arXiv subjects

David M. Ambrose

Publications and source records attributed to David M. Ambrose.

At least 19 recordsLinked to original sources

arXiv2026

Non-decaying weak solutions to the 2D quasi-geostrophic equations

We investigate weak solutions to the two-dimensional quasi-geostrophic equations without dissipation. We establish global existence of weak solutions for temperature bounded and lacking spatial decay and velocity in the space $L^2_{ul}(\mathbb{R}^2)$. Our methods rely on a spectral Serfati identity, which we use to establish uniform $L^2_{ul}$ bounds on a sequence of velocities satisfying the dissipative equations. These bounds, combined with a maximum principle on the scalar temperature, allow us to pass to the zero-dissipation limit, giving global-in-time weak solutions.

math.AP↗

arXiv2026

Nonlinear Instability in the 2D Kuramoto-Sivashinsky equation

In this paper we analyze the Kuramoto-Sivashinsky equation (KSE), a model of flame-front propagation, on a two-dimensional square torus of arbitrary size $2 L$ with $L > π$. In this case, the linearized equation at the origin admits a finite number of growing modes, which corresponds to the positive eigenvalues of the linear operator $- Δ^2 - Δ$. The problem of analyzing the long-time behavior of solutions of the 2D KSE in two spatial dimensions remains largely open; the only global existence results are for sufficiently small tori, or for sufficiently anisotropic and thin domains, due to the lack of good a priori estimates. The main purpose of this paper is to analyze the instability around growing modes at the nonlinear level. More precisely, we consider the maximal growing mode $λ_0$ and we show that there is a finite dimensional manifold of initial data of size $\varepsilon$ arbitrarily small such that the corresponding solutions become of size $O(1)$ over a time-scale of order ${\rm log}(\varepsilon^{- 1})$. The proof is based on several ingredients such as a sharp quantitative construction of an approximate solution bifurcating from the maximal linearly growing mode, a fixed point argument with exponential weights to construct local in time solutions, and a continuation argument based on sharp energy estimates and para-differential calculus.

math.AP↗

arXiv2025

Pseudomeasure distributions for nonseparable, nonlocal mean field games

For a number of important mean field games models, the Hamiltonian is non-local and not additively separable. This means that the distribution of agents appears in the Hamiltonian only in an integral over the whole spatial domain. For mean field games with a class of such Hamiltonians, we prove existence of solutions for the mean field games system of partial differential equations, allowing pseudomeasure data for the distribution of agents. Specifically, this allows the initial distribution of agents to be a sum of Dirac masses. The existence theorem requires a smallness condition on the size of the terminal data for the value function (or, alternatively, on the size of the Hamiltonian); no smallness condition on the size of the initial data or on the size of the time horizon is required. We also prove uniqueness and continuous dependence results under the same type of smallness conditions. We prove continuous dependence under two complementary hypotheses on the initial data: strong convergence of a sequence of pseudomeasures, and weak-$*$ convergence of a sequence of bounded measures.

math.AP↗

arXiv2025

Exact periodic solutions of the generalized Constantin-Lax-Majda equation with dissipation

We present exact pole dynamics solutions to the generalized Constantin-Lax-Majda (gCLM) equation in a periodic geometry with dissipation $-Λ^σ$, where its spatial Fourier transform is $\widehat{Λ^σ}=|k|^σ$. The gCLM equation is a simplified model for singularity formation in the 3D incompressible Euler equations. It includes an advection term with parameter $a$, which allows different relative weights for advection and vortex stretching. There has been intense interest in the gCLM equation, and it has served as a proving ground for the development of methods to study singularity formation in the 3D Euler equations. Several exact solutions for the problem on the real line have been previously found by the method of pole dynamics, but only one such solution has been reported for the periodic geometry. We derive new periodic solutions for $a=0$ and $1/2$ and $σ=0$ and $1$, for which a closed collection of (periodically repeated) poles evolve in the complex plane. Self-similar finite-time blow-up of the solutions is analyzed and compared for the different values of $σ$, and to a global-in-time well-posedness theory for solutions with small data presented in a previous paper of the authors. Motivated by the exact solutions, the well-posedness theory is extended to include the case $a=0$, $σ\geq 0$. Several interesting features of the solutions are discussed.

