2-D Quadratically Quasilinear Wave Equations

Updated 9 August 2025

2-D quadratically quasilinear wave equations are partial differential equations where the principal second derivatives appear linearly while coefficients depend quadratically on the solution u or its gradient.
They exhibit key phenomena such as sharp blow-up, finite/infinite energy propagation, and integrability via quadratic complexes, linking geometry with PDE analysis.
Recent advances employ refined energy methods, ghost weight techniques, and transformation to good unknowns, enabling global regularity even in challenging exterior domain problems.

A 2-D quadratically quasilinear wave equation is a partial differential equation (PDE) in two independent variables (typically time $t$ and one spatial coordinate $x$ ), where the principal (second order) derivatives appear linearly, but the coefficients depend quadratically on the solution $u$ and/or its first derivatives. These equations are distinguished by the critical interplay between nonlinearity and dispersion in low spatial dimensions, leading to phenomena such as sharp blow-up, finite/infinite energy propagation, integrability for special structures, and delicate global regularity questions.

1. Formulation and Basic Structures

The general form of a 2-D quadratically quasilinear wave equation is

$\sum_{i,j=0}^2 g_{ij}(u, \nabla u) \, \partial_{ij} u = 0$

where the coefficients $g_{ij}$ are smooth functions, homogeneous of degree two in $(u, \nabla u)$ , and $\partial_{ij}u$ denotes second derivatives with respect to $(t, x)$ . Particularly, models where $g_{ij}$ depends only on $\nabla u$ or only on $u$ are of significant interest; many studies focus on the case where $g_{ij}$ is quadratic in $\nabla u$ (so-called "quadratically quasilinear" form) (Ding et al., 2013).

A geometric approach links such PDEs to quadratic complexes and conformal structures. In projective geometry, a quadratic line complex naturally induces a conformal structure on a 2D manifold, which after a coordinate identification yields a quasilinear wave equation with coefficients quadratic in $(u_t, u_x)$ (Ferapontov et al., 2012).

2. Null Conditions and Linear Degeneracy

Null Condition

The “null condition” is a structural constraint on the quadratic nonlinearity that ensures certain nonlinear interactions—especially those aligned with the characteristic cone—cancel. Formally, for quadratic nonlinearities $Q(\partial u, \partial^2 u)$ , the null condition requires

$\sum g^{ij}_k w_k w_i w_j = 0$

for every null vector $w = (-1, \cos\theta, \sin\theta)$ (Cheng et al., 2021, Li, 2021). This restriction suppresses the strongest nonlinear wave interactions that lead to finite-time blow-up.

Linear degeneracy is a related notion: a second-order quasilinear wave equation is linearly degenerate if all traveling-wave reductions inherit linear degeneracy, often formalized via invariant conditions such as

$\partial_{(k} f_{ij)} = \phi_{(k} f_{ij)}$

with complete symmetrization over indices, ensuring that the equation derived from a quadratic complex is indeed linearly degenerate (Ferapontov et al., 2012). Linearly degenerate systems typically avoid shock formation and exhibit global regularity.

3. Integrability and Classification by Quadratic Complexes

A key insight is that all linearly degenerate 2-D quasilinear wave equations arise as reductions of a Monge form associated with a quadratic line complex (Ferapontov et al., 2012). The classification of all linearly degenerate equations thus reduces to the algebraic classification of quadratic complexes via their Segre symbols, which encode the Jordan canonical form of the operator $Q\Omega^{-1}$ . For instance, the generic Segre symbol $[111111]$ corresponds to all six eigenvalues being distinct, with other symbols like $[(11)(11)(11)]$ indicating further degeneracy.

Special Segre types correspond to equations for which the underlying conformal structure is conformally flat (vanishing Cotton tensor), leading to integrability via the existence of a linear Lax pair. Thus, the analytic property of linear degeneracy is equivalent (in this geometric correspondence) to the algebraic property of being induced by some quadratic complex.

Segre Symbol	Equation Type	Conformal Structure
[111111]	Generic linearly degenerate PDE	Non-flat
[(11)(11)(11)]	Degenerate, integrable PDE	Conformally flat
[(111)(111)]	Higher degeneracy, integrable	Conformally flat

4. Blow-up, Lifespan, and the Role of the Null Condition

If the null condition fails, then even for small smooth initial data, 2-D quadratically quasilinear wave equations can develop singularities in finite time (Ding et al., 2013). The precise criterion for blow-up involves both the structure of the nonlinearity (i.e., violation of null form cancellation) and a nondegeneracy condition on the initial data, formulated in terms of the Radon transform. The lifespan $T_\varepsilon$ of smooth solutions satisfies

$\lim_{\varepsilon \to 0} \varepsilon T_\varepsilon = \tau_0$

where $\tau_0$ is an explicit data-dependent constant. The mechanism of blow-up is geometric: the solution remains continuous, but its gradient becomes singular due to a degeneracy in a nonlinear change of variables designed to resolve the wavefront.

