Global Weak Solutions in Large-Scale Nonlinear PDEs

• Large-data global weak solutions are defined for nonlinear PDE systems that accommodate arbitrarily large initial data under finite energy constraints.
• The methodology integrates multi-scale approximations, uniform energy estimates, and domain expansion techniques to manage strong nonlinearities and singular couplings.
• Applications include compressible flow in nematic liquid crystals and other continuum mechanics problems, where geometric angle conditions ensure regularity and prevent defect formation.

A large-data global weak solution refers to a solution of a nonlinear partial differential equation (PDE) system that is defined for all time and allows for arbitrarily large (often only restricted by finite energy) initial data. Rigorous mathematical theory for such solutions is crucial, especially in systems with highly nonlinear or possibly singular couplings, which arise in continuum mechanics, fluid dynamics, and materials science. This entry details the mathematical underpinnings and main results concerning large-data global weak solutions, with particular emphasis on the compressible flow of nematic liquid crystals and selected related systems.

1. Large-Data Global Weak Solutions: Definition and General Theory

A large-data global weak solution for a PDE system is a triple (or tuple) of functions representing the physical variables (e.g., density, velocity, director field) that satisfy:

• The governing equations in the sense of distributions (or integrated against test functions),
• Initial and boundary data prescribed in a weak or trace sense,
• Energy inequalities or entropy bounds that are uniform in time,
• A functional framework compatible with potentially large, even unbounded, initial data (for example, high $L^p$-norms).

For the compressible Ericksen–Leslie system modeling nematic liquid crystals in two spatial dimensions, the relevant PDEs are

$$\begin{align*} &\partial_t \rho + \nabla \cdot (\rho u) = 0,\ &\partial_t (\rho u) + \nabla \cdot (\rho u \otimes u) + \nabla p(\rho) = \mu \Delta u + (\mu+\lambda) \nabla \mathrm{div}\, u - \mathrm{div}(\nabla d \otimes \nabla d - \tfrac{1}{2} |\nabla d|^2 I),\ &\partial_t d + u \cdot \nabla d = \Delta d + |\nabla d|^2 d,\quad |d|=1. \end{align*}$$

Here $\rho \ge 0$, $u \in \mathbb{R}^2$ is velocity, $d:\mathbb{R}^2\to S^1$ is the director, $p(\rho) = A\rho^\gamma$, and $(\mu>0, \mu+\lambda\ge 0)$ are viscosities (Jiang et al., 2012).

A global weak solution must satisfy:

• Regularity: $\rho \in L^\infty_t (L^1 \cap L^\gamma)$, $u \in L^2_t H^1$, $\sqrt{\rho}u \in L^\infty_t L^2$, $d \in L^2_t H^2$ with $|d|=1$,
• Distributional validity of the PDEs for smooth test functions,
• Renormalized continuity property for any $b \in C^1$,
• Energy inequality (involving kinetic, internal, and elastic energy) that holds for almost every time.

This framework accommodates arbitrarily large initial data within the finite-energy class.

2. Construction Methodologies and Domain Expansion

To prove large-data global weak solution existence, one typically combines multi-scale approximations, uniform a priori bounds, compactness arguments, and rigidity or coercivity results, with the following primary steps:

1. Bounded Domain Approximation:
   • Solve the system on expanding balls $B_R \subset \mathbb{R}^2$ with Dirichlet boundary conditions (e.g., $d|_{\partial B_R}=e_2$ for some distinguished direction, $u|_{\partial B_R}=0$).
   • Use a three-level approximation: finite-dimensional Galerkin truncation, addition of artificial viscosity, and artificial pressure terms, ensuring strong parabolic regularization and control of possible singularities (Jiang et al., 2012).
2. Uniform Estimates:
   • Uniform $R$-independent energy bounds on kinetic, potential, and elastic contributions via the energy inequality,
   • Additional $L^2$ and $L^4$-norm bounds on director gradients, enabled by special geometric conditions (see below),
   • Control over all possibly singular or supercritical terms.
3. Passage to the Limit:
   • Compactness is obtained using the Aubin–Lions lemma for time-space localizations, weak compactness in relevant topologies, and the direct identification of weak limits in nonlinear terms,
   • Weak-strong convergence methods for nonlinearities, leveraging the structure of the equations and uniform control,
   • Final passage to $R \to \infty$ yields a global solution on the full plane.

