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Generalised dissipative solutions for a non-isothermal phase-field system: existence, weak-strong uniqueness, and long-time behaviour
Aaron Brunk, Marvin Fritz
TL;DR
The paper addresses existence, stability, and long-time behaviour for a thermodynamically consistent non-isothermal phase-field system with two order parameters and inverse temperature. It constructs global generalised dissipative weak solutions using a fully discrete approximation, then proves weak–strong uniqueness on bounded thermodynamic ranges and stationary ω-limit behaviour. The analysis retains possible singular-energy concentration through non-negative defect measures.
Problem
Non-isothermal phase-field evolution must couple heat conduction, latent heat, and temperature-dependent interactions consistently with mass, internal-energy, and entropy balances.
Method
A fully discrete finite-element scheme combines exact conservation, a discrete entropy inequality, availability coercivity, a dimension-adapted positivity barrier, compactness, and relative entropy.
Results
The analysis establishes global generalised dissipative weak solutions, weak–strong uniqueness on bounded thermodynamic state ranges, and stationary ω-limit states with possible non-negative energy defects.
Takeaways & Limitations
The entropy–availability structure yields finite total Onsager dissipation and identifies stationary subsequential long-time states without isolation assumptions for convergence of the entire trajectory.
Takeaways & Limitations
The weak–strong uniqueness theorem retains a bounded thermodynamic state-range assumption, and the regularisation parameters remain fixed rather than being sent to zero.
Abstract
from arXiv · showhide
We study a thermodynamically consistent non-isothermal phase-field system coupling two order parameters and the inverse temperature through a fully non-diagonal Onsager mobility. The model describes the interaction of mass diffusion, heat conduction, and local phase relaxation while conserving mass and internal energy and producing entropy. Global generalised dissipative weak solutions are constructed using a fully discrete approximation. The discrete scheme conserves mass and internal energy and satisfies a discrete entropy inequality. An availability estimate, together with a dimension-adapted barrier, yields strict positivity at fixed mesh and uniform estimates. Compactness then allow passage to the continuous system. Concentration of the singular part of the internal energy is represented by a non-negative defect measure. The entropy-availability structure yields finite dissipation on the infinite time interval and the existence of stationary $ω$-limit states. Finally, a relative-entropy argument establishes weak-strong uniqueness whenever the weak and strong solutions remain in a bounded thermodynamic state range.
1. Introduction
The paper develops a thermodynamically consistent non-isothermal phase-field framework and constructs global generalised dissipative weak solutions. It also proves weak–strong uniqueness and identifies stationary ω-limit states under stated scope conditions.
- The model couples two order parameters and inverse temperature through a non-diagonal Onsager mobility while conserving mass and internal energy and producing entropy.
- The fully discrete scheme exactly conserves global mass and internal energy and satisfies a discrete entropy inequality.
- An availability estimate and dimension-adapted nodal barrier yield positive discrete solutions and uniform estimates, enabling compactness and passage to global weak solutions.
- Possible concentration of the critical singular internal-energy term is represented by a non-negative Radon measure in the limiting energy balance.
- Relative entropy proves weak–strong uniqueness on a common bounded thermodynamic state range, with the concentration defect vanishing.
- Every global weak solution has finite total Onsager dissipation and a sequence converging strongly in H1 to a stationary state, without requiring isolation assumptions for whole-trajectory convergence.
2. Thermodynamic structure and constitutive assumptions
The thermodynamic formulation uses inverse temperature, Gibbs relations, and a positive Onsager structure to derive entropy production. A modified free energy and availability functional provide coercivity, positivity, and compactness tools.
- The Gibbs relation expresses entropy variations through inverse temperature, internal-energy variations, and chemical-potential variations.
- The symmetric uniformly positive definite Onsager operator couples chemical potentials, heat flux, and local relaxation while producing non-negative quadratic entropy dissipation.
- The thermal modification adds singular inverse-temperature and radiative terms; q≥d is the threshold used for the finite-element positivity barrier.
- The modified model fixes ε1 and εq positively, and no vanishing-regularisation limit is asserted.
- Availability is coercive in temperature, inverse-temperature powers, and quartic phase growth: a e−sbulk ≥ c(θ+θ^-q+|z|4)−C.
- Uniform discrete estimates give strong L2 compactness for forward, backward, and affine interpolants along subsequences.
3. Existence and convergence of generalised dissipative weak solutions
The existence proof combines a consistently integrated finite-element scheme with discrete thermodynamic estimates, positivity arguments, and compactness. It produces global generalised dissipative weak solutions under dimension-adapted assumptions.
- The existence theorem applies in dimensions 1≤d≤3 when the Onsager and modified-free-energy assumptions hold with q≥d and admissible initial data.
- The constructive proof derives exact mass and internal-energy conservation together with a discrete entropy inequality.
- A coercive availability estimate and dimension-adapted nodal barrier support a Brouwer degree argument yielding a positive global discrete trajectory.
