Localized Initial Data Research Articles

Existence and Uniqueness of Solutions for a Quasilinear KdV Equation with Degenerate Dispersion

AbstractWe consider a quasilinear KdV equation that admits compactly supported traveling wave solutions (compactons). This model is one of the most straightforward instances of degenerate dispersion, a phenomenon that appears in a variety of physical settings as diverse as sedimentation, magma dynamics and shallow water waves. We prove the existence and uniqueness of solutions with sufficiently smooth, spatially localized initial data. © 2019 Wiley Periodicals, Inc.

Solution of the Two-Dimensional Dirac Equation with a Linear Potential and a Localized Initial Condition

In this paper, we consider the Cauchy problem with localized initial data for a two-dimensional massless Dirac equation with a linear potential.

Global solution for the 3D gravity water waves system above a flat bottom

Given any suitably small, localized, and smooth initial data, in this paper, we prove global regularity for the 3D finite depth gravity water wave system. As a byproduct, we rule out the small, localized traveling waves in 3D, which do exist for the same system in 2D.

Small data global solutions for the Camassa–Choi equations

We consider solutions to the Cauchy problem for an internal-wave model derived by Camassa–Choi (1996 J. Fluid Mech. 313 83–103). This model is a natural generalization of the Benjamin–Ono and intermediate long wave equations for weak transverse effects as in the case of the Kadomtsev–Petviashvili equations for the Korteweg-de Vries equation. For that reason they are often referred to as the KP-ILW or the KP–Benjamin–Ono equations regarding finite or infinite depth respectively. We prove the existence and long-time dynamics of global solutions from small, smooth, spatially localized initial data on . The techniques applied here involve testing by wave packet techniques developed by Ifrim and Tataru in (2015 Nonlinearity 28 2661–75; 2016 Bull. Soc. Math. France 144 369–94).

Life-span of solutions to semilinear wave equation with time-dependent critical damping for specially localized initial data

This paper is concerned with the blowup phenomena for initial value problem of semilinear wave equation with critical time-dependent damping term (DW). The result is the sharp upper bound of lifespan of solution with respect to the small parameter $\ep$ when $p_F(N)\leq p\leq p_0(N+\mu)$, where $p_F(N)$ denotes the Fujita exponent for the nonlinear heat equations and $p_0(n)$ denotes the Strauss exponent for nonlinear wave equation in $n$-dimension with $\mu=0$. Consequently, by connecting the result of D'Abbicco--Lucente--Reissig 2015, our result clarifies the threshold exponent $p_0(N+\mu)$ for dividing blowup phenomena and global existence of small solutions when $N=3$. The crucial idea is to construct suitable test functions satisfying the conjugate linear equation $\pa_t^2\Phi-\Delta \Phi-\pa_t(\frac{\mu}{1+t}\Phi)=0$ of (DW) including the Gauss hypergeometric functions; note that the construction of test functions is different from Zhou--Han in 2014.

Classical well-posedness in dispersive equations with nonlinearities of mild regularity, and a composition theorem in Besov spaces

For both localized and periodic initial data, we prove local existence in classical energy space H^{s}, s>frac{3}{2}, for a class of dispersive equations u_t + (n(u))_x + L u_x = 0 with nonlinearities of mild regularity. Our results are valid for symmetric Fourier multiplier operators L whose symbol is of temperate growth, and n(cdot ) in the local Sobolev space H^{s+2}_{mathrm {loc}}(mathbb {R}). In particular, the results include non-smooth and exponentially growing nonlinearities. Our proof is based on a combination of semigroup methods and a new composition result for Besov spaces. In particular, we extend a previous result for Nemytskii operators on Besov spaces on mathbb {R} to the periodic setting by using the difference–derivative characterization of Besov spaces.