The domain expansion method leverages uniform-in-$R$ a priori bounds to ensure that solutions on larger and larger balls converge (in the sense of distributions and weak topologies) to a global solution in the whole space.

3. Key Analytical Ingredients: Rigidity and Coercivity Theorems

A pivotal advance in the large-data theory for liquid crystal and related flows is the use of dimension-specific rigidity results for the order parameter (here the director $d$). In two dimensions, the Lei–Li–Zhang rigidity theorem is fundamental:

• If $d:\mathbb{R}^2\to S^1$ satisfies $\nabla d \in L^2$, $\|\nabla d\|_{L^2}<C$, and the second component $d_2(x) \ge \delta > 0$ almost everywhere, then

$$\|\nabla d\|_{L^4}^4 \le (1-\omega_0) \|\Delta d\|_{L^2}^2,$$

with $\omega_0 = \omega_0(\delta, C) > 0$, and especially

$$\int |\Delta d + |\nabla d|^2 d|^2 \ge \tfrac{\omega_0}{2} (\|\Delta d\|_{L^2}^2 + \|\nabla d\|_{L^4}^4).$$

This uniform coercivity replaces interpolation inequalities that are only available on bounded domains and precludes the formation of topological defects by enforcing uniform angular nondegeneracy [Proposition 6.2, (Jiang et al., 2012)].

The geometric angle condition ($d_2(x) \ge \delta > 0$ for initial data) preserves non-vanishing of the $d$-field's second component, ensuring the absence of defect concentrations throughout the evolution.

4. A Priori Bounds and Compactness Frameworks

The analysis relies upon:

• Uniform energy bounds:

$$\sup_{0 \leq t \leq T}\int \left( \tfrac12 \rho|u|^2 + Q(\rho) + \tfrac12 |\nabla d|^2 \right) dx + \int_0^T \int \left( \mu|\nabla u|^2 + (\mu+\lambda)|\mathrm{div}\,u|^2 + |\Delta d + |\nabla d|^2 d|^2 \right) dx\,dt < E_0.$$

• Uniform bounds in Sobolev norms for all components via the rigidity theorem and energy balance—for example, control on $\Delta d$ and $\nabla d$ in $L^2$, $\nabla d$ in $L^4$,
• Use of BD–Feireisl structure for the density (originating from compressible Navier–Stokes theory), ensuring tightness in mass and energy and compatibility with vacuum regions,
• Compactness via the Aubin–Lions lemma: from uniform $L^2_t H^1_x$ and $L^2_t H^2_x$ control, strong convergence is extracted locally.

These ingredients ensure that all nonlinear and nonlinear-nondegenerate terms can be passed to the limit in the weak formulation, despite the lack of smallness in the initial data or data norms (Jiang et al., 2012).

5. Global Existence Theorems and Main Results

For the two-dimensional compressible nematic liquid crystal system:

• For initial data with finite energy, density $\rho_0 \in L^1 \cap L^\gamma(\mathbb{R}^2)$, initial momentum $\rho_0 u_0 \in L^1 \cap L^{2\gamma/(\gamma+1)}$, and director field $d_0-e_2 \in H^1(\mathbb{R}^2)$ subject to the geometric angle condition $d_{0,2}(x) \ge \delta > 0$,
• There exists a global-in-time, finite-energy weak solution $(\rho, u, d)$ satisfying all properties outlined above, for any time interval $[0,T]$ [Theorem 1.2, (Jiang et al., 2012)].

Preservation of the director invulnerability ($d_2(x,t) \ge \delta > 0$) for all $t$ ensures the solution remains free from topological singularities.

