We consider the Fokas method expression for the solution of the heat equation on the half line with Dirichlet data and we study in detail its boundary behaviour near the spatiotemporal domain boundaries, i.e., the semi-axes, infinity and the origin, by analyzing the integrals involved. We also study the boundary behaviour of the derivatives of the solution. In particular we give conditions on the data which guarantee the extension of the solution to a smooth function up to the semi-infinite boundaries.
We study the Schwartz class of initial-boundary value (IBV) problems for the integrable Fokas-Lenells equation on the half-line via the Deift-Zhou's nonlinear descent method analysis of the corresponding Riemann-Hilbert problem such that the asymptotics of the Schwartz class of IBV problems as t\to\infty is presented.
Initial-boundary value problems for 1-dimensional `completely integrable' equations can be solved via an extension of the inverse scattering method, which is due to Fokas and his collaborators. A crucial feature of this method is that it requires the values of more boundary data than given for a well-posed problem. In the case of cubic NLS, knowledge of the Dirichet data suffices to make the problem well-posed but the Fokas method also requires knowledge of the values of Neumann data. The study of the Dirichlet to Neumann map is thus necessary before the application of the `Fokas transform'. I…
The linearization of the classical Boussinesq system is solved explicitly in the case of nonzero boundary conditions on the half-line. The analysis relies on the unified transform method of Fokas and is performed in two different frameworks: (i) by exploiting the recently introduced extension of Fokas's method to systems of equations; (ii) by expressing the linearized classical Boussinesq system as a single, higher-order equation which is then solved via the usual version of the unified transform. The resulting formula provides a novel representation for the solution of the linearized classica…
Initial-boundary value problems for 1-dimensional `completely integrable' equations can be solved via an extension of the inverse scattering method, which is due to Fokas and his collaborators. A crucial feature of this method is that it requires the values of more boundary data than given for a well-posed problem. In the case of cubic NLS, knowledge of the Dirichet data suffices to make the problem well-posed but the Fokas method also requires knowledge of the values of Neumann data. The study of the Dirichlet to Neumann map is thus necessary before the application of the `Fokas transform'. I…
The initial-boundary value problem (ibvp) for the $m$-th order dispersion Korteweg-de Vries (KdV) equation on the half-line with rough data and solution in restricted Bourgain spaces is studied using the Fokas Unified Transform Method (UTM). Thus, this work advances the implementation of the Fokas method, used earlier for the KdV on the half-line with smooth data and solution in the classical Hadamard space, consisting of function that are continuous in time and Sobolev in the spatial variable, to the more general Bourgain spaces framework of dispersive equations with rough data on the half-li…
We obtain solution representation formulas for some linear initial boundary value problems posed on the half space that involve mixed spatial derivative terms via the unified transform method (UTM), also known as the Fokas method. We first implement the method on the second order parabolic PDEs; in this case one can alternatively eliminate the mixed derivatives by a linear change of variables. Then, we employ the method to biharmonic problems, where it is not possible to eliminate the cross term via a linear change of variables. A basic ingredient of the UTM is the use of certain invariant map…
It was shown by the second author in (CPAA, 2019) for the biharmonic Schrödinger equation and most recently by Himonas and Mantzavinos (Nonlinearity, 2020) for 2D Schrödinger equation that Fokas method based formulas are capable of defining weak solutions of associated nonlinear initial boundary value problems (ibvps) below the Banach algebra threshold. In view of these results, we revisit the theory of interior-boundary Strichartz estimates for the Schrödinger equation posed on the right half line, considering both Dirichlet and Neumann cases. Finally, we apply these estimates to obtain low r…
We study the local and global wellposedness of the initial-boundary value problem for the biharmonic Schrödinger equation on the half-line with inhomogeneous Dirichlet-Neumann boundary data. First, we obtain a representation formula for the solution of the linear nonhomogenenous problem by using the Fokas method (also known as the \emph{unified transform method}). We use this representation formula to prove space and time estimates on the solutions of the linear model in fractional Sobolev spaces by using Fourier analysis. Secondly, we consider the nonlinear model with a power type nonlinearit…
It is known that the initial-boundary value problem for certain integrable partial differential equations (PDEs) on the half-line with integrable boundary conditions can be mapped to a special case of the Inverse Scattering Method (ISM) on the full-line. This can also be established within the so-called Unified Transform (UT) for initial-boundary value problems with linearizable boundary conditions. In this paper, we show a converse to this statement within the Ablowitz-Kaup-Newell-Segur (AKNS) scheme: the ISM on the full-line can be mapped to an initial-boundary value problem with linearizabl…
This work studies the initial-boundary value problem of the two-dimensional nonlinear Schrödinger equation on the half-plane with initial data in Sobolev spaces and Neumann or Robin boundary data in appropriate Bourgain spaces. It establishes well-posedness in the sense of Hadamard by utilizing the explicit solution formula for the forced linear initial-boundary value problem obtained via Fokas's unified transform, and a contraction mapping argument.
