Space-time Norm Research Articles

Low Mach number limit of the global solution to the compressible Navier–Stokes system for large data in the critical Besov space

In this paper, we consider the compressible Navier–Stokes system around the constant equilibrium states and prove the existence of a unique global solution for arbitrarily large initial data in the scaling critical Besov space provided that the Mach number is sufficiently small, and the incompressible part of the initial velocity generates the global solution of the incompressible Navier–Stokes equation. Moreover, we consider the low Mach number limit and show that the compressible solution converges to the solution of the incompressible Navier–Stokes equation in some space time norms.

A Space-Time Legendre-Petrov-Galerkin Method for Third-Order Differential Equations

In this article, a space-time spectral method is considered to approximate third-order differential equations with non-periodic boundary conditions. The Legendre-Petrov-Galerkin discretization is employed in both space and time. In the theoretical analysis, rigorous proof of error estimates in the weighted space-time norms is obtained for the fully discrete scheme. We also formulate the matrix form of the fully discrete scheme by taking appropriate test and trial functions in both space and time. Finally, extensive numerical experiments are conducted for linear and nonlinear problems, and spectral accuracy is derived for both space and time. Moreover, the numerical results are compared with those computed by other numerical methods to confirm the efficiency of the proposed method.

Well-posedness for a coupled system of Kawahara/KdV type equations with polynomials nonlinearities

<p style='text-indent:20px;'>We consider the initial value problem (IVP) associated to a coupled system of modified Kawahara/KdV type equations with polynomials nonlinearities. For the model in question, the Cauchy problem is of interest, and is shown to be well-posed for given data in a Gevrey spaces. Our results make use of techniques presented in Grujić and Kalisch, who studied the Gevrey regularity for a class of water-wave models and the well-posedness of a IVP associated to a general equation. The proof relies on estimates in space-time norms adapted to the linear part of the equations. In particular, estimates in Bourgain spaces are proven for the linear and nonlinear terms of the system and the main result is obtained by a contraction principle. The class of system in view generalizes the system of modified Kawahara/KdV type equations studied by Kondo and Pes, which contains a number of systems arising in the modeling of waves in fluids, stability and instability of solitary waves and models for wave propagation in physical systems where both nonlinear and dispersive effects are important.</p>

Black holes in the quadratic-order extended vector–tensor theories

We investigate the static and spherically black hole solutions in the quadratic-order extended vector–tensor theories without suffering from the Ostrogradsky instabilities, which include the quartic-order (beyond-)generalized Proca theories as the subclass. We start from the most general action of the vector–tensor theories constructed with up to the quadratic-order terms of the first-order covariant derivatives of the vector field, and derive the Euler–Lagrange equations for the metric and vector field variables in the static and spherically symmetric backgrounds. We then substitute the spacetime metric functions of the Schwarzschild, Schwarzschild–de Sitter/anti-de Sitter, Reissner–Nordström-type, and Reissner–Nordström–de Sitter/anti-de Sitter-type solutions and the vector field with the constant spacetime norm into the Euler–Lagrange equations, and obtain the conditions for the existence of these black hole solutions. These solutions are classified into the two cases 1) the solutions with the vanishing vector field strength; the stealth Schwarzschild and the Schwarzschild–de Sitter/anti-de Sitter solutions, and 2) those with the nonvanishing vector field strength; the charged stealth Schwarzschild and the charged Schwarzschild–de Sitter/anti-de Sitter solutions, in the case that the tuning relation among the coupling functions is satisfied. In the latter case, if this tuning relation is violated, the solution becomes the Reissner–Nordström-type solution. We show that the conditions for the existence of these solutions are compatible with the degeneracy conditions for the class-A theories, and recover the black hole solutions in the generalized Proca theories as the particular cases.

A space-time certified reduced basis method for quasilinear parabolic partial differential equations

In this paper, we propose a certified reduced basis (RB) method for quasilinear parabolic problems with strongly monotone spatial differential operator. We provide a residual-based a posteriori error estimate for a space-time formulation and the corresponding efficiently computable bound for the certification of the method. We introduce a Petrov-Galerkin finite element discretization of the continuous space-time problem and use it as our reference in a posteriori error control. The Petrov-Galerkin discretization is further approximated by the Crank-Nicolson time-marching problem. It allows to use a POD-Greedy approach to construct the reduced-basis spaces of small dimensions and to apply the Empirical Interpolation Method (EIM) to guarantee the efficient offline-online computational procedure. In our approach, we compute the reduced basis solution in a time-marching framework while the RB approximation error in a space-time norm is controlled by our computable bound. Therefore, we combine a POD-Greedy approximation with a space-time Galerkin method.