These results are robust with respect to the initial energy size, and no smallness assumption on the data is required aside from the geometric angle restriction.

For other large-data systems, related strategies based on tensor coercivity, BD-type entropies, or variable-reduction methods have yielded corresponding large-data global weak solution existence results for compressible quantum MHD, two-fluid models, Oldroyd-B and bead-spring kinetic polymeric models, energy-critical dispersive equations, and generalized Boussinesq-structure systems (Guo et al., 2016, Vasseur et al., 2017, Barrett et al., 2016, He et al., 2022, Cheng et al., 2023, Zhang et al., 13 Feb 2025).

6. Extensions, Comparison, and Technical Nuances

Key distinguishing features and technical hurdles in these frameworks include:

• Dimension-Specific Effects: Rigidity/coercivity theorems such as those by Lei–Li–Zhang are sharp in $d=2$. In higher dimensions, compensatory techniques (e.g., higher integrability, defect measures) are required, as in $d=3$ treatments for related systems (Wang et al., 2011).
• Vacuum and Singularities: Admissibility of initial vacuum, singular potentials, or degenerate viscosities, and passage through these regions via renormalized solutions and BD inequalities.
• Defect/Angle Conditions: In nematic or director-field-driven systems, geometric constraints are both necessary and effective in excluding singularities and preserving regularity.
• Nonlinear Smoothing: For some compressible and shallow-water variants, nonlinear parabolic regularization counteracts hyperbolic degeneracy, enabling large-data weak solution construction without smallness.
• Critical-Scale Data and Function Spaces: Weak solution concepts have been adapted to non-energy classes (e.g., $L^{3,\infty}$, $BMO^{-1}$, critical Besov spaces), with compactness and approximate schemes tailored to the scale-invariance and lack of Sobolev embedding at criticality (Barker et al., 2016, Yi et al., 2017, Haspot, 2012, Haspot, 2013).
• Variable Reduction/Compactness: For coupled fluid mixtures, variable reduction exploiting convex structure and transport properties facilitates strong convergence despite multi-variable nonlinearities (Vasseur et al., 2017).

7. Key Lemmas, Propositions, and Future Directions

Central technical ingredients enabling these results include:

• Local Cauchy-problem well-posedness for the parabolic–transport equation for $d$ under angle constraints (Proposition 6.1, (Jiang et al., 2012)),
• Domain expansion and global approximations preserving geometric constraints, a priori bounds, and enabling passage from bounded to unbounded domains (Propositions 6.3 and 6.4),
• Rigidity/coercivity inequalities (Lei–Li–Zhang, used in Proposition 6.2),
• Compactness via strong convergence in energy or entropy variables, variable splitting, and compensated compactness,
• Weak–strong uniqueness principles in certain frameworks (via monotonicity or perturbative arguments) (Cheng et al., 2023).

Research directions include extending these methods to non-isothermal models, nontrivial boundary geometry, fully coupled multi-physics fluids, and the sharp analysis of solution regularity and possible defect quantization in higher dimensions or with weaker geometric constraints. Cross-pollination from recent advances in critical-function-space well-posedness, kinetic models, and nonlinear dissipation mechanics continues to expand both the theoretical framework and the range of large-data global weak solution results.

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References:

• "Global Weak Solutions to the Equations of Compressible Flow of Nematic Liquid Crystals in Two Dimensions" (Jiang et al., 2012)
• "Global existence of weak solutions for generalized quantum MHD equation" (Guo et al., 2016)
• "Global weak solution to the viscous two-fluid model with finite energy" (Vasseur et al., 2017)
• "Existence of large-data finite-energy global weak solutions to a compressible Oldroyd-B model" (Barrett et al., 2016)
• "Existence of Large-Data Global Weak Solutions to Kinetic Models of Nonhomogeneous Dilute Polymeric Fluids" (He et al., 2022)
• "On global solutions to the Navier-Stokes system with large $L^{3,\infty}$ initial data" (Barker et al., 2016)