In this note, we give an elementary proof of the lack of null controllability for the heat equation on the half line by employing the machinery inherited by the unified transform, known also as the Fokas method. This approach also extends in a uniform way to higher dimensions and different initial-boundary value problems governed by the heat equation, suggesting a novel methodology for studying problems related to controllability.
The Majda-Biello system models the interaction of Rossby waves. It consists of two coupled KdV equations one of which has a parameter $α$ as coefficient of its dispersion. This work studies this system on the half line with Robin, Neumann, and Dirichlet boundary data. It shows that for $0<α<1$ or $1<α<4$ all these problems are well-posed for initial data in Sobolev spaces $H^s$, $s\ge 0$. For $α=1$ or $α>4$ well-posedness holds for Dirichlet data if $s>-3/4$, while for Neumann and Robin data it depends on the sign of the parameters involved in the data. For $α=4$ well-posedness of all problems…
We study the large time behaviour of the solution of a linear dispersive PDEs posed on a finite interval, when the prescribed boundary conditions are time periodic. We use the approach pioneered in Fokas & Lenells 2012 for nonlinear integrable PDEs. and then applied to linear problems on the half-line in Fokas & van der Weele 2021, to characterise necessary conditions for the solution of such a problem to be periodic, at least in an asymptotic sense. We then fully describe the periodicity properties of the solution in three important illustrative examples, recovering known results for the seco…
The initial-boundary value problem (IBVP) for the nonlinear Schrödinger (NLS) equation on the half-plane with nonzero boundary data is studied by advancing a novel approach recently developed for the well-posedness of the cubic NLS on the half-line, which takes advantage of the solution formula produced by the unified transform of Fokas for the associated linear IBVP. For initial data in Sobolev spaces on the half-plane and boundary data in Bourgain spaces arising naturally when the linear IBVP is solved with zero initial data, the present work provides a local well-posedness result for NLS in…
In this paper, we prove dispersion estimates for the boundary integral operator associated with the fourth order Schrödinger equation posed on the half line. Proofs of such estimates for domains with boundaries are rare and generally require highly technical approaches, as opposed to our simple treatment which is based on constructing a boundary integral operator of oscillatory nature via the Fokas method. Our method is uniform and can be extended to other higher order partial differential equations where the main equation possibly involves more than one spatial derivatives.
The initial-boundary value problem for the Kundu--Eckhaus equation on the half-line is considered in this paper by using the Fokas method. We will show that the solution $u(x,t)$ can be expressed in terms of the solution of a matrix Riemann--Hilbert problem formulated in the complex $k$-plane. Furthermore, based on a nonlinear steepest descent analysis of the associated Riemann--Hilbert problem, we can give the precise asymptotic formulas for the solution of the Kundu--Eckhaus equation on the half-line.
