Low Regularity Initial Data Research Articles

Global Regularity for Gravity Unstable Muskat Bubbles

In this paper, we study the dynamics of fluids in porous media governed by Darcy’s law: the Muskat problem. We consider the setting of two immiscible fluids of different densities and viscosities under the influence of gravity in which one fluid is completely surrounded by the other. This setting is gravity unstable because along a portion of the interface, the denser fluid must be above the other. Surprisingly, even without capillarity, the circle-shaped bubble is a steady state solution moving with vertical constant velocity determined by the density jump between the fluids. Taking advantage of our discovery of this steady state, we are able to prove global in time existence and uniqueness of dynamic bubbles of nearly circular shapes under the influence of surface tension. We prove this global existence result for low regularity initial data. Moreover, we prove that these solutions are instantly analytic and decay exponentially fast in time to the circle.

Low regularity solutions for the Vlasov–Poisson–Landau/Boltzmann system

In the paper, we are concerned with the nonlinear Cauchy problem on the Vlasov–Poisson–Landau/Boltzmann system around global Maxwellians in a torus or a finite channel. The main goal is to establish the global existence and large time behaviour of small amplitude solutions for a class of low regularity initial data. The molecular interaction type is restricted to the case of hard potentials for two classical collision operators because of the effect of the self-consistent forces. The result extends the one by Duan–Liu–Sakamoto-Strain (Duan et al 2021 Commun. Pure Appl. Math. 74 932–1020) for the pure Landau/Boltzmann equation to the case of the VPL and VPB systems.

A Fourier Integrator for the Cubic Nonlinear Schrödinger Equation with Rough Initial Data

Standard numerical integrators suffer from an order reduction when applied to nonlinear Schrodinger equations with low-regularity initial data. For example, standard Strang splitting requires the b...

Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in 2D

This paper is concerned with the Cauchy problem of the Klein-Gordon-Zakharov system with very low regularity initial data. We prove the bilinear estimates which are crucial to get the local in time well-posedness. The estimates are established by the Fourier restriction norm method. We utilize the nonlinear version of the classical Loomis-Whitney inequality.

Optimal Time Decay of Navier–Stokes Equations with Low Regularity Initial Data

In this paper, we study the optimal time decay rate of isentropic Navier-Stokes equations under the low regularity assumptions about initial data. In the previous works about optimal time decay rate, the initial data need to be small in $H^{[N/2]+2}(\mathbb{R}^{N})$. Our work combined negative Besov space estimates and the conventional energy estimates in Besov space framework which is developed by R. Danchin. Though our methods, we can get optimal time decay rate with initial data just small in $\dot{B}^{N/2-1, N/2+1} \cap \dot{B}^{N/2-1, N/2}$ and belong to some negative Besov space(need not to be small). Finally, combining the recent results in \cite{zhang2014} with our methods, we can only need the initial data to be small in homogeneous Besov space $\dot{B}^{N/2-2, N/2} \cap \dot{B}^{N/2-1}$ to get the optimal time decay rate in space $L^{2}$.

Low regularity solutions to the Chern-Simons-Dirac and the Chern-Simons-Higgs equations in the Lorenz gauge

ABSTRACTIn this paper, we address the problem of local well-posedness of the Chern-Simons-Dirac (CSD) and the Chern-Simons-Higgs (CSH) equations in the Lorenz gauge for low regularity initial data. One of our main contributions is the uncovering of a null structure of (CSD). Combined with the standard machinery of Xs, b spaces, we obtain local well-posedness of (CSD) for initial data . Moreover, it is observed that the same techniques applied to (CSH) lead to a quick proof of local well-posedness for initial data , , which improves the previous result of Selberg and Tesfahun.

Low regularity well-posedness for Gross-Neveu equations

We address the problem of local and global well-posedness of Gross-Neveu (GN) equations for low regularity initial data. Combined with the standard machinery of $X_R$, $Y_R$ and $X^{s,b}$ spaces, we obtain local-wellposedness of (GN) for initial data $u, v \in H^s$ with $s\geq 0$. To prove the existence of global solution for the critical space $L^2$, we show non concentration of $L^2$ norm.

The local well-posedness for nonlinear fourth-order Schrödinger equation with mass-critical nonlinearity and derivative

We study the Cauchy problem of the nonlinear fourth-order Schrödinger equation with mass-critical nonlinearity and derivative: i u t +a u x x x x +b u 2 u ¯ x x +c | u | 8 u=0, x∈R, t∈R, where a, b, and c are real numbers. We obtain the local well-posedness for the Cauchy problem with low regularity initial value data by the Fourier restriction norm method.

Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data

In the present paper, we consider the Cauchy problem of a system of quadratic derivative nonlinear Schr\"odinger equations which was introduced by M. Colin and T. Colin (2004) as a model of laser-plasma interaction. The local existence of the solution of the system in the Sobolev space $H^{s}$ for $s>d/2+3$ is proved by M. Colin and T. Colin. We prove the well-posedness of the system with low regularity initial data. For some cases, we also prove the well-posedness and the scattering at the scaling critical regularity by using $U^{2}$ space and $V^{2}$ space which are applied to prove the well-posedness and the scattering for KP-II equation at the scaling critical regularity by Hadac, Herr and Koch (2009).