Asymptotic solutions of the one-dimensional linearized Korteweg–de Vries equation with localized initial data

The Cauchy problem with localized initial data for the linearized Korteweg–de Vries equation is considered. In the case of constant coefficients, exact solutions for the initial function in the form of the Gaussian exponential are constructed. For a fairly arbitrary localized initial function, an asymptotic (with respect to the small localization parameter) solution is constructed as the combination of the Airy function and its derivative. In the limit as the parameter tends to zero, this solution becomes the exactGreen function for the Cauchy problem. Such an asymptotics is also applicable to the case of a discontinuous initial function. For an equation with variable coefficients, the asymptotic solution in a neighborhood of focal points is expressed using special functions. The leading front of the wave and its asymptotics are constructed.

Numerical study of blow-up and stability of line solitons for the Novikov–Veselov equation

We study numerically the evolution of perturbed Korteweg–de Vries solitons and of well localized initial data by the Novikov–Veselov (NV) equation at different levels of the ‘energy’ parameter E. We show that as , NV behaves, as expected, similarly to its formal limit, the Kadomtsev–Petviashvili equation. However at intermediate regimes, i.e. when is not very large, more varied scenarios are possible, in particular, blow-ups are observed. The mechanism of the blow-up is studied.

Uniform asymptotics of the boundary values of the solution in a linear problem on the run-up of waves on a shallow beach

We consider the Cauchy problem with spatially localized initial data for a two-dimensional wave equation with variable velocity in a domain Ω. The velocity is assumed to degenerate on the boundary ∂Ω of the domain as the square root of the distance to ∂Ω. In particular, this problems describes the run-up of tsunami waves on a shallow beach in the linear approximation. Further, the problem contains a natural small parameter (the typical source-to-basin size ratio) and hence admits analysis by asymptotic methods. It was shown in the paper “Characteristics with singularities and the boundary values of the asymptotic solution of the Cauchy problem for a degenerate wave equation” [1] that the boundary values of the asymptotic solution of this problem given by a modified Maslov canonical operator on the Lagrangian manifold formed by the nonstandard characteristics associatedwith the problemcan be expressed via the canonical operator on a Lagrangian submanifold of the cotangent bundle of the boundary. However, the problem as to how this restriction is related to the boundary values of the exact solution of the problem remained open. In the present paper, we show that if the initial perturbation is specified by a function rapidly decaying at infinity, then the restriction of such an asymptotic solution to the boundary gives the asymptotics of the boundary values of the exact solution in the uniform norm. To this end, we in particular prove a trace theorem for nonstandard Sobolev type spaces with degeneration at the boundary.

The Bramson logarithmic delay in the cane toads equations

We study a non-local reaction-diffusion-mutation equation modelling the spreading of a cane toads population structured by a phenotypical trait responsible for the spatial diffusion rate. When the trait space is bounded, the cane toads equation admits travelling wave solutions as shown in an earlier work of the first author and V. Calvez. Here, we prove a Bramson type spreading result: the lag between the position of solutions with localized initial data and that of the travelling waves grows as ( 3 / ( 2 λ ∗ ) ) log ⁡ t (3/(2\lambda ^*))\log t . This result relies on a present-time Harnack inequality which allows one to compare solutions of the cane toads equation to those of a Fisher-KPP type equation that is local in the trait variable.

Propagation of a linear wave created by a spatially localized perturbation in a regular lattice and punctured Lagrangian manifolds

The following results are obtained for the Cauchy problem with localized initial data for the crystal lattice vibration equations with continuous and discrete time: (i) the asymptotics of the solution is determined by Lagrangian manifolds with singularities (“punctured” Lagrangian manifolds); (ii) Maslov’s canonical operator is defined on such manifolds as a modification of a new representation recently obtained for the canonical operator by the present authors together with A. I. Shafarevich (Dokl. Ross. Akad. Nauk 46 (6), 641–644 (2016)); (iii) the projection of the Lagrangian manifold onto the configuration plane specifies a bounded oscillation region, whose boundary (which is naturally referred to as the leading edge front) is determined by the Hamiltonians corresponding to the limit wave equations; (iv) the leading edge front is a special caustic, which possibly contains stronger focal points. These observations, together with earlier results, lead to efficient formulas for the wave field in a neighborhood of the leading edge front.

