibly non monotone pressure law, in two different situations: in the first one we suppose that the viscosities µ(ρ), and λ(ρ) are density-dependent and satisfy the Bresch-Desjardins condition, in the second one we consider constant densities. In the two cases, we prove that the problem admits a global weak solution, provided that the polytropic index γ satisfy γ > 1. . 76N10,35Q30.

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For the isentropic compressible fluids in one-space dimension, we prove that the Navier-Stokes equations with density-dependent viscosity have neither forward nor backward self-similar strong solutions with finite kinetic energy. Moreover, we obtain the same result for the nonisentropic compressible gas flow, that is, for the fluid dynamics of the Navier-Stokes equations coupled with a transport equation of entropy. These results generalize those in Guo and Jiang's work (2006) where the one-dimensional compressible fluids with constant viscosity are considered.

High Regularity of Solutions of Compressible Navier-Stokes Equations

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We study the Navier-Stokes equations for compressible barotropic fluids in a bounded or unbounded domain Ω of R 3. The initial density may vanish in an open subset of Ω or to be positive but vanish at space infinity. We first prove the local existence of solutions (ρ (j) , u (j)) in C([0, T * ]; H 2(k−j)+3 × D 1 0 ∩ D 2(k−j)+3 (Ω)), 0 ≤ j ≤ k, k ≥ 1 under the assumptions that the data satisfy compatibility conditions and that the initial density is sufficiently small. To control the nonnegativity or decay at infinity of density, we need to establish a boundary value problem of (k+1)-coupled elliptic system which may not be in general solvable. The smallness condition of initial density is necessary for the solvability, which is not necessary in case that the initial density has positive lower bound. Secondly, we prove the global existence of smooth radial solutions of isentropic compressible Navier-Stokes equations on a bounded annulus or a domain which is the exterior of a ball under a smallness condition of initial density.

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The purpose of this work is to investigate the problem of global existence of weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids. A certain class of density and temperature dependent viscosity and conductivity coefficients is considered. This result extends the work of P.-L. Lions [Compacité des solutions des équations de Navier-Stokes compressibles isentropiques. C.R. Acad. Sci. Paris, Sér I. 317, (1993), p. 115-120] restricted to barotropic flows, and provides weak solutions "à la Leray" to a full compressible model -including the internal energy evolution equation with thermal conduction effects -. A partial answer is therefore given to this currently widely open problem, described for instance in [Mathematical topics in fluid dynamics, Vol.2, Compressible models. Oxford Science Publication, Oxford, (1998)]. The proof is based on the generalization to the temperature dependent case, of a new mathematical entropy equality derived by the authors in [Some diffusive capillary models of Korteweg type. C. R. Acad. Sciences, Paris, Section Mécanique. Vol. 332, no 11 (2004), p 881-886]. The actual construction of approximate solutions, based on additional regularizing effects such as capillarity, will be provided in [12].

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The present note extends to smooth enough bounded domains recent results about barotropic compressible Navier-Stokes systems with density dependent viscosity coefficients. We show how to get the existence of global weak solutions for both classical Dirichlet and Navier boundary conditions on the velocity, under appropriate constraints on the initial density profile and domain curvature. An additional turbulent drag term in the momentum equation is used to handle the construction of approximate solutions.

Global well-posedness of regular solutions to the three-dimensional isentropic compressible Navier-Stokes equations with degenerate viscosities and vacuum

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In this paper, the Cauchy problem for the three-dimensional (3-D) isentropic compressible Navier-Stokes equations with degenerate viscosities is considered. By introducing some new variables and making use of the " quasi-symmetric hyperbolic"-"degenerate elliptic" coupled structure to control the behavior of the fluid velocity, we prove the global-in-time well-posedness of regular solutions with vacuum for a class of smooth initial data that are of small density but possibly large velocities. Here the initial mass density is required to decay to zero in the far field, and the spectrum of the Jacobi matrix of the initial velocity are all positive. The result here applies to a class of degenerate density-dependent viscosity coefficients, is independent of the BD-entropy, and seems to be the first on the global existence of smooth solutions which have large velocities and contain vacuum state for such degenerate system in three space dimensions.

Global well-posedness for compressible Navier-Stokes equations with highly oscillating initial velocity

Communications on Pure and Applied Mathematics, 2010

proved the global well-posedness of the incompressible Navier-Stokes equations for a class of highly oscillating data. In this paper, we prove the global well-posedness for the compressible Navier-Stokes equations in the critical functional framework with the initial data close to a stable equilibrium. Especially, this result allows us to construct global solutions for the highly oscillating initial velocity. The proof relies on a new estimate for the hyperbolic/parabolic system with convection terms. By Lemma 2.1, we have

On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities

manuscripta mathematica, 2006

We study the Navier-Stokes equations for compressible barotropic fluids in a bounded or unbounded domain Ω of R 3. We first prove the local existence of solutions (ρ, u) in C([0, T * ]; (ρ ∞ + H 3 (Ω)) × (D 1 0 ∩ D 3)(Ω)) under the assumption that the data satisfies a natural compatibility condition. Then deriving the smoothing effect of the velocity u in t > 0, we conclude that (ρ, u) is a classical solution in (0, T * *) × Ω for some T * * ∈ (0, T * ]. For these results, the initial density needs not be bounded below away from zero and may vanish in an open subset (vacuum) of Ω.

Weak solutions of the compressible isentropic Navier-Stokes equations

Applied Mathematics Letters, 1999

This note is devoted to some properties of weak solutions to the compressible isentropic Navier-Stokes equations in dimension 2 or 3. First, the restrictions on the pressure growth for the known global existence and stability results are illustrated through explicit radial solutions. On the other hand, an improved global time regularity is derived for the momentum.