- Strong compactness of phase variables and inverse temperature identifies lower-order thermal nonlinearities and permits convergence to a global generalised dissipative weak solution.
- The analysed scheme uses exact L2 integration; fixed quadrature can introduce residuals into discrete energy and entropy identities and requires separate analysis.
3.2. Discrete availability, full norms, and time-derivative estimates
Uniform availability estimates provide mesh- and timestep-independent state and dissipation bounds, alongside temperature-tail and discrete time-derivative controls. These estimates do not imply mesh-uniform pointwise temperature separation.
- Uniform estimates: Uniformly bounded initial availability yields estimates independent of h, τ, and n.The availability representation and ellipticity of the Onsager operator provide time-uniform state bounds and dissipation control.
- Full norms: Summation in time gives the full L2(0,T;H1) estimate, while Poincaré and gradient dissipation complete the chemical-potential H1 bounds.
- Scope: No mesh-uniform pointwise lower or upper temperature bound is asserted.
- Time derivatives: Discrete time-derivative estimates follow from the balance equations, bounded mobility, and discrete dual-norm arguments.
3.3. Initial approximation, positivity, and fixed-mesh solvability
The construction combines thermodynamically consistent initial approximations with a dimension-adapted positivity barrier and a Brouwer degree argument. This yields positive solutions at every fixed-mesh time step and global discrete trajectories with uniform estimates.
- Initial approximation: Smooth positive approximations preserve convergence of thermal compositions, logarithms, reciprocal powers, phase polynomials, energies, and entropies.Nodal interpolation remains strictly positive for sufficiently small h, and a vanishing constant correction enforces the exact mass constraint.
- Positivity barrier: The fixed-mesh positivity barrier requires q≥d and is sharp for the stated finite-element argument.For q<d, a nodal value can approach zero while the inverse-power integral remains bounded.
- Fixed-mesh solvability: For every fixed h>0 and τ>0, the nonlinear time-step problem has a solution with strictly positive inverse temperature.Coercive bounds and the nodal barrier provide the compactness needed for the degree continuation argument.
- Global discrete trajectory: Repeated one-step solvability produces a strictly positive fully discrete solution on every time grid, with estimates uniform in h and τ.
- Scope: The fixed-mesh positivity constant is used only in the finite-dimensional degree argument, not in the continuum compactness passage.The degree argument establishes existence but not uniqueness of the nonlinear algebraic solution at each time step.
3.4. Temporal interpolants and phase compactness
Discrete Aubin–Lions compactness yields strong convergence of the phase variables, with uniform-in-time H1 bounds for both phase components. The phase-energy interpolants are compact through interpolation and polynomial structure.
- Phase variables: The discrete phase variables converge strongly in Lp((0,T)×Ω) for every 1≤p<6.
- Phase variables: Both phase components remain uniformly bounded in L∞(0,T;H1(Ω)).
- Compactness mechanism: Strong L2 convergence combined with uniform L∞(0,T;L6) bounds upgrades convergence by interpolation.
- Phase energy: The phase energy converges strongly for both time interpolants after controlling phase differences and the cubic factor.
3.5. Strong compactness of the inverse temperature
Time-translation control and the thermal energy structure establish strong compactness of the inverse temperature. The limit belongs to L2(0,T;H1(Ω)) and is positive almost everywhere, with strong L2 convergence obtained by uniform integrability.
- Thermal structure: The logarithmic thermal term supplies a global monotonicity estimate suited to the consistent Galerkin scheme.
- Temporal control: The internal-energy interpolants satisfy time-translation estimates derived from discrete bounds and affine interpolation.
- Limit temperature: A subsequence converges to θ∈L2(0,T;H1(Ω)) with θ>0 almost everywhere.
- Strong convergence: Uniform integrability and convergence in measure yield strong L2 convergence of the forward and backward temperature interpolants.
3.6. Thermal nonlinearities and the internal-energy defect
Lower-order thermal terms converge strongly, while the critical inverse-power energy may lose compactness only through concentration represented by a non-negative measure.
- Thermal nonlinearities: For every exponent r<q, the lower-order thermal nonlinearities converge strongly, including the relevant logarithmic terms.The proof uses uniform integrability, almost-everywhere convergence, and Vitali’s theorem.
- Thermal nonlinearities: Strong convergence of the inverse temperature eliminates oscillation defects in bounded truncations of the critical singular energy.The truncations converge strongly in L1 by dominated convergence, so the associated Young measure is a Dirac mass.
- Internal-energy defect: Only concentration remains possible for the untruncated critical inverse-power energy, and it is recorded by a non-negative Radon measure.The measure has a positive time-fibre representative and no atoms in time.
- Convergence: The fully discrete problem admits global trajectories for every mesh size and time step, and arbitrary sequences with h,τ→0 have convergent subsequences without a coupling condition.The limits include strong phase convergence and measure-valued convergence of the critical singular energies.
