Global-in-time existence of finite energy weak solutions to a relaxed Navier-Stokes-Korteweg model

Abstract: We consider a parabolic relaxation formulation of the compressible Navier-Stokes-Korteweg system and prove the global-in-time existence of finite energy weak solutions to the associated initial-boundary-value-problem. Our proof is based on a three-level approximation scheme, a weak compactness property of the effective viscous flux, and parabolic regularity estimates. Our result holds for a broad variety of non-monotone pressure functions and generalizes the corresponding results known for the compressible Navier-Stokes equations to the relaxation system.

• The paper proves global-in-time existence of finite-energy weak solutions in bounded two- and three-dimensional domains for every pressure-growth exponent γ > 1, including non-monotone Van der Waals laws.
• The authors combine three-level regularization, Galerkin approximation, parabolic estimates, generalized effective-viscous-flux compactness, and renormalized continuity equations to establish strong density convergence.
• The relaxation energy controls density through quadratic coupling and gradient terms, overcoming compactness difficulties and providing an existence foundation for relaxation limits and future homogenization studies.

The model and its motivation

This paper by Oschmann and Wendt establishes the global-in-time existence of finite energy weak solutions for the initial-boundary-value problem (IBVP) associated with a parabolic relaxation formulation of the isothermal compressible Navier–Stokes–Korteweg equations (NSKE) in spatial dimension $d \in \{2,3\}$ (2608.12969). The NSKE describe a diffuse-interface model for a homogeneous compressible viscous two-phase fluid, with density $\rho$ and velocity $u$ governed by a continuity equation and a momentum equation containing a third-order differential operator $\kappa \rho \nabla \Delta \rho$ modeling capillary effects. For two-phase applications, the pressure function $p$ is typically of Van-der-Waals type: monotonically increasing on $[0,a] \cup [b,\infty)$ and decreasing on $(a,b)$, where $(a,b)$ is called the spinodal region.

Two analytical difficulties distinguish the NSKE from the compressible Navier–Stokes equations (NSE): the third-order operator complicates compactness arguments in a weak solution framework, and the non-monotone pressure renders density compactness delicate. Numerically, the Jacobian of the first-order flux has complex eigenvalues in the spinodal region, precluding standard hyperbolicity-based finite-volume methods.

The relaxation system (rNSKE) addresses these issues by introducing an artificial unknown $c$, the relaxation parameter, satisfying an additional linear equation:

$\partial_t \rho + \operatorname{div}(\rho u) = 0,$

$\rho$0

$\rho$1

Here $\rho$2 and $\rho$3 are coupling coefficients; the case $\rho$4 yields an elliptic relaxation equation (Rohde's system), while $\rho$5 gives the parabolic version of Hitz–Keim–Munz–Rohde. Crucially, the momentum equation can be rewritten using the artificial pressure $\rho$6, which becomes non-decreasing for Van-der-Waals-type $\rho$7 when $\rho$8 is large enough. In the relaxation limit $\rho$9 (with $u$0, or $u$1 and $u$2), the rNSKE formally approach the NSKE, and rigorous convergence results exist both in smooth settings and, for $u$3, in the class of finite energy weak solutions. However, those convergence results postulated global-in-time existence of such weak solutions — a gap this paper fills.

Main result

The main theorem asserts that for bounded domains $u$4, $u$5, with $u$6 boundary, any admissible pressure function with growth rate $u$7, and initial data $u$8, $u$9, $\kappa \rho \nabla \Delta \rho$0 satisfying the standard vacuum compatibility conditions, there exists a finite energy weak solution on $\kappa \rho \nabla \Delta \rho$1 for arbitrary $\kappa \rho \nabla \Delta \rho$2. Here $\kappa \rho \nabla \Delta \rho$3.