Conversely, under the null condition, global existence for small data was established first for the Cauchy problem (Li, 2021) and, more recently, for exterior domain problems (Hou et al., 11 Nov 2024, Hou et al., 5 Aug 2025). The null condition ensures that quadratic interactions do not accumulate sufficiently to cause breakdown, even in 2D where the dispersive decay is only $t^{-1/2}$ .

5. Energy Methods, Weighted Estimates, and Good Unknowns

Due to slow decay in 2D, standard energy techniques must be refined:

Weighted energy estimates: These take the form

$\int e^{p(r-t)} \left[ |\partial u|^2 + |q'(r-t)| |T u|^2 \right] dx$

where $T = \partial_r - \partial_t$ is a tangential (good) derivative and the weight $q$ enhances control near the light cone (Li, 2021, Cheng et al., 2021). The “ghost weight” method, introduced by Alinhac, exploits these for stronger decay.

Avoidance of Lorentz boosts: In contrast to higher dimensions, successful 2D proofs (with or without boundaries) now routinely avoid Lorentz boost vector fields (Cheng et al., 2021, Cheng et al., 2021). The preferred commutator set consists of time, space derivatives, angular momentum, and scaling, with adjusted weights to compensate for the lack of Lorentz symmetry.
Good unknowns and divergence structures: For the exterior domain problem, crucial advances have been made by identifying new variables (good unknowns) and divergence structures. After appropriate redefinitions (often via subtracting quadratic terms localized near the cone), the reduced equation has a nonlinearity that either gains extra spatial decay (e.g., $|x|^{-1}$ factor) or is cubic/order three or higher in $u$ (Hou et al., 5 Aug 2025, Hou et al., 11 Nov 2024). This renders the nonlinear term manageable within available decay rates for sharp energy closure.

6. Decay Estimates, Boundary Effects, and Extension to Exterior Domains

Achieving global-in-time existence for 2-D quadratically quasilinear wave equations in exterior domains was a longstanding open problem, due to the critical slow decay and the complications introduced by boundaries.

The method of proof involves:

Transforming the equation via good unknowns, neutralizing the effects of boundary-induced losses.
Establishing sharp pointwise decay estimates for solutions and especially for “good derivatives” (tangential to the light cone), leveraging Littlewood–Paley theory and spatial cutoffs (Hou et al., 11 Nov 2024, Hou et al., 5 Aug 2025).
Utilizing divergence structures to absorb problematic terms by integration by parts.
Deriving local (interior) energy decay in the absence of KSS-type variable coefficient estimates available in higher dimensions.

Refined estimates for the local energy, $L^2$ norms, and pointwise estimates such as

$|u(t,x)| \lesssim \langle x \rangle^{-1/2} \langle t-|x| \rangle^{-1},\quad |Zu(t,x)| \lesssim \langle x \rangle^{-1/2} \langle t-|x| \rangle^{-\varepsilon}$

are now established for all times in exterior domains for null-form-structured quadratic nonlinearities (Hou et al., 11 Nov 2024, Hou et al., 5 Aug 2025). These results extend the known effective theory from the whole space to obstacle (exterior) domains, resolving a major conjecture.

7. Integrability, Hamiltonian Structure, and Applications

In special settings, 2-D quadratically quasilinear wave equations can be integrable. From the perspective of Hamiltonian flows, integrability is characterized by the existence of infinitely many commuting symmetries—often realized via solutions to a wave equation with nonconstant speed (Manganaro et al., 2022). These commuting flows arise naturally in models such as shallow water equations or certain elasticity systems, and the wave equations themselves correspond to conservation laws or symmetry generators.

For equations induced from quadratic complexes with conformally flat structure (special Segre cases), one finds linear Lax pairs and associated integrability (Ferapontov et al., 2012). Physical applications include nonlinear optics, elastodynamics, relativistic membranes, and electromagnetic theory, particularly when higher order null conditions or geometric structures appear in the problem's Lagrangian (Ding et al., 30 Jul 2024).

This summary synthesizes key constructions (quadratic complexes, null conditions, divergence structures), analytical techniques (energy methods, ghost weight, good unknowns), and fundamental results (blow-up, global regularity, integrability) that form the current research landscape for 2-D quadratically quasilinear wave equations. The recent extensions to general exterior domains and advances in geometric and analytic control are pivotal for further developments in both theory and applications.