- Time representatives: The limiting solution has time representatives and product rules sufficient to identify the total internal-energy measure and its initial trace.The total energy is represented continuously in a negative Sobolev space and weak-star continuously as a measure.
3.7. Passage to the limit and proof to the main theorem
The compactness limits identify the constitutive laws, Onsager terms, entropy inequality, balance laws, and conservation properties, yielding the global generalised dissipative weak solution.
- Passage to the limit: The constitutive nonlinearities and state-dependent Onsager expressions pass to the limit using strong coefficient convergence and weak convergence of flux variables.The mobility blocks converge strongly enough to identify all three limiting Onsager expressions.
- Balance laws: Discrete summation by parts yields the continuous internal-energy identity and the two phase balance laws.The regular internal energy converges strongly, while the singular part converges through the defect measure.
- Entropy inequality: The discrete entropy inequality passes to the limit by strong convergence of mobility square roots, lower semicontinuity, and convergence of the initial entropy.The resulting entropy inequality has the regularity required in the weak-solution definition.
- Conservation laws: Spatially constant tests recover conservation of mass and total internal energy for the limiting solution.Continuous representatives of the phase variable and total internal-energy measure make these conservation laws well defined.
3.8. Vanishing defect measure
The concentration defect vanishes when the critical singular energies satisfy additional compactness or mass-convergence information.
- Defect-removal criteria: The defect measure vanishes if the critical singular energies are uniformly integrable.Almost-everywhere convergence and Vitali’s theorem then give strong L1 convergence of the singular energies.
- Defect-removal criteria: The defect measure also vanishes if the singular-energy space–time masses converge.Testing the weak-star measure convergence with the constant function 1 forces the non-negative defect to be zero.
4. Weak–strong uniqueness and long-time behavior
Relative entropy proves weak–strong uniqueness on a common bounded thermodynamic range, while finite dissipation yields stationary ω-limit states without isolation assumptions.
- Weak–strong uniqueness: The stability theorem is restricted to solutions remaining in a bounded state range because compactness alone provides integral control, not mesh-uniform pointwise separation of inverse temperature.On that range, entropy-variable remainders are quadratic and the state-dependent mobility can be retained.
- Weak–strong uniqueness: Bounded-range coercivity makes the relative functional equivalent to squared differences of the state variables.The proof uses smooth inverse thermodynamic coordinates and a penalised bulk integrand with uniformly positive Schur complement.
- Weak–strong uniqueness: The relative-entropy inequality controls Onsager dissipation, interfacial terms, and the energy defect for a weak solution compared with a strong solution.The functional is formulated through availability and includes the measure-valued energy component.
- Weak–strong uniqueness: With identical initial data, Gronwall’s lemma implies that the bounded weak and strong solutions coincide almost everywhere and that the defect measure vanishes.The conclusion is stated in Theorem 4.6 under local C1 regularity of the mobility.
- Long-time behaviour: Every global weak solution has finite total dissipation and a sequence of times converging to a stationary ω-limit state.Along this sequence, both phase variables and inverse temperature converge strongly in H1; the limiting inverse temperature is a positive spatial constant.
- Long-time behaviour: The asymptotic energy measure may combine a pre-existing defect with new concentration of the critical inverse-power energy.This additional concentration is retained as a non-negative measure in the stationary limit description.
5. Summary and Outlook
The paper develops an existence, stability, and long-time theory using availability, a dimension-adapted singular thermal contribution, and fully discrete thermodynamic estimates. The resulting solutions accommodate a thermal defect measure, satisfy weak–strong uniqueness in bounded state ranges, and possess stationary ω-limit states, while several extensions remain open.
- Availability and a dimension-adapted singular thermal contribution address positivity, nonlinear thermal constitutive relations, and weak compactness of internal energy.
- The discrete construction preserves mass and internal energy exactly, satisfies a discrete entropy inequality, and yields strict inverse-temperature positivity on each fixed mesh when q≥d.A Brouwer degree construction provides global discrete trajectories.
- Only the critical singular thermal term εqθ^-q may concentrate in the limit, and this loss is represented by a non-negative defect measure λ.The defect disappears under additional equiintegrability of the singular energy.
- Finite total Onsager dissipation and uniform energy-variable bounds yield subsequences converging strongly in H1(Ω) to stationary states with equilibrium chemical potentials and constant positive inverse temperature.An asymptotic thermal-energy defect may remain along the selected sequence.
- The analysis identifies stationary ω-limit states but does not generally establish convergence of the full trajectory to a single equilibrium.
- Open problems include removing fixed regularisation parameters, dropping the bounded-range hypothesis for weak–strong uniqueness, and upgrading subsequential to full long-time convergence.The paper points to uniform parameter estimates, essential–residual decompositions, and Łojasiewicz–Simon inequalities as possible directions.