Paradifferential Weight in Quasilinear PDEs

• Paradifferential weight is a symbol-valued function designed to rebalance variable coefficients in quasilinear PDEs, preventing derivative loss in energy estimates.
• It employs precise exponent matching (e.g., -54 and +14) to achieve cancellations in the modified energy method for systems like hydroelastic waves.
• Defined within the Hörmander class, these weights ensure boundedness and controlled commutator estimates across Sobolev spaces.

A paradifferential weight is a symbol-valued function, typically of the form $m(\alpha, \xi)$ or $n(\alpha, \xi)$ involving spatial and frequency variables, inserted into a paradifferential operator to re-balance variable coefficients in the analysis of quasilinear PDEs. Paradifferential weights are employed to compensate for the loss of derivatives and control the dynamics introduced by the principal quasilinear terms, especially when proving energy estimates in low regularity regimes. They play a central role in the energy method for dispersive and fluid-structure interaction equations, such as the two-dimensional hydroelastic waves, ensuring precise cancellations between undesirable terms and nonlinear source contributions (Wan et al., 26 Dec 2025).

1. Definition and Symbol Class

Let $J(\alpha) = |1 + W_\alpha(\alpha)|^2$ denote the Jacobian determinant corresponding to a conformal coordinate transformation, where $W_\alpha$ is the derivative of a conformal map. The core paradifferential weights are chosen as powers of $J(\alpha)$: $$m(\alpha, \xi) = J(\alpha)^{-54}, \qquad n(\alpha, \xi) = J(\alpha)^{14}$$ These symbols belong to the zero-th order Hörmander class $\Gamma^0_\rho(\mathbb{R})$ for $\rho > \frac{1}{2}$. In detail, for all $k \leq \lfloor \rho \rfloor$ and fixed $|\xi| \geq \frac{1}{2}$, the derivatives $\partial_\xi^k m(\cdot, \xi)$ and $\partial_\xi^k n(\cdot, \xi)$ lie in $W^{\rho, \infty}(\mathbb{R})$, uniformly bounded in the $\xi$ variable. This ensures the appropriate regularity and boundedness required for the symbolic calculus in the subsequent analysis.

2. Paradifferential Operators and Energy Forms

Using the symbols above, the associated paradifferential operators are defined by

$$T_{J^{-54}} : u(\alpha) \mapsto \operatorname{Op}\big(\chi(\xi - \eta, \eta) J^{-54}(\alpha)\,\hat{u}(\eta)\big); \qquad T_{J^{14}} : u(\alpha) \mapsto \operatorname{Op}\big(\chi(\xi - \eta, \eta) J^{14}(\alpha)\,\hat{u}(\eta)\big)$$

with $\chi$ denoting the standard low-high frequency cutoff. By symbolic calculus, both $T_{J^{-54}}$ and $T_{J^{14}}$ are bounded on Sobolev spaces $H^s$ and Zygmund classes $C^s_*$, provided the control norms of $J$ are finite. Their principal use lies in constructing modified quadratic energies that robustly handle the non-constant leading coefficients in the differentiated equations.

In the quasilinear setting, the modified energy at the $H^0$-level is defined as: $$E_{\mathrm{lin}}(w, r) = \int \Im\left(T_{J^{-54}} w_\alpha \,\overline{w_\alpha}\right) + \Re \left(r\,T_{J^{14}} \overline{r}\right) + \Re(w\,\overline{w})\,d\alpha$$ where $w, r$ denote the differentiated state variables associated with the system.

3. Motivation: Quasilinear Cancellations and Energy Closure

Paradifferential weights are selected to resolve difficulties arising from integration by parts in the time derivative of energy identities. In systems such as hydroelastic waves, the leading quasilinear term introduces spatially varying coefficients multiplying high derivatives, typically of the form $T_{J^{-1} |D|^{5/2}}$. Direct integration by parts loses control due to $\partial_t J$, necessitating the insertion of a compensating weight.