A space-time spectral method for multi-dimensional Sobolev equations

In this paper, a space-time spectral method is applied to approximate the linear multi-dimensional Sobolev equations. The Legendre-Galerkin method and a dual-Petrov-Galerkin discretization are employed in space and time, respectively. Being different from the general Legendre-Galerkin method, the technique adopted in this study lies in diagonalization of the mass matrix by using Fourier-like basis functions to save the computing time and memory. Moreover, the detailed stability analysis and error estimates are provided in the weighted space-time norms. We also formulate the matrix forms of the fully discrete scheme. Finally, extensive numerical tests are implemented to verify the theoretical results and demonstrate the efficiency of our method.

Well-Posedness for a Coupled System of Kawahara/KdV Type Equations

We consider the initial value problem (IVP) associated to a coupled system of Kawahara/KdV type equations. We prove the well-posedness results for given data in a Gevrey spaces. The proof relies on estimates in space-time norms adapted to the linear part of the equations. In particular, estimates in Bourgain spaces are proven for the linear and nonlinear terms of the system and the main result is obtained by a contraction principle. The class of system in view contains a number of systems arising in the modeling of waves in fluids, stability and instability of solitary waves and models for wave propagation in physical systems where both nonlinear and dispersive effects are important. The techniques presented in this work were based in Grujic and Kalisch, who studied the Gevrey regularity for a class of water-wave models and the well-posedness of a IVP associated to a general equation, whose the initial data belongs to Gevrey spaces.

Scattering for the one-dimensional Klein–Gordon equation with exponential nonlinearity

We consider the asymptotic behavior of solutions to the Cauchy problem for the defocusing nonlinear Klein–Gordon equation (NLKG) with exponential nonlinearity in the one spatial dimension with data in the energy space [Formula: see text]. We prove that any energy solution has a global bound of the [Formula: see text] space-time norm, and hence, scatters in [Formula: see text] as [Formula: see text]. The proof is based on the argument by Killip–Stovall–Visan (Trans. Amer. Math. Soc. 364(3) (2012) 1571–1631). However, since well-posedness in [Formula: see text] for NLKG with the exponential nonlinearity holds only for small initial data, we use the [Formula: see text]-norm for some [Formula: see text] instead of the [Formula: see text]-norm, where [Formula: see text] denotes the [Formula: see text]th order [Formula: see text]-based Sobolev space.

A QSC method for fractional subdiffusion equations with fractional boundary conditions and its application in parameters identification

A quadratic spline collocation (QSC) method combined with L1 time discretization, named QSC-L1, is proposed to solve fractional subdiffusion equations with artificial boundary conditions. A novel norm-based stability and convergence analysis is carefully discussed, which shows that the QSC-L1 method is unconditionally stable in a discrete space–time norm, and has a convergence order O(τ2−α+h2), where τ and h are the temporal and spatial step sizes, respectively. Then, based on fast evaluation of the Caputo fractional derivative (see, Jiang et al., 2017), a fast version of QSC-L1 which is called QSC-FL1 is proposed to improve the computational efficiency. Two numerical examples are provided to support the theoretical results. Furthermore, an inverse problem is considered, in which some parameters of the fractional subdiffusion equations need to be identified. A Levenberg–Marquardt (L–M) method equipped with the QSC-FL1 method is developed for solving the inverse problem. Numerical tests show the effectiveness of the method even for the case that the observation data is contaminated by some levels of random noise.

