Global solutions to an initial-boundary value problem for a model of convection driven by surface tension

Abstract: This paper establishes the global existence of non-negative weak solutions to a two-dimensional, fourth-order nonlinear degenerate parabolic equation modeling surface-tension-driven convection in thin fluid films. First, we construct a regularized approximate problem and prove its solvability via the Galerkin method. Utilizing energy and entropy functionals alongside a singular entropy condition $1/h_0 \in L<sup>1(Ω)$, we secure uniform a priori bounds for higher-order spatial and time derivatives. These bounds enable the use of the Aubin-Lions lemma and Gagliardo-Nirenberg inequalities to achieve strong compactness and essential $L<sup>6$-integrability. Furthermore, we adopt the Alber-Zhu framework to rigorously define higher-order local weak derivatives and pass to the limit. Finally, we prove the limit function is non-negative, confirming it as a global weak solution to the original problem.

• The paper establishes global weak solutions for the two-dimensional Davis equation under no-flux boundaries when h₀ ∈ H¹(Ω), h₀ ≥ 0, and 1/h₀ ∈ L¹(Ω).
• The authors combine κ-regularization, Galerkin approximation, energy and singular entropy estimates, and Aubin–Lions compactness to control the degenerate fourth-order flux and pass to the limit.
• The resulting solution remains positive almost everywhere with a uniform reciprocal L¹ bound, while uniqueness, positive lower bounds, rupture, and waiting-time behavior remain unresolved.

The model and the main result

The paper studies the two-dimensional initial-boundary value problem for the thin-film evolution equation derived by Davis via long-wave asymptotics of the Navier–Stokes system coupled with heat transfer:

$$h_t = -\nabla\cdot\left(\alpha_1 h^3\nabla\Delta h - \alpha_2 h^3\nabla h + \alpha_3 h^2\nabla h\right),$$

on a bounded smooth domain $\Omega\subset\mathbb{R}^2$, with no-flux boundary conditions and initial data $h_0$. Here $\alpha_1 = 1/(3Ca)$, $\alpha_2 = Ga/3$, and $\alpha_3 = Ma/2$ encode capillary, gravitational, and Marangoni effects. The equation is a fourth-order degenerate parabolic PDE: the principal mobility $h^3$ vanishes at $h=0$, and no maximum principle is available. Prior existence theory covered either one-dimensional settings, the case without lower-order convection ($\psi\equiv 0$) in multi-dimensions (Dal Passo–Garcke–Grün; Grün; Bertsch et al.), or periodic/unbounded domains for power-law mobilities. The authors' own prior work established the one-dimensional analogue under no-flux conditions; the two-dimensional case on bounded domains with both second-order convective terms was open.

The main theorem asserts: if $h_0 \in H^1(\Omega)$, $\Omega\subset\mathbb{R}^2$0 a.e., and $\Omega\subset\mathbb{R}^2$1, then for every $\Omega\subset\mathbb{R}^2$2 there exists a global weak solution $\Omega\subset\mathbb{R}^2$3 satisfying

• $\Omega\subset\mathbb{R}^2$4,
• $\Omega\subset\mathbb{R}^2$5,
• $\Omega\subset\mathbb{R}^2$6 for a.e. $\Omega\subset\mathbb{R}^2$7, for all $\Omega\subset\mathbb{R}^2$8,
• $\Omega\subset\mathbb{R}^2$9 and $h_0$0.

The solution concept uses the Alber–Zhu notion of local weak $h_0$1-derivatives: since $h_0$2 only slice-wise in time, the positivity set $h_0$3 need not be open in space-time, so $h_0$4 is defined as an $h_0$5-th local weak derivative on $h_0$6 via exhaustion by measurable subsets, and the degenerate flux $h_0$7 enters the weak formulation integrated over $h_0$8 only.

Regularized problem and Galerkin approximation

The degeneracy is removed by replacing $h_0$9 with $\alpha_1 = 1/(3Ca)$0, yielding a uniformly parabolic approximate problem. Galerkin approximations are built from Neumann eigenfunctions of the Laplacian; the resulting ODE system has locally Lipschitz right-hand side, so Picard–Lindelöf gives local classical solutions, extended globally by uniform-in-$\alpha_1 = 1/(3Ca)$1 estimates: $\alpha_1 = 1/(3Ca)$2 and $\alpha_1 = 1/(3Ca)$3 bounds on $\alpha_1 = 1/(3Ca)$4 and $\alpha_1 = 1/(3Ca)$5, plus $\alpha_1 = 1/(3Ca)$6 bounded in $\alpha_1 = 1/(3Ca)$7. The Aubin–Lions compactness lemma (with $\alpha_1 = 1/(3Ca)$8, $\alpha_1 = 1/(3Ca)$9) yields strong convergence in $\alpha_2 = Ga/3$0 and in $\alpha_2 = Ga/3$1; a Gagliardo–Nirenberg interpolation then upgrades this to strong convergence in $\alpha_2 = Ga/3$2 — precisely the integrability needed to pass to the limit in the cubic and quadratic convective terms via Hölder pairing. Weak-strong convergence arguments identify all flux limits, giving a global weak solution to the regularized problem for each fixed $\alpha_2 = Ga/3$3.