A pressure function is admissible if it is continuous, nonnegative, vanishing at zero, and satisfies the growth bounds $\kappa \rho \nabla \Delta \rho$4 and $\kappa \rho \nabla \Delta \rho$5. This class encompasses isentropic laws $\kappa \rho \nabla \Delta \rho$6, general monotone laws with polynomial growth of $\kappa \rho \nabla \Delta \rho$7, and non-monotone Van-der-Waals-type pressures. A key structural observation is that admissible pressures admit a decomposition $\kappa \rho \nabla \Delta \rho$8 into a monotone part $\kappa \rho \nabla \Delta \rho$9 with $p$0 and a non-monotone part $p$1 that is smooth and compactly supported; the corresponding potential $p$2 is convex while $p$3 need not be.

The definition of finite energy weak solution mirrors the NSE theory: weak satisfaction of all three equations, initial data attained a.e., and the energy inequality

$p$4

with energy functional

$p$5

Notably, the result holds for the entire regime $p$6, which is strictly larger than the regime $p$7 known for the compressible NSE via the Lions–Feireisl–Novotný–Petzeltová theory. This improvement is attributable to the quadratic terms $p$8 and $p$9 in the energy, which provide an a priori bound on the density directly in terms of the initial data. Furthermore, no restrictions are imposed on the coupling coefficients: the artificial pressure $[0,a] \cup [b,\infty)$0 is not required to be monotone, although monotonicity can be arranged for sufficiently large $[0,a] \cup [b,\infty)$1.

Proof strategy: three-level approximation

The proof adapts the approximation scheme of Feireisl–Novotný–Petzeltová to the rNSKE. Three new difficulties arise relative to the NSE: a different energy structure, possible pressure non-monotonicity, and a nonlinear coupling between the momentum equation and the parabolic equation for $[0,a] \cup [b,\infty)$2. It is a priori unclear whether the artificial viscosity regularization of the density is compatible with the rNSKE; the paper shows that the error term it produces in the energy inequality can be absorbed by the corresponding dissipative term.

The scheme regularizes the system with two parameters: an artificial viscosity $[0,a] \cup [b,\infty)$3 in the continuity equation and momentum equation, and an artificial pressure term $[0,a] \cup [b,\infty)$4 in $[0,a] \cup [b,\infty)$5.

Existence for the doubly regularized system

For fixed $[0,a] \cup [b,\infty)$6 and $[0,a] \cup [b,\infty)$7 (with $[0,a] \cup [b,\infty)$8 playing the role of the viscosity coefficient), existence proceeds via a Galerkin approximation of the momentum equation while the continuity and parabolic equations are solved strongly. Local-in-time existence follows from a Banach fixed point argument built on solution operators $[0,a] \cup [b,\infty)$9 (transport equation for $(a,b)$0) and $(a,b)$1 (linear parabolic equation for $(a,b)$2). Globalization uses the energy identity, whose right-hand side terms $(a,b)$3 and $(a,b)$4 are controlled by Young's inequality and Grönwall's lemma; crucially, the dissipative terms $(a,b)$5 and $(a,b)$6 absorb the error introduced by the artificial viscosity.

Passing $(a,b)$7 in the Galerkin index requires uniform bounds independent of $(a,b)$8: energy-based bounds, parabolic regularity estimates giving $(a,b)$9 bounded in $(a,b)$0 and $(a,b)$1, and $(a,b)$2 estimates for the continuity equation yielding exponents $(a,b)$3 and $(a,b)$4. Strong convergence of $(a,b)$5 in $(a,b)$6 and of $(a,b)$7 in $(a,b)$8 follows from Aubin–Lions compactness, allowing identification of all nonlinear weak limits including $(a,b)$9.

Vanishing artificial viscosity

For fixed $c$0, the limit $c$1 requires improved equi-integrability of $c$2, obtained by testing the momentum equation against Bogovskiĭ-operator-based multipliers. Strong convergence of the density then rests on a weak compactness relation for the effective viscous flux, derived via a general proposition proved in the appendix. That proposition extends the classical statement of Novotný–Straskraba by allowing one additional force term in the momentum equation and one additional source term in the continuity equation, and covers both $c$3 and $c$4; the authors note this extension is of independent interest beyond the present application.

