Global Weak Solutions for PDEs

• Global weak solutions are rigorously defined formulations for PDEs that address non-smooth, nonlinear, and singular problems where classical solutions may not exist.
• They leverage energy and entropy inequalities to provide a priori control and ensure the physical consistency of systems such as fluid dynamics and kinetic models.
• Approximation methods like Galerkin projection, artificial viscosity, and compactness arguments are critical for constructing solutions and managing discontinuities.

A global weak solution is a mathematically rigorous notion of solution for partial differential equations (PDEs), often arising in nonlinear, non-smooth, or singular contexts where classical (strong) solutions may not exist for all time or for general initial data. The “global” aspect refers to existence on an arbitrary time interval $[0,T]$ (or $[0,\infty)$), while “weak” indicates that equations are satisfied in a suitably integrated sense, often via duality or test functions rather than pointwise. Global weak solutions are central to the analysis of nonlinear PDEs in fluid dynamics, kinetic theory, geometric flows, and mathematical physics, providing a robust concept that accommodates discontinuities, singularities, and low-regularity data.

1. Variational and Weak Formulations

A weak solution is formulated by multiplying the governing PDEs against smooth test functions and integrating by parts, often aiming to move derivatives from the (possibly rough) solution onto the smooth test functions. This process ensures that solutions need not possess classical derivatives everywhere. Formally, for an evolution equation $\partial_t u + \mathcal{N}(u)=0$, a weak solution $u$ satisfies for all suitable test functions $\phi$,

$$\int_0^T \int_{\Omega} u \, \partial_t \phi + \mathcal{N}^*(u, \phi) \,dx\,dt + \int_{\Omega} u_0\, \phi(0)\,dx = 0.$$

The nature of $\mathcal{N}^*$ and the choice of admissible test functions depend on the system; e.g., for the relativistic BGK kinetic equation, the weak form involves test functions in $$C^1_c([0,T)\times\mathbb{R}^3_x\times\mathbb{R}^3_q)$$ and measures several couplings of $f$ and derived macroscopic fields (Calvo et al., 2019). In geometric and dispersive flows, such as harmonic map flows or nonlinear Schrödinger equations, an analogous distributional formulation is essential (Han et al., 2018, Cheng et al., 2023).

2. Energy and Entropy Inequalities

A defining feature of global weak solutions is the a priori control of natural conserved or dissipated quantities. Typical structures include energy, entropy, or Lyapunov functionals:

• Energy dissipation: In many systems, energy inequalities of the form

$E(u(t)) + \int_0^t D[u](s)\,ds \leq E(u_0)$

are available, where $E$ is the system's energy and $D[u]$ is the dissipation mechanism, e.g., enstrophy in Navier–Stokes, Dirichlet energy in harmonic maps, kinetic entropy in BGK or kinetic-fluid models (Han et al., 2018, Calvo et al., 2019, Chu et al., 2012, Cieślak et al., 2022).

• Entropy dissipation: For kinetic equations, a relative entropy or $H$-theorem is often essential for compactness and passage to the limit (Calvo et al., 2019, Chen et al., 2022).

The enforcement of such inequalities in the weak solution framework typically relies on lower semicontinuity and uniform bounds derived from approximations of smooth solutions.

3. Construction via Approximation and Compactness

Almost all existence results for global weak solutions proceed via construction of approximate solutions—either regularized, truncated, or Galerkin-projected—and systematic extraction of convergent subsequences using compactness arguments:

• Artificial viscosity or pressure: Adding higher-order parabolic or pressure regularizations to ensure boundedness and precompactness of critical norms. Passage to the limit is performed sequentially, with artificial parameters ($\epsilon,\delta$) sent to zero and uniform estimates carefully propagated (Chu et al., 2012, Tu et al., 2015, Li et al., 2018).
• Entropy or energy truncations: In kinetic or nonlocal models, velocity or temperature cutoffs in the Maxwellian (BGK) or momentum truncations (e.g. in Burgers–Vlasov, relativistic models) facilitate the use of classical existence theories and control tail contributions (Calvo et al., 2019, Yu et al., 2020).
• Compensated compactness and renormalization: Core tools such as the DiPerna–Lions theory for transport equations, effective viscous flux identities (Lions–Feireisl) for compressible flows, and Aubin–Lions or Arzelà–Ascoli compactness lemmata are ubiquitous (Han et al., 2018, Vasseur et al., 2017, Chu et al., 2012).
• Functional analytic frameworks: Weak solutions reside in spaces allowing