Uniform a priori estimates

Three independent-of-$\alpha_2 = Ga/3$4 estimates form the core of the paper:

Energy estimate. The functional $\alpha_2 = Ga/3$5, with $\alpha_2 = Ga/3$6 containing a shifted logarithmic term, dissipates monotonically along solutions; completing the square shows the dissipation controls $\alpha_2 = Ga/3$7. Coercivity of $\alpha_2 = Ga/3$8 (via the inequality $\alpha_2 = Ga/3$9) yields the uniform bound $\alpha_3 = Ma/2$0.

Entropy estimate. Testing against $\alpha_3 = Ma/2$1, where $\alpha_3 = Ma/2$2, produces the strictly convex entropy $\alpha_3 = Ma/2$3 converging as $\alpha_3 = Ma/2$4 to $\alpha_3 = Ma/2$5 for $\alpha_3 = Ma/2$6 and $\alpha_3 = Ma/2$7 for $\alpha_3 = Ma/2$8. The Marangoni term contributes $\alpha_3 = Ma/2$9 on the right-hand side; this is absorbed because $h^3$0 grows logarithmically while $h^3$1 grows like $h^3$2, so $h^3$3. Gronwall's lemma then gives uniform bounds on $h^3$4 and on $h^3$5, hence $h^3$6. This entropy control is what enforces positivity in the limit; it depends essentially on the assumption $h^3$7, which guarantees $h^3$8.

Time-derivative estimate. Combining the above with Gagliardo–Nirenberg bounds on $h^3$9 gives $h=0$0, closing the Aubin–Lions framework at the $h=0$1-level.

Passing to the limit and non-negativity

As $h=0$2, a subsequence converges strongly in $h=0$3 for any $h=0$4 (Aubin–Lions with $h=0$5, $h=0$6), strongly in $h=0$7, and consequently strongly in $h=0$8; the powers $h=0$9 converge strongly in the corresponding Lebesgue spaces.

Identification of the degenerate flux proceeds by domain decomposition. On exhausting sets $\psi\equiv 0$0 constructed via Egorov's theorem, uniform convergence forces $\psi\equiv 0$1, so the energy bound implies $\psi\equiv 0$2 is bounded in $\psi\equiv 0$3; a distributional argument using the Lebesgue differentiation theorem shows the weak limit equals $\psi\equiv 0$4 slice-wise, and patching across $\psi\equiv 0$5 yields the Alber–Zhu local derivative on $\psi\equiv 0$6. On the singular set $\psi\equiv 0$7, a Hölder estimate combined with strong $\psi\equiv 0$8 convergence shows the $\psi\equiv 0$9-norm of the flux on $h_0 \in H^1(\Omega)$0 is at most $h_0 \in H^1(\Omega)$1; letting $h_0 \in H^1(\Omega)$2 proves the weak limit $h_0 \in H^1(\Omega)$3 vanishes a.e. on $h_0 \in H^1(\Omega)$4.

Positivity follows from a double Fatou argument rather than a contradiction argument (unavailable without a maximum principle). First, Fatou's lemma applied to $h_0 \in H^1(\Omega)$5 at times of uniform convergence gives $h_0 \in H^1(\Omega)$6; second, temporal continuity $h_0 \in H^1(\Omega)$7 extends this to all $h_0 \in H^1(\Omega)$8 via a second application of Fatou's lemma along approximating times. Since $h_0 \in H^1(\Omega)$9 on $\Omega\subset\mathbb{R}^2$00, $\Omega\subset\mathbb{R}^2$01 a.e. for all $\Omega\subset\mathbb{R}^2$02, and $\Omega\subset\mathbb{R}^2$03 yields the uniform reciprocal bound $\Omega\subset\mathbb{R}^2$04. A consequence is that $\Omega\subset\mathbb{R}^2$05 has measure zero, so the local weak formulation over $\Omega\subset\mathbb{R}^2$06 coincides with a global one and the identification $\Omega\subset\mathbb{R}^2$07 holds throughout $\Omega\subset\mathbb{R}^2$08. The proof of the main theorem is completed by passing to the limit in the weak formulation term by term.

Limitations and open questions

The paper leaves uniqueness open: the regularity $\Omega\subset\mathbb{R}^2$09 is insufficient for $\Omega\subset\mathbb{R}^2$10-contraction or difference estimates on the degenerate operator $\Omega\subset\mathbb{R}^2$11. The condition $\Omega\subset\mathbb{R}^2$12 does not produce a positive lower bound for $\Omega\subset\mathbb{R}^2$13 — it yields only strict positivity a.e. together with the reciprocal $\Omega\subset\mathbb{R}^2$14 bound — so questions of waiting-time behavior, finite speed of propagation, and film rupture (touchdown at positive times) remain outside the scope of the existence result. Whether the entropy method extends to rougher initial data lacking $\Omega\subset\mathbb{R}^2$15 regularity, or to higher-order asymptotic models with additional terms, is not addressed.

Conclusion

The paper completes the program initiated in the authors' one-dimensional work by establishing global existence of non-negative weak solutions to the two-dimensional Davis equation under no-flux boundary conditions. The argument combines a $\Omega\subset\mathbb{R}^2$16-regularization with Galerkin approximation, energy and singular entropy estimates tied to the assumption $\Omega\subset\mathbb{R}^2$17, Aubin–Lions/Gagliardo–Nirenberg compactness delivering the critical $\Omega\subset\mathbb{R}^2$18 integrability, the Alber–Zhu local derivative framework to handle the non-open degeneracy set, and a double Fatou argument replacing the missing maximum principle. The result provides the rigorous analytical foundation required to validate the Davis model's predictions of dry-spot formation and coarsening in surface-tension-driven convection.