math.AP↗

arXiv2025

Non-Decaying Solutions to the 2D Dissipative Quasi-Geostrophic Equations

We consider the surface quasi-geostrophic equation in two spatial dimensions, with subcritical diffusion (i.e. with fractional diffusion of order $2α$ for $α>\frac{1}{2}$.) We establish existence of solutions without assuming either decay at spatial infinity or spatial periodicity. One obstacle is that for $L^{\infty}$ data, the constitutive law may not be applicable, as Riesz transforms are unbounded. However, for $L^{\infty}$ initial data for which the constitutive law does converge, we demonstrate that there exists a unique solution locally in time, and that the constitutive law continues to hold at positive times. In the case that $α\in(\frac{1}{2},1]$ and that the initial data has some smoothness (specifically, if the data is in $C^{2}$), we demonstrate a maximum principle and show that this unique solution is actually classical and global in time. Then, a density argument allows us to show that mild solutions with only $L^{\infty}$ data are also global in time, and also possess this maximum principle. Finally, we introduce a related problem in which we replace the usual constitutive law for the surface quasi-geostrophic equation with a generalization of Sertfati type, and prove the same results for this relaxed model.

math.AP↗

arXiv2025

Coherent Structures in Flame Fronts

We study traveling waves in a coordinate-free model of flame fronts. The flame front is the interface between the burnt and unburnt phases of a gas undergoing combustion. The front therefore moves in a preferred direction, as the unburnt gas is consumed. In the horizontally periodic, vertically unbounded setting, we prove the existence of waves of permanent form which are traveling in the vertical direction. We also compute these waves. The analysis and computation use the framework of traveling waves in the arclength parameterization as previously developed by two of the authors and Wright.

math.AP↗

arXiv2025

Horizontally periodic generalized surface quasigeostrophic patches and layers

We study solutions to the $α$-SQG equations, which interpolate between the incompressible Euler and surface quasi-geostrophic equations. We extend prior results on existence of bounded patches, proving propagation of $H^k$-regularity of the patch boundary, $k \ge 3$, for finite time for patches that are periodic in one spatial dimension. Such periodic patches also encompass layers, or two-sided fronts. As the authors have treated the Euler case in prior work, we now primarily focus on the range of $α$ for which $α$-SQG lies strictly between the Euler and SQG equations.

math.AP↗

arXiv2024

Local well-posedness of the Benjamin-Ono equation with spatially quasiperiodic data

We consider the Benjamin-Ono equation in the spatially quasiperiodic setting. We establish local well-posedness of the initial value problem with initial data in quasiperiodic Sobolev spaces. This requires developing some of the fundamental properties of Sobolev spaces and the energy method for quasiperiodic functions. We discuss prospects for global existence. We demonstrate that while conservation laws still hold, these quantities no longer control the associated Sobolev norms, thereby preventing the establishment of global results by usual arguments.

math.AP↗

arXiv2024

Asymptotics of two-dimensional hydroelastic waves: The zero mass, zero bending limit

We consider two-dimensional hydroelastic waves, in which a free fluid surface separates two fluids of infinite vertical extent. Elastic effects are accounted for at the interface, with a parameter measuring the elastic bending force and another parameter measuring the mass of the elastic sheet. In prior work, the authors have demonstrated well-posedness of this initial value problem in Sobolev spaces. We now take the limit as these two parameters vanish. Since the size of the time interval of existence given by this prior theory vanishes as the mass and bending parameters go to zero, we now establish estimates which are uniform with respect to these parameters. We may then make an additional estimate which demonstrates that the solutions form a Cauchy sequence as the parameters go to zero, so that the limit may be taken. This demonstrates that the vortex sheet with surface tension is the zero mass, zero bending limit of hydroelastic waves in two spatial dimensions.