We study the diffusion (or heat) equation on a finite 1-dimensional spatial domain, but we replace one of the boundary conditions with a "nonlocal condition", through which we specify a weighted average of the solution over the spatial interval. We provide conditions on the regularity of both the data and weight for the problem to admit a unique solution, and also provide a solution representation in terms of contour integrals. The solution and well-posedness results rely upon an extension of the Fokas (or unified) transform method to initial-nonlocal value problems for linear equations; the n…
We consider the heat equation with spatially variable thermal conductivity and homogeneous Dirichlet boundary conditions. Using the Method of Fokas or Unified Transform Method, we derive solution representations as the limit of solutions of constant-coefficient interface problems where the number of subdomains and interfaces becomes unbounded. This produces an explicit representation of the solution, from which we can compute the solution and determine its properties. Using this solution expression, we can find the eigenvalues of the corresponding variable-coefficient eigenvalue problem as roo…
This paper is concerned with the impermeable wall problem for an ideal polytropic model of non-viscous and heat-conductive gas in one-dimensional half space. It is shown that the 3-rarefaction wave is stable under some smallness conditions. The proof is given by an elementary energy method and the key point is to control the boundary terms due to the less dissipativity of the system.
The classical half line Robin problem for the heat equation may be solved via a spatial Fourier transform method. In this work, we study the problem in which the static Robin condition $bq(0,t)+q_x(0,t)=0$ is replaced with a dynamic Robin condition; $b=b(t)$ is allowed to vary in time. We present a solution representation, and justify its validity, via an extension of the Fokas transform method. We show how to reduce the problem to a variable coefficient fractional linear ordinary differential equation for the Dirichlet boundary value. We implement the fractional Frobenius method to solve this…
The heat equation with inverse square potential on both half-lines of $\mathbb{R}$ is discussed in the presence of \emph{bridging} boundary conditions at the origin. The problem is the lowest energy (zero-momentum) mode of the transmission of the heat flow across a Grushin-type cylinder, a generalisation of an almost Riemannian structure with compact singularity set. This and related models are reviewed, and the issue is posed of the analysis of the dispersive properties for the heat kernel generated by the underlying positive self-adjoint operator. Numerical integration is shown that provides…
A reaction-diffusion equation with power nonlinearity formulated either on the half-line or on the finite interval with nonzero boundary conditions is shown to be locally well-posed in the sense of Hadamard for data in Sobolev spaces. The result is established via a contraction mapping argument, taking advantage of a novel approach that utilizes the formula produced by the unified transform method of Fokas for the forced linear heat equation to obtain linear estimates analogous to those previously derived for the nonlinear Schrödinger, Korteweg-de Vries and "good" Boussinesq equations. Thus, t…
A differential-difference operator is used to model the heat equation on a finite graph analogue of Poincaré's upper half-plane. Finite analogues of the classical theta functions are shown to be solutions to the heat equation in this setting.
The solution of an initial-boundary value problem for a linear evolution partial differential equation posed on the half-line can be represented in terms of an integral in the complex (spectral) plane. This representation is obtained by the {\em unified transform} introduced by Fokas in the 90's. On the other hand, it is known that many initial-boundary value problems can be solved via a classical transform pair, constructed via the spectral analysis of the associated spatial operator. For example, the Dirichlet problem for the heat equation can be solved by applying the Fourier sine transform…
We consider the inhomogeneous heat equation on the half-space $\mathbb R_{+}^{d}$ with Neumann boundary conditions. We prove a space-time Gevrey regularity of the solution, with a radius of analyticity uniform up to the boundary of the half-space. We also address the case of homogeneous Robin boundary conditions. Our results generalize the case of homogeneous Dirichlet boundary conditions established by Kukavica and Vicol in [10].
The existence and uniqueness of solutions to the Yang-Mills heat equation over domains in Euclidean three space was proven in a previous paper for initial data lying in the Sobolev space of order one-half, which is the critical Sobolev index for this equation. In the present paper the stability of these solutions will be established. The variational equation, which is only weakly parabolic, and has highly singular coefficients, will be shown to have unique strong solutions up to addition of a vertical solution. Initial data will be taken to be in Sobolev class one-half. The proof relies on an …
The Short Wave equation on the half-line by the Unified Transform Method
University of Vienna（维也纳大学）；B. Verkin Institute for Low Temperature Physics and Engineering（B.维尔金低温物理与工程研究所）；V.N.Karazin Kharkiv National University（V.N.卡拉津哈尔科夫国立大学）
We consider the nonlinear half laplacian heat equation $$ u_t+(-Δ)^{\frac{1}{2}} u-|u|^{p-1}u=0,\quad \mathbb{R}^n\times (0, T). $$ We prove that all blows-up are type I, provided that $n \leq 4$ and $ 1<p<p_{*} (n)$ where $ p_{*} (n)$ is an explicit exponent which is below $\frac{n+1}{n-1}$, the critical Sobolev exponent. Central to our proof is a Giga-Kohn type monotonicity formula for half laplacian and a Liouville type theorem for self-similar nonlinear heat equation. This is the first instance of a monotonicity formula at the level of the nonlocal equation, without invoking the extension …
We show the existence and the uniqueness of initial traces of nonnegative solutions to a semilinear heat equation on a half space of ${\mathbb R}^N$ under the zero Dirichlet boundary condition. Furthermore, we obtain necessary conditions and sufficient conditions on the initial data for the solvability of the corresponding Cauchy--Dirichlet problem. Our necessary conditions and sufficient conditions are sharp and enable us to find optimal singularities of initial data for the solvability of the Cauchy--Dirichlet problem.
