Project A13 • Dispersive estimates for wave equations with low regularity coefficients

Principal investigators

  Prof. Dr. Dorothee Frey (7/2021 - )
  Prof. Dr. Roland Schnaubelt (7/2021 - )

Project summary

Wave equations play a fundamental role in physics. For the investigation of wave propagation in inhomogeneous media or the modeling of nonlinear wave phenomena one is led to the study of wave equations with low regularity coefficients. Yet the mathematical investigation of their qualitative behavior remains highly challenging, as the low regularity of the coefficients partly impedes the application of classical analytical methods such as the Fourier transform and Fourier restriction theory. In this project we propose a new approach to Strichartz and dispersive estimates for wave and wave-type equations in low regularity settings that exploits structural properties of the operators involved rather than solely relying on regularity properties of the coefficients.

In our work in the second funding period (project start was in 07/2021) we have in particular shown that even though the existing results on wave-type equations with $C^s$ coefficients with $s<2$ are sharp in general, there exist classes of structured Lipschitz coefficients for which the available fixed-time $L^p$ and Strichartz estimates can be considerably improved.

Fixed-time $L^p$ estimates and Strichartz estimates for wave equations

We investigate linear wave equations of the form \begin{align*} \begin{cases} \partial_t^2 u +A u =0, \\ u(0,\cdot) =f, \quad u_t(0,\cdot) = g, \end{cases} \end{align*} on $\mathbb{R}\times\mathbb{R}^d$, where $A$ is a second order elliptic operator in either divergence or non-divergence form. In [FP20], we obtained fixed-time $L^p$ estimates for operators of the form \begin{equation*} A=-\sum_{j=1}^d b_j \partial_j (a_j \partial_j) \end{equation*} with coefficients $a_j,b_j \in C^{0,1}(\mathbb{R}^d)$ that only depend on $x_j$ for $j=1,\ldots,d$. The structural assumption we impose means that the stiffness $a_j$ for the deformation $\partial_j u$ in $x_j$-direction only depends on $x_j$. Compared to the wave equation with smooth coefficients, there is no loss of regularity in the initial data. In [Fre23], we establish analog results for lower order perturbations of $A$ by showing boundedness of multiplication operators on certain Hardy spaces that are adapted to the solution operator of the linear wave equation.

Under the same structural assumptions on the coefficients we obtained in [FS22] local-in-time Strichartz estimates with precise dependence on the Lipschitz norm of the coefficients. Global Strichartz estimates have been shown for coefficients with derivatives having small $L^1$ norm. Analogous results hold in the case of Schrödinger equations. We have also extended our estimates to coefficients that are merely locally in $BV$, using approximation results of Burq–Planchon. Our Strichartz estimates lead to local wellposedness in $L^2(\mathbb{R}^d)$ of (sub-)critical wave equations with power-type nonlinearities. As a consequence of the dispersive estimates, we furthermore obtain new results on spectral multipliers and Bochner–Riesz means for the operator $A$ as above.

Semilinear wave equations with slowly decaying data

In [RS23] wellposedness results for the cubic semilinear wave equation on $\mathbb{R}^2$ with slowly decaying data have been established. The data and solutions belong to adapted $L^p$ based Besov spaces with $p>2$, whereas in standard $L^p$ spaces one cannot expect such results as the wave group does not map $L^p$ into $L^p$. The results rely on new local smoothing estimates in these Besov spaces which are proved by $\ell^2$ decoupling. Using wave packet decompositions and polynomial partitioning, [Sch22] establishes new local smoothing estimates for wave equations on compact Riemannian manifolds.

Maxwell systems

We also treat related questions for the linear Maxwell system \begin{equation*} \partial_t (\varepsilon\mathbf{E}) = \text{curl} \mathbf{H}-\mathbf{J}, \qquad \partial_t (\mu\mathbf{H}) = -\text{curl} \mathbf{E}, \end{equation*} on $\mathbb{R}^3$ for the electric and magnetic fields $\mathbf{E}(t,x)\in\mathbb{R}^3$ and $\mathbf{H}(t,x)\in\mathbb{R}^3$, the current $\mathbf{J}(t,x)\in\mathbb{R}^3$, and the positive definite permittivities $\varepsilon(x)\in\mathbb{R}^{3\times3}_{\mathrm{sym}}$ and permeabilities $\mu(x)\in\mathbb{R}^{3\times3}_{\mathrm{sym}}$. The electric charge is $\rho= \nabla\cdot (\varepsilon\mathbf{E})$. In the two dimensional case on $\mathbb{R}^2$ one has fields $\mathbf{E}(t,x),\mathbf{J}(t,x)\in\mathbb{R}^2$ and $\mathbf{H}(t,x)\in\mathbb{R}$. In project A5 Strichartz estimates for the Maxwell system have been established in several situations, where the results are sharp for the two dimensional case, and for scalar $\varepsilon$ and $\mu$ on $\mathbb{R}^3$. For coefficients in $C^s$ with $s<2$ the same loss $\sigma$ in regularity as for the scalar wave equation in earlier works of Tataru has been obtained.

In [ScS22] we have studied the 2D Maxwell system on $\mathbb{R}^2$ in the case of structured Lipschitz coefficients $\varepsilon(x_1,x_2)=\operatorname{diag}(\varepsilon(x_2),\varepsilon(x_1))$ and $\mu=1$. We have proven frequency-localized Strichartz estimates with no loss in regularity compared to $C^2$ coefficients. If the permittivity $\varepsilon$ is slightly more regular and belongs to the first-order Besov space $B^1_{2,\infty}(\mathbb{R}^2)$, we have derived loss-less Strichartz estimates, including the inhomogeneity $\mathbf{J}$ on the right-hand side which may cause a nontrivial charge $\rho(t)= \rho(0)-\int_0^t \nabla\cdot\mathbf{J}(s) ds.$ The additional charge term is needed to compensate the degeneracy of the principal symbol, which is a fundamental difference to the wave case.


  1. . Oscillatory integral operators with homogeneous phase functions. J. Anal. Math., 547–582, December . URL Online first. [preprint] [bibtex]

  2. and . Strichartz estimates for Maxwell equations in media: the structured case in two dimensionals. Arch. Math., 121:425–436, October . URL https://10.1007/s00013-023-01898-3. [preprint] [bibtex]

  3. and . Strichartz estimates for equations with structured Lipschitz coefficients. J. Evol. Equ., 23(3):45, 34pp., June . URL [preprint] [bibtex]

  4. and . Nonlinear wave equations with slowly decaying initial data. J. Differential Equations, 350:152–188, March . URL https://10.1016/j.jde.2022.12.022. [preprint] [bibtex]


  1. . Multiplication operators on a class of Hardy spaces for Fourier integral operators. CRC 1173 Preprint 2023/1, Karlsruhe Institute of Technology, January . [bibtex]

  2. and . $L^p$ estimates for wave equations with specific $C^{0,1}$ coefficients. CRC 1173 Preprint 2020/29, Karlsruhe Institute of Technology, October . Revised version from April 2021. [bibtex]

Former staff
Name Title Function
Dr. Postdoctoral researcher