The compactness relation feeds into the Feireisl technique for non-monotone pressures: exploiting convexity of $c$5 and $c$6 for suitable $c$7, together with renormalized solutions of the continuity equation, one obtains $c$8, and a Grönwall argument forces equality, hence strong convergence in $c$9 and identification $\partial_t \rho + \operatorname{div}(\rho u) = 0,$0. Passing to the limit in the energy inequality requires care because the limit energy contains only $\partial_t \rho + \operatorname{div}(\rho u) = 0,$1 rather than the convexified $\partial_t \rho + \operatorname{div}(\rho u) = 0,$2; this is resolved using the distributional identity $\partial_t \rho + \operatorname{div}(\rho u) = 0,$3 to convert the $\partial_t \rho + \operatorname{div}(\rho u) = 0,$4 remainder back into potential form.

Vanishing artificial pressure

The final limit $\partial_t \rho + \operatorname{div}(\rho u) = 0,$5 faces weaker uniform bounds than the previous step, so the Bogovskiĭ test function must be constructed with the multiplier $\partial_t \rho + \operatorname{div}(\rho u) = 0,$6, $\partial_t \rho + \operatorname{div}(\rho u) = 0,$7, yielding the improved estimate

$\partial_t \rho + \operatorname{div}(\rho u) = 0,$8

which implies $\partial_t \rho + \operatorname{div}(\rho u) = 0,$9 in $\rho$00. In dimension $\rho$01 the same result holds with $\rho$02.

Strong convergence of the density now requires the specific cut-off functions $\rho$03 and potentials $\rho$04 introduced by Feireisl, since $\rho$05 is genuinely non-monotone and the uniform bounds are weaker. Applying the effective viscous flux proposition with test functions built from $\rho$06, combining with renormalized formulations of the continuity equation for both approximate and limit densities, and passing $\rho$07 reproduces the entropy-type inequality driving strong convergence in $\rho$08. The energy inequality passes to the limit via weak lower semicontinuity of convex functionals and the convergence $\rho$09 in $\rho$10, completing the proof.

Limitations and open questions

Several qualifications attach to the result. First, the theorem concerns the relaxed system, not the original NSKE; while formal asymptotics and rigorous results in restricted settings support the relaxation limit, extending the rigorous convergence of Chaudhuri–Rohde–Wendt to settings beyond those treated there remains tied to the existence theory established here. Second, the analysis assumes constant shear and bulk viscosities; the density-dependent viscosity setting in which global weak solutions to the genuine NSKE are known (Antonelli–Spirito, Bresch–Desjardins–Lin) lies outside the present framework. Third, uniqueness of finite energy weak solutions is not addressed, consistent with the state of the art even for the compressible NSE. Fourth, the domain boundary is required to be $\rho$11, and the initial density must be strictly positive in the regularized stages (approached via mollification with lower bounds $\rho$12); whether the construction extends to rougher domains or handles vacuum more directly is left open. Finally, the authors identify two concrete open directions: deriving homogenization results for the rNSKE in perforated domains (building on the single existing homogenization result for the NSKE, which replaced the third-order operator with a convolution), and developing a weak solution framework with global existence for the non-isothermal relaxation system proposed by Keim–Munz–Rohde.

Conclusion

The paper proves global-in-time existence of finite energy weak solutions to the IBVP for the parabolically relaxed Navier–Stokes–Korteweg system in two and three dimensions, for a broad class of admissible pressures including non-monotone Van-der-Waals laws, throughout the full growth range $\rho$13. The proof combines the Feireisl–Novotný–Petzeltová approximation scheme, a generalized effective viscous flux compactness property valid for systems with additional source and force terms, parabolic regularity for the relaxation variable, and the cut-off techniques developed for non-monotone pressures. The result supplies the missing existence ingredient for prior relaxation-limit and mixture-model analyses that presupposed such solutions, and provides the foundation for subsequent homogenization studies of the relaxed system.