math.AP↗

arXiv2024

Kinetic-type Mean Field Games with Non-separable Local Hamiltonians

We prove well-posedness of a class of kinetic-type Mean Field Games, which typically arise when agents control their acceleration. Such systems include independent variables representing the spatial position as well as velocity. We consider non-separable Hamiltonians without any structural conditions, which depend locally on the density variable. Our analysis is based on two main ingredients: an energy method for the forward-backward system in Sobolev spaces, on the one hand and on a suitable vector field method to control derivatives with respect to the velocity variable, on the other hand. The careful combination of these two techniques reveals interesting phenomena applicable for Mean Field Games involving general classes of drift-diffusion operators and nonlinearities. While many prior existence theories for general Mean Field Games systems take the final datum function to be smoothing, we can allow this function to be non-smoothing, i.e. also depending locally on the final measure. Our well-posedness results hold under an appropriate smallness condition, assumed jointly on the data.

math.AP↗

arXiv2024

Improved regularity and analyticity of Cannone-Karch solutions of the three-dimensional Navier-Stokes equations on the torus

We consider the three-dimensional Navier-Stokes equations, with initial data having second derivatives in the space of pseudomeasures. Solutions of this system with such data have been shown to exist previously by Cannone and Karch. As the Navier-Stokes equations are a parabolic system, the solutions gain regularity at positive times. We demonstrate an improved gain of regularity at positive times as compared to that demonstrated by Cannone and Karch. We further demonstrate that the solutions are analytic at all positive times, with lower bounds given for the radius of analyticity.

math.AP↗

arXiv2024

The velocity field and Birkhoff-Rott integral for non-decaying, non-periodic vortex sheets

The Birkhoff-Rott integral expresses the fluid velocity on a vortex sheet. This integral converges if certain quantities decay at horizontal infinity, but can also be summed over periodic images in the horizontally periodic case. However, non-decaying, non-periodic cases are also of interest, such as the interaction of periodic wavetrains with non-commensurate periods (i.e. spatially quasiperiodic solutions), or non-periodic disturbances to periodic wavetrains. We therefore develop a more general single formula for the Birkhoff-Rott integral, which unifies and extends the cases of decay and periodicity. We verify that under some reasonable conditions this new version of the Birkhoff-Rott integral is the restriction to the vortex sheet of an incompressible, irrotational velocity field, with continuous normal component but with a jump in tangential velocity across the vortex sheet. We give a number of examples of non-decaying, non-periodic sheet positions and sheet strengths for which our assumptions may be verified. While we develop this in the case of two-dimensional fluids, the methodology applies equally well to three-dimensional fluids.

physics.flu-dyn↗

arXiv2023

Existence and analyticity of solutions of the Kuramoto-Sivashinsky equation with singular data

We prove existence of solutions to the Kuramoto-Sivashinsky equation with low-regularity data, in function spaces based on the Wiener algebra and in pseudomeasure spaces. In any spatial dimension, we allow the data to have its antiderivative in the Wiener algebra. In one spatial dimension, we also allow data which is in a pseudomeasure space of negative order. In two spatial dimensions, we also allow data which is in a pseudomeasure space one derivative more regular than in the one-dimensional case. In the course of carrying out the existence arguments, we show a parabolic gain of regularity of the solutions as compared to the data. Subsequently, we show that the solutions are in fact analytic at any positive time in the interval of existence.

math.AP↗

arXiv2022

Contour dynamics and global regularity for periodic vortex patches and layers

We study vortex patches for the 2D incompressible Euler equations. Prior works on this problem take the support of the vorticity (i.e., the vortex patch) to be a bounded region. We instead consider the horizontally periodic setting. This includes both the case of a periodic array of bounded vortex patches and the case of vertically bounded vortex layers. We develop the contour dynamics equation for the boundary of the patch in this horizontally periodic setting, and demonstrate global $C^{1,ε}$ regularity of this patch boundary. In the process of formulating the problem, we consider different notions of periodic solutions of the 2D incompressible Euler equations, and demonstrate equivalence of these.