Well-posedness for the three-dimensional compressible liquid crystal flows

This paper is concerned with the initial-boundary value problem for the three-dimensional compressible liquid crystal flows. The system consists of the Navier-Stokes equations describing the evolution of a compressible viscous fluid coupled with various kinematic transport equations for the heat flow of harmonic maps into $\mathbb{S}^2$. Assuming the initial density has vacuum and the initial data satisfies a natural compatibility condition, the existence and uniqueness is established for the local strong solution with large initial data and also for the global strong solution with initial data being close to an equilibrium state. The existence result is proved via the local well-posedness and uniform estimates for a proper linearized system with convective terms.

Characteristics with singularities and the boundary values of the asymptotic solution of the Cauchy problem for a degenerate wave equation

We consider the Cauchy problem with spatially localized initial data for the twodimensional wave equation degenerating on the boundary of the domain. This problem arises, in particular, in the theory of tsunami wave run-up on a shallow beach. Earlier, S. Yu. Dobrokhotov, V. E. Nazaikinskii, and B. Tirozzi developed a method for constructing asymptotic solutions of this problem. The method is based on a modified Maslov canonical operator and on characteristics (trajectories) unbounded in the momentum variables; such characteristics are nonstandard from the viewpoint of the theory of partial differential equations. In a neighborhood of the velocity degeneration line, which is a caustic of a special form, the canonical operator is defined via the Hankel transform, which arises when applying Fock’s quantization procedure to the canonical transformation regularizing the above-mentioned nonstandard characteristics in a neighborhood of the velocity degeneration line (the boundary of the domain). It is shown in the present paper that the restriction of the asymptotic solutions to the boundary is determined by the standard canonical operator, which simplifies the asymptotic formulas for the solution on the boundary dramatically; for initial perturbations of special form, the solutions can be expressed via simple algebraic functions.

On the dispersionless Kadomtsev–Petviashvili equation with arbitrary nonlinearity and dimensionality: exact solutions, longtime asymptotics of the Cauchy problem, wave breaking and shocks

We study the generalization of the dispersionless Kadomtsev–Petviashvili (dKP) equation in dimensions and with nonlinearity of degree , a model equation describing the propagation of weakly nonlinear, quasi one-dimensional waves in the absence of dispersion and dissipation, and arising in several physical contexts, like acoustics, plasma physics, hydrodynamics and nonlinear optics. In 2 + 1 dimensions and with quadratic nonlinearity, this equation is integrable through a novel inverse scattering transform, and it has been recently shown to be a prototype model equation in the description of the two-dimensional wave breaking of localized initial data. In higher dimensions and with higher nonlinearity, the generalized dKP equations are not integrable, but their invariance under motions on the paraboloid allows one to construct in this paper a family of exact solutions describing waves constant on their paraboloidal wave front and breaking simultaneously in all points of it, developing after breaking either multivaluedness or single-valued discontinuous profiles (shocks). Then such exact solutions are used to build the longtime behavior of the solutions of the Cauchy problem, for small and localized initial data, showing that wave breaking of small initial data takes place in the longtime regime if and only if . Lastly, the analytic aspects of such wave breaking are investigated in detail in terms of the small initial data, in both cases in which the solution becomes multivalued after breaking or it develops a shock. These results, contained in the 2012 master’s thesis of one of the authors (FS) [], generalize those obtained in [] for the dKP equation in dimensions with quadratic nonlinearity, and are obtained following the same strategy.

The Lifespan of Small Data Solutions to the KP-I

Author(s): Harrop-Griffiths, Benjamin; Ifrim, Mihaela; Tataru, Daniel | Abstract: We show that for small, localized initial data there exists a global solution to the KP-I equation in a Galilean-invariant space using the method of testing by wave packets.