The exponents $-54$ and $+14$ are engineered so that, in the time derivative of the quadratic energy, the highest-order contributions stemming from $D_t J$ cancel against the principal third-order elastic nonlinearities. The choice of $J^{-54}$ for the $w$-energy and $J^{14}$ for the $r$-energy ensures exact matching of principal para-coefficients between different parts of the coupled system. This nontrivial arithmetic of exponents is dictated by the structure of the differentiated equations and verified via expansion of the cubic terms.

4. Symbol Estimates and Commutator Bounds

The paradifferential weights satisfy several crucial estimates. The symbol seminorms

$$M^0_\rho(J^{-54}) + M^0_\rho(J^{14}) \lesssim \mathcal{A}_{74}$$

are uniformly controlled via a function $\mathcal{A}_{74}$ (the control norm). Pointwise, $J(\alpha)$ and $J(\alpha)^{-1}$ remain uniformly comparable to $1$, and their high-regularity norms (e.g., $C^{74}_*, W^{74,4}$) are bounded.

For any $s \geq 0$ and smooth $u(\alpha)$, the commutators obey

$$\|[D_t, T_{J^{-54}}]u\|_{H^{s-1}} + \|[D_t, T_{J^{14}}]u\|_{H^{s-1}} \lesssim \mathcal{A}_0 \|u\|_{H^s}$$

These estimates are fundamental for the control of time derivatives of paradifferential energies, ensuring that no derivative loss or non-perturbative remainder is incurred in the energy argument.

5. Integration into Higher-Order Modified Energies

The quadratic energy $E_{\mathrm{lin}}(w, r)$ does not by itself eliminate all non-perturbative cubic remainders after differentiation. To neutralize remaining problematic terms, specifically at cubic order, one introduces a correction of the form: $$E^{3}_{\mathrm{cor}}(w, r) = \Re \int \big( A(R, T_{J^{14}} w, \overline{r}) + B(R, T_{J^{14}(1 - \overline{Y})^2} r, \overline{w}) + C(W, T_{J^{-54}(1-Y)} w, \overline{w}) + D(W, T_{J^{14}(1-Y)} r, \overline{r}) \big) \, d\alpha$$ with $A, B, C, D$ determined by solving a cubic symbol-matching system, achieving precise cancellation of the leading third-order terms in the time derivative of the full modified energy. Final quartic corrections are added to handle remaining resonances, leading to a fully modified energy

$$E_{\mathrm{mod}}(w, r) = E_{\mathrm{lin}} + E^{3}_{\mathrm{cor}} + E^{4}_{\mathrm{cor}}$$

that closes all perturbative error terms in $H^s$.

6. Summary of Paradifferential Weight Properties

The table below summarizes the principal features of the paradifferential weights employed in this setting:

Symbol	Exponent	Symbol Class
$J^{-54}(\alpha)$	$-54$	$\Gamma^0_\rho$ ($\rho>\frac12$)
$J^{14}(\alpha)$	$14$	$\Gamma^0_\rho$ ($\rho>\frac12$)

These weights are specifically constructed for the re-balancing of quasilinear coefficients in the para-energy, with the property that their symbol seminorms and regularity are controlled by simple functions of the control norm $\mathcal{A}_{74}$. Their homogeneity in $\xi$ is trivial (independent of frequency), and all essential estimates (boundedness, commutators, coefficient comparability) hold uniformly.

7. Significance within the Broader Paradifferential Calculus

The paradifferential weight framework as used in the analysis of hydroelastic waves (Wan et al., 26 Dec 2025) is indispensable for treating systems wherein quasilinear effects disrupt naive energy methods, particularly at low regularity thresholds. By leveraging these tailored weights, it becomes possible to prove sharp local well-posedness results in spaces $\mathcal{H}^s$ for exponents $s > \frac{3}{4}$, well below the scaling heuristics suggest. This approach systematically neutralizes principal-order nonlinear obstructions and may be viewed as a canonical strategy for extracting hidden cancellations in highly nonlinear dispersive PDEs. The general methodology has clear links to other paradifferential renormalization techniques, but the explicit construction and algebraic matching of energy weights, as exemplified by the $J^{-54}, J^{14}$ pair, represents a refined implementation suited to critical-regularity quasilinear systems.