math.AP↗

arXiv2022

Global existence and singularity formation for the generalized Constantin-Lax-Majda equation with dissipation: The real line vs. periodic domains

The question of global existence versus finite-time singularity formation is considered for the generalized Constantin-Lax-Majda equation with dissipation $-Λ^σ$, where $\widehat {Λ^σ}=|k|^σ$, both for the problem on the circle $x \in [-π,π]$ and the real line. In the periodic geometry, two complementary approaches are used to prove global-in-time existence of solutions for $σ\geq 1$ and all real values of an advection parameter $a$ when the data is small. We also derive new analytical solutions in both geometries when $a=0$, and on the real line when $a=1/2$, for various values of $ σ$. These solutions exhibit self-similar finite-time singularity formation, and the similarity exponents and conditions for singularity formation are fully characterized. We revisit an analytical solution on the real line due to Schochet for $a=0$ and $σ=2$, and reinterpret it terms of self-similar finite-time collapse. The analytical solutions on the real line allow finite-time singularity formation for arbitrarily small data, even for values of $σ$ that are greater than or equal to one, thereby illustrating a critical difference between the problems on the real line and the circle. The analysis is complemented by accurate numerical simulations, which are able to track the formation and motion singularities in the complex plane. The computations validate and extend the analytical theory.

math.AP↗

arXiv2022

Well-posedness of mean field games master equations involving non-separable local Hamiltonians

In this paper we construct short time classical solutions to a class of master equations in the presence of non-degenerate individual noise arising in the theory of mean field games. The considered Hamiltonians are non-separable and $local$ functions of the measure variable, therefore the equation is restricted to absolutely continuous measures whose densities lie in suitable Sobolev spaces. Our results hold for smooth enough Hamiltonians, without any additional structural conditions as convexity or monotonicity.

math.AP↗

arXiv2022

Existence of Solutions to Fluid Equations in Hölder and Uniformly Local Sobolev Spaces

We establish short-time existence of solutions to the surface quasi-geostrophic equation in both the Hölder spaces $C^r(\mathbb{R}^2)$ for $r>1$ and the uniformly local Sobolev spaces $H^s_{ul}(\mathbb{R}^2)$ for $s\geq 3$. Using methods similar to those for the surface quasi-geostrophic equation, we also obtain short-time existence for the three-dimensional Euler equations in uniformly local Sobolev spaces.

math.AP↗

arXiv2022

Numerical Algorithms for Water Waves with Background Flow over Obstacles and Topography

We present two accurate and efficient algorithms for solving the incompressible, irrotational Euler equations with a free surface in two dimensions with background flow over a periodic, multiply-connected fluid domain that includes stationary obstacles and variable bottom topography. One approach is formulated in terms of the surface velocity potential while the other evolves the vortex sheet strength. Both methods employ layer potentials in the form of periodized Cauchy integrals to compute the normal velocity of the free surface, are compatible with arbitrary parameterizations of the free surface and boundaries, and allow for circulation around each obstacle, which leads to multiple-valued velocity potentials but single-valued stream functions. We prove that the resulting second-kind Fredholm integral equations are invertible, possibly after a physically motivated finite-rank correction. In an angle-arclength setting, we show how to avoid curve reconstruction errors that are incompatible with spatial periodicity. We use the proposed methods to study gravity-capillary waves generated by flow around several elliptical obstacles above a flat or variable bottom boundary. In each case, the free surface eventually self-intersects in a splash singularity or collides with a boundary. We also show how to evaluate the velocity and pressure with spectral accuracy throughout the fluid, including near the free surface and solid boundaries. To assess the accuracy of the time evolution, we monitor energy conservation and the decay of Fourier modes and compare the numerical results of the two methods to each other. We implement several solvers for the discretized linear systems and compare their performance. The fastest approach employs a graphics processing unit (GPU) to construct the matrices and carry out iterations of the generalized minimal residual method (GMRES).

physics.flu-dyn↗