Single point gradient blow-up on the boundary for a Hamilton-Jacobi equation with $p$-Laplacian diffusion

We study the initial-boundary value problem for the Hamilton-Jacobi equation with nonlinear diffusion $u_t=\Delta _p u+|\nabla u|^q$ in a two-dimensional domain for $q>p>2$. It is known that the spatial derivative of solutions may become unbounded in finite time while the solutions themselves remain bounded. We show that, for suitably localized and monotone initial data, the gradient blow-up occurs at a single point of the boundary. Such a result was known up to now only in the case of linear diffusion ($p=2$). The analysis in the case $p>2$ is considerably more delicate.

Dynamics of nonnegative solutions of one-dimensional reaction–diffusion equations with localized initial data. Part I: A general quasiconvergence theorem and its consequences

ABSTRACTWe consider the Cauchy problem where f is a locally Lipschitz function on ℝ with f(0) = 0, and u0 is a nonnegative function in C0(ℝ), the space of continuous functions with limits at ± ∞ equal to 0. Assuming that the solution u is bounded, we study its large-time behavior from several points of view. One of our main results is a general quasiconvergence theorem saying that all limit profiles of u(·, t) in are steady states. We also prove convergence results under additional conditions on u0. In the bistable case, we characterize the solutions on the threshold between decay to zero and propagation to a positive steady state and show that the threshold is sharp for each increasing family of initial data in C0(ℝ).

Born–Oppenheimer Approximation for an Atom in Constant Magnetic Fields

We obtain a reduction scheme for the study of the quantum evolution of an atom in constant magnetic fields using the method developed by Martinez, Nenciu and Sordoni based on the construction of almost invariant subspace. In Martinez-Sordoni \cite{MaSo2} such a case is also studied but their reduced Hamiltonian includes the vector potential terms. In this paper, using the center of mass coordinates and constructing the almost invariant subspace different from theirs, we obtain the reduced Hamiltonian which does not include the vector potential terms. Using the reduced evolution we also obtain the asymptotic expantion of the evolution for a specific localized initial data, which verifies the straight motion of an atom in constatnt magnetic fields.

Local non-negative initial data scalar characterization of the Kerr solution

For any vacuum initial data set, we define a local, non-negative scalar quantity which vanishes at every point of the data hypersurface if and only if the data are {\em Kerr initial} data. Our scalar quantity only depends on the quantities used to construct the vacuum initial data set which are the Riemannian metric defined on the initial data hypersurface and a symmetric tensor which plays the role of the second fundamental form of the embedded initial data hypersurface. The dependency is {\em algorithmic} in the sense that given the initial data one can compute the scalar quantity by algebraic and differential manipulations, being thus suitable for an implementation in a numerical code. The scalar could also be useful in studies of the non-linear stability of the Kerr solution because it serves to measure the deviation of a vacuum initial data set from the Kerr initial data in a local and algorithmic way.

Erratum to: “Asymptotic Solutions of the Cauchy Problem with Localized Initial Conditions for Linearized Two-Dimensional Boussinesq-Type Equations with Variable Coefficient”

In the paper, asymptotic solutions of the Cauchy problem with localized initial data for the two-dimensional wave equation (with variable speed) which is also perturbed by (spatially) variable weakly dispersive components are constructed. We consider both the case of normal dispersion occurring in the linearized Boussinesq equation for water waves over smoothly changing bottom and the case of anomalous dispersion arising when studying the wave equation with rapidly oscillating velocity. With regard to the fact that the front of the solution has focal points and self-intersection points, we present formulas based on the modified Maslov canonical operator in the case of initial perturbations of a rather general form which decrease at infinity. For perturbations of special form, we express the asymptotic behavior of a solution in the vicinity of the front, using derivatives of the sum of squares of the Airy functions Ai and Bi.