A New Length Estimate for Curve Shortening Flow and Low Regularity Initial Data

In this paper we introduce a geometric quantity, the r-multiplicity, that controls the length of a smooth curve as it evolves by curve shortening flow (CSF). The length estimates we obtain are used to prove results about the level set flow in the plane. If K is locally-connected, connected and compact, then the level set flow of K either vanishes instantly, fattens instantly or instantly becomes a smooth closed curve. If the compact set in question is a Jordan curve J, then the proof proceeds by using the r-multiplicity to show that if γ n is a sequence of smooth curves converging uniformly to J, then the lengths $${\fancyscript{L}({\gamma_n}_t)}$$ , where γ n t denotes the result of applying CSF to γ n for time t, are uniformly bounded for each t > 0. Once the level set flow has been shown to be smooth we prove that the Cauchy problem for CSF has a unique solution if the initial data is a finite length Jordan curve.

The Lagrangian Averaged Navier–Stokes equation with rough data in Sobolev spaces

The Lagrangian Averaged Navier–Stokes equation is a recently derived approximation to the Navier–Stokes equation. In this article we prove the existence of short time solutions to the incompressible, isotropic Lagrangian Averaged Navier–Stokes equation with low regularity initial data in Sobolev spaces Ws,p(Rn) for 1<p<∞. For L2-based Sobolev spaces, we obtain global existence results. More specifically, we achieve local existence with initial data in the Sobolev space Hn/2p,p(Rn). For initial data in H3/4,2(R3), we obtain global existence, improving on previous global existence results, which required data in H3,2(R3).

Existence of Solutions for the Debye-Hückel System with Low Regularity Initial Data

In this paper we study existence of solutions for the Cauchy problem of the Debye-H\"{u}ckel system with low regularity initial data. By using the Chemin-Lerner time-space estimate for the heat equation, we prove that there exists a unique local solution if the initial data belongs to the Besov space $\dot{B}^{s}_{p,q}(\mathbb{R}^{n})$ for $-3/2<s\leq-2+\frac{n}{2}$, $p=\frac{n}{s+2}$ and $1\leq q\leq \infty$, and furthermore, if the initial data is sufficiently small then the solution is global. This result improves the regularity index of the initial data space in previous results on this model.

Orbital stability for periodic standing waves of the Klein–Gordon–Zakharov system and the beam equation

The existence and stability of spatially periodic waves \({\left(e^{i{\omega}t}\varphi_\omega, \psi_\omega\right)}\) in the Klein–Gordon–Zakharov (KGZ) system are studied. We show a local existence result for low regularity initial data. Then, we construct a one-parameter family of periodic dnoidal waves for (KGZ) system when the period is bigger than \({\sqrt{2}\pi}\) . We show that these waves are stable whenever an appropriate function satisfies the standard Grillakis–Shatah–Strauss (Grillakis et al. J Funct Anal 74(1):160–197, 1987; Grillakis et al. J Funct Anal 94(2):308–348, 1990) type condition. We compute the intervals for the parameter ω explicitly in terms of L and by taking the limit L → ∞ we recover the previously known stability results for the solitary waves in the whole line case. For the beam equation, we show the existence of spatially periodic standing waves and show that orbital stability holds if an appropriate functional satisfies Grillakis–Shatah–Strauss type condition.

On cauchy problem of the Benney-Lin equation with low regularity initial data

In this work we prove that the initial value problem of the Benney-Lin equation ut + uxxx + β(uxx + uxxxx) + ηuxxxxx + uux = 0 (x ∈ ℝ, t ≥ 0), where β > 0 and η ∈ ℝ, is locally well-posed in Sobolev spaces Hs(R) for s ≥ −7/5. The method we use to prove this result is the bilinear estimate method initiated by Bourgain.

Quadratic nonlinear derivative Schrödinger equations - Part 2

In this paper we consider the local well-posedness theory for the quadratic nonlinear Schrodinger equation with low regularity initial data in the case when the nonlinearity contains derivatives. We work in 2 + 1 dimensions and prove a local well-posedness result for small initial data with low regularity.

Well-posedness of the Cauchy problem of a water wave equation with low regularity initial data

In this paper we investigate local well-posedness of the IVP of a class of fifth-order nonlinear dispersive equations including the equation of Benney modeling the interaction of short and long water waves. By introducing the modified Sobolev spaces H ( s , ω ) ( R ) and the modified Bourgain space X s , ω , b ( R 2 ) , and making bilinear estimates in the space X s , ω , b ( R 2 ) , we prove that the IVP of this class of equations is locally well-posed in H ( s , 1 4 ) ( R ) for any s > − 1 4 .

The Einstein-Scalar Field Constraints on Asymptotically Euclidean Manifolds*

By using the conformal method, solutions of the Einstein-scalar field gravitational constraint equations are obtained. Handling scalar fields is a bit more challenging than handling matter fields such as fluids, Maxwell fields or Yang-Mills fields, because the scalar field introduces three extra terms into the Lichnerowicz equation, rather than just one. The proofs are constructive and allow for arbitrary dimension (> 2) as well as low regularity initial data.

The cauchy problem for a shallow water type equation

We prove existence and uniqueness of local and global solutions of the periodic Cauchy problem for a higher order shallow water type equation under low regularity initial data. Using Fourier analysis we first prove local estimates in appropriate spaces and then use a contraction mapping argument and a conserved norm to get global existence.