On weak solutions to the 1d compressible Navier-Stokes equations: a Lipschitz continuous dependence on data in weaker norms and an error of their homogenization

We deal with the global in time weak solutions to the 1D compressible Navier-Stokes system of equations for large discontinuous initial data and nonhomogeneous boundary conditions of three standard types. We prove the Lipschitz-type continuous dependence of the solution $(η,u,θ)$, in a norm slightly stronger than $L^{2,\infty}(Q)\times L^2(Q)\times L^2(Q)$, on the initial data $(η^0,u^0,e^0)$ in a norm of $L^2(Ω)\times H^{-1}(Ω)\times H^{-1}(Ω)$-type and also on the free terms in all the equations in some dual norms. Here $η$, $u$ and $θ$ are the specific volume, velocity and absolute temperature as well as $η^0$, $u^0$ and $e^0$ are the initial specific volume, velocity and specific total energy, and $Q=Ω\times (0,T)$. We also apply this result to the case of discontinuous rapidly oscillating, with the period $\varepsilon$, initial data and free terms and derive an estimate $O(\varepsilon)$ for the difference between the solutions to the Navier-Stokes equations and their Bakhvalov-Eglit two-scale homogenized version with averaged data.

💡 Research Summary

The paper addresses two intertwined problems for the one‑dimensional compressible Navier‑Stokes system with large, possibly discontinuous data: (i) a quantitative Lipschitz‑type continuous dependence of weak solutions on the initial data and source terms, and (ii) an error estimate for the two‑scale homogenization of the system when the data oscillate rapidly with period ε.

1. Functional setting and auxiliary linear results.
The authors work in a framework that is weaker than the classical H¹‑based theory. The initial specific volume η⁰ belongs to L²(Ω), while the initial velocity u⁰ and specific total energy e⁰ belong to H⁻¹(Ω). The source terms in the momentum and energy equations are assumed to lie in dual spaces associated with a family of anisotropic Sobolev‑type spaces V²;κ,m*. Two propositions (Proposition 1 and Proposition 2) give a priori estimates for linear one‑dimensional parabolic problems with mixed boundary conditions (Dirichlet, Neumann, Robin). The estimates control the L²∩L^∞(0,T;H⁻¹) norm of the solution by the H⁻¹ norm of the initial data, the L^{4/3} norm of the Dirichlet data, the L¹ norm of the Neumann data, and appropriate dual norms of the lower‑order terms. The proofs rely on integration by parts identities, the definition of the dual norms, and a Gronwall‑Bellman lemma (Lemma 1) that bounds an L^r‑norm by a lower‑order L^{r₁}‑norm.

2. Lipschitz continuous dependence for the full Navier‑Stokes system.
Using the linear theory, the authors treat the full nonlinear system (mass, momentum, and internal‑energy balances) in Lagrangian coordinates. By rewriting the equations in a form that isolates the linear diffusion operators (κ D v) and treating the nonlinear terms (pressure, viscous stress, heat flux) as perturbations, they construct a five‑step “length‑estimate” argument. This yields the main theorem: for two weak solutions (η,u,θ) and (˜η,˜u,˜θ) the quantity

‖η−˜η‖{L^{2,∞}(Q)} + ‖u−˜u‖{L²(Q)} + ‖θ−˜θ‖_{L²(Q)}

is bounded by a constant C (depending only on bounds for κ, the time horizon T, and the norms of the data) multiplied by the sum of the H⁻¹‑type differences of the initial data and the dual‑norm differences of the source terms. The result holds uniformly for the three standard boundary conditions and improves earlier theorems by allowing discontinuous data and weaker norms for the solution.

3. Rapidly oscillating data and two‑scale homogenization.
The second part considers initial data and source terms of the form f(x, x/ε) with ε‑periodic dependence in the fast variable y = x/ε. The authors perform a two‑scale asymptotic expansion, introducing macroscopic variables (x,t) and microscopic variables (y,τ). The limit system is the Bakhvalov‑Eglit homogenized equations, where the coefficients and the averaged initial data appear as spatial averages over the periodic cell. A subtle point is the averaging of the temperature initial data, which requires a κ‑weighted mean rather than a simple arithmetic mean.

Applying the Lipschitz continuity theorem to the difference between the ε‑problem and the homogenized problem, and exploiting the periodicity to bound the oscillatory remainders, they obtain the error estimate

‖(η_ε,u_ε,θ_ε) − (η̄,ū,θ̄)‖_{L^{2,∞}×L²×L²} = O(ε).

Moreover, in stronger norms they prove

‖·‖{C(0,T;L²)} = O(ε^{1/2}), ‖·‖{L^∞(Q)} = O(ε^{1/4}).

These rates improve upon earlier barotropic results (which typically give O(ε^{1/2}) only) and hold despite the presence of nonlinear heat conduction and source terms.

4. Significance and outlook.
The paper provides a rigorous quantitative link between small perturbations of initial/boundary data and the resulting change in weak solutions, a tool valuable for sensitivity analysis, numerical error control, and optimal control of compressible flows. The homogenization error estimate demonstrates that the two‑scale Bakhvalov‑Eglit model faithfully reproduces the original dynamics up to O(ε) even for discontinuous data, opening the way to efficient multiscale simulations. The authors suggest extensions to multidimensional settings and to non‑periodic microstructures as promising future directions.