Stabilization of wave dynamics by moving boundary

A wave equation on a time-dependent domain is considered. The shape of the domain changes according to a prescribed space/time-dependent velocity field. On the moving boundary the solution satisfies zero Dirichlet condition. It is known that if the domain keeps expanding at a “subsonic” speed, then the associated finite energy decays uniformly. Here, the scenario of interest is when the domain remains bounded and undergoes phases of expansion and contraction. Although the energy identity in this case is not necessarily conservative, it is shown that the L2 space–time norm of the normal trace remains a priori bounded at small normal speeds of the boundary, analogously to the classical Dirichlet wave problem on a static domain. In addition, it is demonstrated that small normal velocity but very large acceleration of the boundary is compatible with the known existence theory, provided the magnitude of the deformations is relatively small. An “adaptive” boundary movement control is proposed and implemented numerically. The control action is dynamically computed from the normal trace data and dissipates the energy by means of small deformations of the domain only.

Black holes in the generalized Proca theory

We investigate static and spherically symmetric black hole solutions in the generalized Proca theory which corresponds to the generalization of the shift-symmetric scalar–tensor Horndeski theory to the vector–tensor theory. Any solution obtained in this paper possesses a constant spacetime norm of the vector field, $$X:=-\frac{1}{2}g^{\mu \nu }A_\mu A_\nu =X_0=\mathrm{constant}$$ . The solutions in the theory with generalized quartic coupling $$G_4(X)$$ generalize the stealth Schwarzschild and the Schwarzschild- (anti-) de Sitter solutions obtained in the theory with the nonminimal coupling to the Einstein tensor $$G^{\mu \nu } A_\mu A_\nu $$ . While in the vector–tensor theory with the coupling $$G^{\mu \nu }A_\mu A_\nu $$ the electric charge does not explicitly affect the spacetime geometry, in more general cases with nonzero $$G_{4XX}(X_0)\ne 0$$ this property does not hold in general. The solutions in the theory with generalized cubic coupling $$G_3(X)$$ are given by the Schwarzschild- (anti-) de Sitter spacetime, where the dependence on $$G_3(X)$$ does not appear in the metric function.

Quantitative decay rates for dispersive solutions to the Einstein-scalar field system in spherical symmetry

In this paper, we study the future causally geodesically complete solutions of the spherically symmetric Einstein-scalar field system. Under the a priori assumption that the scalar field $\phi$ scatters locally in the scale-invariant bounded-variation (BV) norm, we prove that $\phi$ and its derivatives decay polynomially. Moreover, we show that the decay rates are sharp. In particular, we obtain sharp quantitative decay for the class of global solutions with small BV norms constructed by Christodoulou. As a consequence of our results, for every future causally geodesically complete solution with sufficiently regular initial data, we show the dichotomy that either the sharp power law tail holds or that the spacetime blows up at infinity in the sense that some scale invariant spacetime norms blow up.

A sharp scattering condition for focusing mass-subcritical nonlinear Schrödinger equation

This article is concerned with time global behavior of solutionsto focusing mass-subcritical nonlinear Schrödinger equation of power type withdata in a critical homogeneous weighted $L^2$ space. We givea sharp sufficient condition for scattering by proving existence ofa threshold solution which does not scatter at least forone time direction and of which initial data attains minimumvalue of a norm of the weighted $L^2$ space inall initial value of non-scattering solution. Unlike in the mass-criticalor -supercritical case, ground state is not a threshold. Thisis an extension of previous author's result to the casewhere the exponent of nonlinearity is below so-called Strauss number.A main new ingredient is a stability estimate in aLorenz-modified-Bezov type spacetime norm.

Duality-based two-level error estimation for time-dependent PDEs: Application to linear and nonlinear parabolic equations

We introduce a duality-based two-level error estimator for linear and nonlinear time-dependent problems. The error measure can be a space-time norm, energy norm, final-time error or other error related functional. The general methodology is developed for an abstract nonlinear parabolic PDE and subsequently applied to linear heat and nonlinear Cahn-Hilliard equations. The error due to finite element approximations is estimated with a residual weighted approximate-dual solution which is computed with two primal approximations at nested levels. We prove that the exact error is estimated by our estimator up to higher-order remainder terms. Numerical experiments confirm the theory regarding consistency of the dual-based two-level estimator. We also present a novel space-time adaptive strategy to control errors based on the new estimator.

Strichartz estimates and maximal operators for the wave equation in [formula omitted

We prove sharp Strichartz-type estimates in three dimensions, including some which hold in reverse space–time norms, for the wave equation with potential. These results are also tied to maximal operator estimates studied by Rogers–Villarroya, of which we prove a sharper version. As a sample application, we use these results to prove the local well-posedness and the global well-posedness for small initial data of semilinear wave equations in R3 with quintic or higher monomial nonlinearities.