Convergence of the solutions of the compressible to the solutions of the incompressible Navier-Stokes equations

Advances in Applied Mathematics, 1991

We study the slightly compressible Navier-Stokes equations. We first consider the Cauchy problem, periodic in space. Under appropriate assumptions on the initial data, the solution of the compressible equations consists-to first order-of a solution of the incompressible equations plus a function which is highly oscillatory in time. We show that the highly oscillatory part (the sound waves) can be described by wave equations, at least locally in time. We also show that the bounded derivative principle is valid; i.e., the highly oscillatory part can be suppressed by initialization. Besides the Cauchy problem, we also consider an initial-boundary value problem. At the inflow boundary, the viscous term in the Navier-Stokes equations is important. We consider the case where the compressible pressure is prescribed at inflow. In general, one obtains a boundary layer in the pressure; in the velocities a boundary layer is not present to first approximation. 8

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References (25)

  1. Herbert Amann. Compact embeddings of vector-valued Sobolev and Besov spaces. Glas. Mat. Ser. III, 35(55)(1):161-177, 2000. Dedicated to the memory of Branko Najman.
  2. Didier Bresch and Benoît Desjardins. Sur un modèle de Saint- Venant visqueux et sa limite quasi-géostrophique. C. R. Math. Acad. Sci. Paris, 335(12):1079-1084, 2002.
  3. Didier Bresch and Benoît Desjardins. Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model. Comm. Math. Phys., 238(1-2):211- 223, 2003.
  4. Didier Bresch and Benoît Desjardins. Some diffusive capillary models of korteweg type. C. R. Math. Acad. Sci. Paris, Section Mécanique, 332(11):881-886, 2004.
  5. Didier Bresch and Benoît Desjardins. Existence globale de solutions pour les équations de navier-stokes compressibles complètes avec conduction thermique. Submitted, 2005.
  6. Didier Bresch, Benoît Desjardins, and Chi-Kun Lin. On some com- pressible fluid models: Korteweg, lubrication, and shallow water systems. Comm. Partial Differential Equations, 28(3-4):843-868, 2003.
  7. Sydney Chapman and T. G. Cowling. The mathematical theory of non-uniform gases. An account of the kinetic theory of viscosity, thermal conduction and diffusion in gases. Third edition, prepared in co-operation with D. Burnett. Cambridge University Press, Lon- don, 1970.
  8. C. M. Dafermos. The second law of thermodynamics and stability. Arch. Rational Mech. Anal., 70(2):167-179, 1979.
  9. Eduard Feireisl. On the motion of a viscous, compressible, and heat conducting fluid. Indiana Univ. Math. J., 53(6):1705-1738, 2004.
  10. Eduard Feireisl, Antonín Novotný, and Hana Petzeltová. On the existence of globally defined weak solutions to the Navier-Stokes equations. J. Math. Fluid Mech., 3(4):358-392, 2001.
  11. David Hoff. Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial data. Trans. Amer. Math. Soc., 303(1):169-181, 1987.
  12. David Hoff. Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data. J. Differential Equations, 120(1):215-254, 1995.
  13. David Hoff. Global solutions of the equations of one-dimensional, compressible flow with large data and forces, and with differing end states. Z. Angew. Math. Phys., 49(5):774-785, 1998.
  14. David Hoff and Joel Smoller. Non-formation of vacuum states for compressible Navier-Stokes equations. Comm. Math. Phys., 216(2):255-276, 2001.
  15. A. V. Kazhikhov and V. V. Shelukhin. Unique global solution with respect to time of initial-boundary value problems for one- dimensional equations of a viscous gas. Prikl. Mat. Meh., 41(2):282- 291, 1977.
  16. Pierre-Louis Lions. Mathematical topics in fluid mechanics. Vol. 2, volume 10 of Oxford Lecture Series in Mathematics and its Appli- cations. The Clarendon Press Oxford University Press, New York, 1998. Compressible models, Oxford Science Publications.
  17. Tai-Ping Liu, Zhouping Xin, and Tong Yang. Vacuum states for compressible flow. Discrete Contin. Dynam. Systems, 4(1):1-32, 1998.
  18. Akitaka Matsumura and Takaaki Nishida. The initial value prob- lem for the equations of motion of compressible viscous and heat- conductive fluids. Proc. Japan Acad. Ser. A Math. Sci., 55(9):337- 342, 1979.
  19. Antoine Mellet and Alexis Vasseur. On the compressible navier- stokes equation. Preprint, 2005.
  20. Mari Okada, Šárka Matušů-Nečasová, and Tetu Makino. Free boundary problem for the equation of one-dimensional motion of compressible gas with density-dependent viscosity. Ann. Univ. Fer- rara Sez. VII (N.S.), 48:1-20, 2002.
  21. Denis Serre. Solutions faibles globales des équations de Navier- Stokes pour un fluide compressible. C. R. Acad. Sci. Paris Sér. I Math., 303(13):639-642, 1986.
  22. Denis Serre. Sur l'équation monodimensionnelle d'un fluide visqueux, compressible et conducteur de chaleur. C. R. Acad. Sci. Paris Sér. I Math., 303(14):703-706, 1986.
  23. V. A. Solonnikov. The solvability of the initial-boundary value problem for the equations of motion of a viscous compressible fluid. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 56:128-142, 197, 1976. Investigations on linear operators and theory of functions, VI.
  24. Tong Yang, Zheng-an Yao, and Changjiang Zhu. Compressible Navier-Stokes equations with density-dependent viscosity and vac- uum. Comm. Partial Differential Equations, 26(5-6):965-981, 2001.
  25. Tong Yang and Changjiang Zhu. Compressible Navier-Stokes equa- tions with degenerate viscosity coefficient and vacuum. Comm. Math. Phys., 230(2):329-363, 2002.