We study an initial value problem for the one-dimensional non-stationary linear Schrödinger equation with a point singular potential. In our approach, the problem is considered as a system of coupled initial-boundary value (IBV) problems on two half-lines, to which we apply the unified approach to IBV problems for linear and integrable nonlinear equations, also known as the Fokas unified transform method. Following the ideas of this method, we obtain the integral representation of the solution of the initial value problem. Since the unified approach is known as providing efficient solutions to…
We introduce a heat flow associated to half-harmonic maps, which have been introduced by Da Lio and Rivière. Those maps exhibit integrability by compensation in one space dimension and are related to harmonic maps with free boundary. We consider a new flow associated to these harmonic maps with free boundary which is actually motivated by a rather unusual heat flow for half-harmonic maps. We construct then weak solutions and prove their partial regularity in space and time via a Ginzburg-Landau approximation. The present paper complements the study initiated by Struwe and Chen-Lin.
This paper is devoted to studying the inflow problem for an ideal polytropic model with non-viscous gas in one-dimensional half space. We showed the existence of the boundary layer in different areas. By employing the energy method, we also proved the unique global-in-time solution existed and the asymptotic stability of both the boundary layer and the superposition with the 3-rarefaction wave under some smallness conditions.
In this paper, we consider the initial-boundary value problem to the compressible Navier-Stokes equations for ideal gases without heat conduction in the half space or outside a fixed ball in $\mathbb R^N$, with $N\geq1$. We prove that any classical solutions $(ρ, u, θ)$, in the class $C^1([0,T]; H^m(Ω))$, $m>[\frac N2]+2$, with bounded from below initial entropy and compactly supported initial density, which allows to touch the physical boundary, must blow-up in finite time, as long as the initial mass is positive. This paper extends the classical reault by Xin [CPAM, 1998], in which the Cauch…
The first half of this work gives a survey of the fractional Laplacian (and related operators), its restricted Dirichlet realization on a bounded domain, and its nonhomogeneous local boundary conditions, as treated by pseudodifferential methods. The second half takes up the associated heat equation with homogeneous Dirichlet condition. Here we recall recently shown sharp results on interior regularity and on $L_p$-estimates up to the boundary, as well as recent Hölder estimates. This is supplied with new higher regularity estimates in $L_2$-spaces using a technique of Lions and Magenes, and hi…
We investigate the initial-boundary value problem for the general three-component nonlinear Schrodinger (gtc-NLS) equation with a 4x4 Lax pair on a finite interval by extending the Fokas unified approach. The solutions of the gtc-NLS equation can be expressed in terms of the solutions of a 4x4 matrix Riemann-Hilbert (RH) problem formulated in the complex k-plane. Moreover, the relevant jump matrices of the RH problem can be explicitly found via the three spectral functions arising from the initial data, the Dirichlet-Neumann boundary data. The global relation is also established to deduce two …
We derive the solution representation for a large class of nonlocal boundary value problems for linear evolution PDEs with constant coefficients in one space variable. The prototypical such PDE is the heat equation, for which problems of this form model physical phenomena in chemistry and for which we formulate and prove a full result. We also consider the third order case, which is much less studied and has been shown by the authors to have very different structural properties in general. The nonlocal conditions we consider can be reformulated as \emph{multipoint conditions}, and then an expl…