A space–time variational approach to hydrodynamic stability theory

We present a hydrodynamic stability theory for incompressible viscous fluid flows based on a space–time variational formulation and associated generalized singular value decomposition of the (linearized) Navier–Stokes equations. We first introduce a linear framework applicable to a wide variety of stationary- or time-dependent base flows: we consider arbitrary disturbances in both the initial condition and the dynamics measured in a ‘data’ space–time norm; the theory provides a rigorous, sharp (realizable) and efficiently computed bound for the velocity perturbation measured in a ‘solution’ space–time norm. We next present a generalization of the linear framework in which the disturbances and perturbation are now measured in respective selected space–time semi-norms ; the semi-norm theory permits rigorous and sharp quantification of, for example, the growth of initial disturbances or functional outputs. We then develop a (Brezzi–Rappaz–Raviart) nonlinear theory which provides, for disturbances which satisfy a certain (rather stringent) amplitude condition, rigorous finite-amplitude bounds for the velocity and output perturbations. Finally, we demonstrate the application of our linear and nonlinear hydrodynamic stability theory to unsteady moderate Reynolds number flow in an eddy-promoter channel.

Derivation of the Cubic NLS and Gross–Pitaevskii Hierarchy from Manybody Dynamics in d = 3 Based on Spacetime Norms

We derive the defocusing cubic Gross-Pitaevskii (GP) hierarchy in dimension $d=3$, from an $N$-body Schr\"{o}dinger equation describing a gas of interacting bosons in the GP scaling, in the limit $N\rightarrow\infty$. The main result of this paper is the proof of convergence of the corresponding BBGKY hierarchy to a GP hierarchy in the spaces introduced in our previous work on the well-posedness of the Cauchy problem for GP hierarchies, \cite{chpa2,chpa3,chpa4}, which are inspired by the solutions spaces based on space-time norms introduced by Klainerman and Machedon in \cite{klma}. We note that in $d=3$, this has been a well-known open problem in the field. While our results do not assume factorization of the solutions, consideration of factorized solutions yields a new derivation of the cubic, defocusing nonlinear Schr\"odinger equation (NLS) in $d=3$.

Stability and Unconditional Uniqueness of Solutions for Energy Critical Wave Equations in High Dimensions

In this paper we establish a complete local theory for the energy-critical nonlinear wave equation (NLW) in high dimensions ℝ × ℝ d with d ≥ 6. We prove the stability of solutions under the weak condition that the perturbation of the linear flow is small in certain space-time norms. As a by-product of our stability analysis, we also prove local well-posedness of solutions for which we only assume the smallness of the linear evolution. These results provide essential technical tools that can be applied towards obtaining the extension to high dimensions of the analysis of Kenig and Merle [17] of the dynamics of the focusing (NLW) below the energy threshold. By employing refined paraproduct estimates we also prove unconditional uniqueness of solutions for d ≥ 6 in the natural energy class. This extends an earlier result by Planchon [26].

A new proof of existence of solutions for focusing and defocusing Gross-Pitaevskii hierarchies

We consider the cubic and quintic Gross-Pitaevskii (GP) hierarchies in d ≥ 1 dimensions, for focusing and defocusing interactions. We present a new proof of existence of solutions that does not require the a priori bound on the spacetime norm, which was introduced in the work of Klainerman and Machedon, [19], and used in our earlier work, [6].

Global well-posedness and scattering for a nonlinear Klein–Gordon system in low dimensions

This article investigates the global well-posedness and the scattering for a nonlinear Klein–Gordon system in spatial dimensions 1 and 2. We establish a Morawetz estimate for this system which is similar to the Morawetz estimate established by Nakanishi [K. Nakanishi, Energy scattering for nonlinear Klein–Gordon and Schrödinger equations in spatial dimensions 1 and 2, J. Funct. Anal. 169(1), pp. 201–225], combining this Morawetz estimate with the induction on energy argument developed by Bourgain [J. Bourgain, Global well-posedness of defocusing 3D critical NLS in the radial case, J. Am. Math. Soc. 12 (1999), pp. 145–171], the bound of a certain space-time norm and scattering result are obtained.