Unconditional Uniqueness Research Articles

Unconditional uniqueness and non-uniqueness for Hardy–Hénon parabolic equations

We study the problems of uniqueness for Hardy–Hénon parabolic equations, which are semilinear heat equations with the singular potential (Hardy type) or the increasing potential (Hénon type) in the nonlinear term. To deal with the Hardy–Hénon type nonlinearities, we employ weighted Lorentz spaces as solution spaces. We prove unconditional uniqueness and non-uniqueness, and we establish uniqueness criterion for Hardy–Hénon parabolic equations in the weighted Lorentz spaces. The results extend the previous works on the Fujita equation and Hardy equations in Lebesgue spaces.

Diffusive stability and self-similar decay for the harmonic map heat flow

In this paper we study the harmonic map heat flow on the euclidean space Rd and we show an unconditional uniqueness result for maps with small initial data in the homogeneous Besov space B˙p,∞dp(Rd) where d<p<∞. As a consequence we obtain decay rates for solutions of the harmonic map flow of the form ‖∇u(t)‖L∞(Rd)≤Ct−12.Additionally, under the assumption of a stronger spatial localization of the initial conditions, we show that the temporal decay happens in a self-similar way. We also explain that similar results hold for the biharmonic map heat flow and the semilinear heat equation with a power-type nonlinearity.

On monotonicity conditions for mean field games

In this paper we propose two new monotonicity conditions that could serve as sufficient conditions for uniqueness of Nash equilibria in mean field games. In this study we aim for unconditional uniqueness that is independent of the length of the time horizon, the regularity of the starting distribution of the agents, or the regularization effect of a non-degenerate idiosyncratic noise. Through a rich class of simple examples we show that these new conditions are not only in dichotomy with each other, but also with the two widely studied monotonicity conditions in the literature, the Lasry–Lions monotonicity and displacement monotonicity conditions.

Unconditional uniqueness for the Benjamin–Ono equation

Unconditional uniqueness for the Benjamin–Ono equation

Global Well-Posedness of the 4-D Energy-Critical Stochastic Nonlinear Schrödinger Equations with Non-Vanishing Boundary Condition

We consider the energy-critical stochastic cubic nonlinear Schrödinger equation on R4 with additive noise, and with the non-vanishing boundary conditions at spatial infinity. By viewing this equation as a perturbation to the energy-critical cubic nonlinear Schrödinger equation on R4, we prove global well-posedness in the energy space. Moreover, we establish unconditional uniqueness of solutions in the energy space.

On uniqueness of mild solutions for Boussinesq equations in Morrey-type spaces

We prove the unconditional uniqueness of mild solutions for the Boussinesq equations in critical weak-Morrey spaces for dimension n⩾3. Our approach relies on duality and block spaces, as well as suitable heat and bilinear estimates in our setting. In comparison with previous results, we provide a new uniqueness class of mild solutions for the system.

Well-posedness of mild solutions to the drift-diffusion and the vorticity equations in amalgam spaces

We consider the Cauchy problem of the drift-diffusion and the vorticity equations. Both equations involve the Poisson equation and a nonlocal effect of the Green's function influences the solution to the problem. In this paper, we study the well-posedness of the drift-diffusion and the vorticity equations by using amalgam spaces of Lebesgue spaces. Moreover, we show the unconditional uniqueness of mild solutions to the drift-diffusion equation in amalgam spaces.

The unconditional uniqueness for the energy-supercritical NLS

We consider the cubic and quintic nonlinear Schr\"{o}dinger equations (NLS) under the $\mathbb{R}^{d}$ and $\mathbb{T}^{d}$ energy-supercritical setting. Via a newly developed unified scheme, we prove the unconditional uniqueness for solutions to NLS at critical regularity for all dimensions. Thus, together with [18,19], the unconditional uniqueness problems for $H^{1}$-critical and $H^{1}$-supercritical cubic and quintic NLS are completely and uniformly resolved at critical regularity for these domains. One application of our theorem is to prove that defocusing blowup solutions of the type in [54] is the only possible $C([0,T);\dot{H}^{s_{c}})$ solution if exist in these domains.

Local well-posedness for the gKdV equation on the background of a bounded function

We prove the local well-posedness for the generalized Korteweg–de Vries equation in $H^s(\mathbb{R})$, $s>1/2$, under general assumptions on the nonlinearity $f(x)$, on the background of an $L^\infty\_{t,x}$-function $\Psi(t,x)$, with $\Psi(t,x)$ satisfying some suitable conditions. As a consequence of our estimates, we also obtain the unconditional uniqueness of the solution in $H^s(\mathbb{R})$. This result not only gives us a framework to solve the gKdV equation around a Kink, for example, but also around a periodic solution, that is, to consider localized non-periodic perturbations of a periodic solution. As a direct corollary, we obtain the unconditional uniqueness of the gKdV equation in $H^s(\mathbb{R})$ for $s>1/2$. We also prove global existence in the energy space $H^1(\mathbb{R})$, in the case where the nonlinearity satisfies $\vert f''(x)\vert\lesssim 1$.

