Global well-posedness of non-heat conductive compressible Navier–Stokes equations in 1D

Abstract

In this paper, the initial-boundary value problem of the 1D full compressible Navier-Stokes equations with positive constant viscosity but with zero heat conductivity is considered. Global well-posedness is established for any H 1 initial data. The initial density is assumed only to be nonnegative, and, thus, is not necessary to be uniformly away from vacuum. Comparing with the well-known result of Kazhikhov-Shelukhin (Kazhikhov, A. V.; Shelukhin, V. V.: Unique global solution with respect to time of initial boundary value problems for one-dimensional equations of a viscous gas, J. Appl. Math. Mech., 41 (1977), 273-282.), the heat conductive coefficient is zero in this paper, and the initial vacuum is allowed. 1 2 JINKAI LIThe compressible Navier-Stokes equations have been extensively studied. In the absence of vacuum, i.e., the case that the density has a uniform positive lower bound, the local well-posedness was proved long time ago by Nash [40], Itaya [18],, Tani [44], Valli [45], and Lukaszewicz [35]; uniqueness was proved even earlier by Graffi [14] and Serrin [43]. Global well-posedness of strong solutions in 1D has been well-known since the works by Kanel [22], and Kazhikhov [23]; global existence and uniqueness of weak solutions was also established, see, e.g., 53], Chen-Hoff-Trivisa [1], and Jiang-Zlotnik [21], and see for the result on the large time behavior. The corresponding global well-posedness results for the multi-dimensional case were established only for small perturbed initial data around some non-vacuum equilibrium or for spherically symmetric large initial data, see, e.g., , Ponce [41], Valli-Zajaczkowski [46], Deckelnick [9], Jiang [19], Hoff [15], Kobayashi-Shibata [25], Danchin [7], Chen-Miao-Zhang [2], Chikami-Danchin [3], Dachin-Xu [8], Fang-Zhang-Zi [10], and the references therein.In the presence of vacuum, that is the density may vanish on some set or tends to zero at the far field, global existence of weak solutions to the isentropic compressible Navier-Stokes equations was first proved by Lions [33,34], with adiabatic constant γ ≥ 9 5 , and later generalized by Feireisl-Novotný-Petzeltová [11] to γ > 3 2 , and further by Jiang-Zhang [20] to γ > 1 for the axisymmetric solutions. For the full compressible Navier-Stokes equations, global existence of the variational weak solutions was proved by Feireisl [12,13], which however is not applicable for the ideal gases. Local wellposedness of strong solutions to the full compressible Navier-Stokes equations, in the presence of vacuum, was proved by Cho-Kim [6], see also Salvi-Straškraba [42], Cho-Choe-Kim [4], and Cho-Kim [5] for the isentropic case. Same to the nonvacuum case, the global well-posedness in 1D also holds for the vacuum case, for arbitrary large initial data, see the recent work by the author [27]. Generally, one can only expect the solutions in the homogeneous Sobolev spaces, see Li-Wang-Xin [26]. Global existence of strong solutions to the multi-dimensional compressible Navier-Stokes equations, with small initial data, ...

Global strong solutions to the Cauchy problem of the planar non-resistive magnetohydrodynamic equations with large initial data

“…which reduces to the one in [24,25] if removing those terms involving h and ω. The key is to get the L ∞ (0, T ; L 2 ) estimate of G, which is expected to be achieved by testing (2.4) with JG.…”

“…In order to prove the global existence, one has to carry out suitable a priori estimates which are finite up to any finite time. Since system (2.2) contains the compressible Navier-Stokes equations without heat conductivity as a subsystem, we attempt to adopt the arguments in Li [24] and Li-Xin [25] to achieve these a priori estimates. As already shown in [24,25], the effective viscous flux, i.e., the quantity λ u y J − P there, plays a central role in the proof.…”

“…Since system (2.2) contains the compressible Navier-Stokes equations without heat conductivity as a subsystem, we attempt to adopt the arguments in Li [24] and Li-Xin [25] to achieve these a priori estimates. As already shown in [24,25], the effective viscous flux, i.e., the quantity λ u y J − P there, plays a central role in the proof. It is reasonable to believe that it is also the case for system (2.2).…”

On global classical solutions to 1D compressible Navier‐Stokes equations with density‐dependent viscosity and vacuum

“…For the initial density away from vacuum, there are many results concerning the global existence and large-time dynamics of solutions to the one-dimensional(1D) problem, see previous studies [1][2][3][4][5][6][7] and the references therein. When the vacuum is allowed initially, as emphasized in many papers related to compressible fluid dynamics, [8][9][10][11][12][13][14][15][16][17][18][19][20][21] the presence of vacuum is one of the major difficulties in discussing the well-posedness of solutions to the compressible Navier-Stokes equations. For the case of density-dependent viscosity, Liu-Xin-Yang 17 illustrates that the viscosity depends on the temperature for the non-isentropic case and thus on the density for the isentropic case.…”

“…9 Remark 4. In Theorem 1, we obtain the time-independent upper bound of the density (11) and the large-time behavior of the velocity (12), which are in sharp contrast to Ding et al 9 where the corresponding a priori estimates depend on time. Moreover, Theorem 1 also generalized the similar results in other studies 6,7 without initial vacuum to the case that the initial density admits vacuum.…”