Unconditional uniqueness for the periodic Benjamin-Ono equation by normal form approach

We show unconditional uniqueness of solutions to the Cauchy problem associated with the Benjamin-Ono equation under the periodic boundary condition with initial data given in Hs for s>1/6. This improves the previous unconditional uniqueness result in H1/2 by Molinet and Pilod (2012). Our proof is based on a gauge transform and integration by parts in the time variable.

Unconditional uniqueness for the energy-critical nonlinear Schrödinger equation on

AbstractWe consider the$\mathbb {T}^{4}$cubic nonlinear Schrödinger equation (NLS), which is energy-critical. We study the unconditional uniqueness of solutions to the NLS via the cubic Gross–Pitaevskii hierarchy, an uncommon method for NLS analysis which is being explored [24, 35] and does not require the existence of a solution in Strichartz-type spaces. We proveU-Vmultilinear estimates to replace the previously used Sobolev multilinear estimates. To incorporate the weaker estimates, we work out new combinatorics from scratch and compute, for the first time, the time integration limits, in the recombined Duhamel–Born expansion. The new combinatorics and theU-Vestimates then seamlessly conclude the$H^{1}$unconditional uniqueness for the NLS under the infinite-hierarchy framework. This work establishes a unified scheme to prove$H^{1}$uniqueness for the$ \mathbb {R}^{3}/\mathbb {R}^{4}/\mathbb {T}^{3}/\mathbb {T}^{4}$energy-critical Gross–Pitaevskii hierarchies and thus the corresponding NLS.

Well-posedness and unconditional uniqueness of mild solutions to the Keller–Segel system in uniformly local spaces

Well-posedness and unconditional uniqueness of mild solutions to the Keller–Segel system in uniformly local spaces

Normal form approach to the one-dimensional periodic cubic nonlinear Schr\xf6dinger equation in almost critical Fourier-Lebesgue spaces

In this paper, we study the one-dimensional cubic nonlinear Schrödinger equation (NLS) on the circle. In particular, we develop a normal form approach to study NLS in almost critical Fourier-Lebesgue spaces. By applying an infinite iteration of normal form reductions introduced by the first author with Z. Guo and S. Kwon (2013), we derive a normal form equation which is equivalent to the renormalized cubic NLS for regular solutions. For rough functions, the normal form equation behaves better than the renormalized cubic NLS, thus providing a further renormalization of the cubic NLS. We then prove that this normal form equation is unconditionally globally well-posed in the Fourier-Lebesgue spaces ℱLp({cal F}{L^p}(mathbb{T})), 1 ≤ p < ∞. By inverting the transformation, we conclude global well-posedness of the renormalized cubic NLS in almost critical Fourier-Lebesgue spaces in a suitable sense. This approach also allows us to prove unconditional uniqueness of the (renormalized) cubic NLS in ℱLp({cal F}{L^p}(mathbb{T})) for 1 leq p leq {3 over 2}.

Unconditional uniqueness for the periodic modified Benjamin–Ono equation by normal form approach

Abstract We show that the solution (in the sense of distribution) to the Cauchy problem with the periodic boundary condition associated with the modified Benjamin–Ono equation is unique in $L^\infty _t(H^s(\mathbb{T} ))$ for $s&amp;gt;1/2$. The proof is based on the analysis of a normal form equation obtained by infinitely many reduction steps using integration by parts in time after a suitable gauge transform.

Nonlinear smoothing and unconditional uniqueness for the Benjamin–Ono equation in weighted Sobolev spaces

We consider the Benjamin–Ono equation on the real line for initial data in weighted Sobolev spaces. After the application of the gauge transform, the flow is shown to be Lipschitz continuous and to present a nonlinear smoothing effect. As a consequence, unconditional uniqueness for the Benjamin–Ono equation is proved.

Some uniqueness theorems of solutions for the problemsof elasticity

In order to construct the well-posed mathematical models of the elastic problems, it is necessary to study the mathematical properties of solutions, including existence, uniqueness, and stability (the continuous dependence on boundary conditions). The uniqueness theorem of solutions provides the methods for solving the problem. The uniqueness of the solutions is one of the most basic and important issues of elasticity theory. In the three-dimensional elasticity theory, unconditional uniqueness is not expected. Therefore, the uniqueness theorem of solutions is established under certain conditions, such as the restrictions on the elasticity tensor, the strain energy function and elastic deformation range. This study reviews the background and history of the uniqueness theorem in elasticity. The focuses are on the uniqueness theorems of solutions in the boundary-value problems of the linear elasticity theory, nonlinear elasticity theory of finite deformation, and elasticity theory with initial stress field. The classical proofs of uniqueness theorems are also given. We also present some unsolved problems on the solution uniqueness in elasticity.

Unconditional uniqueness results for the nonlinear Schrödinger equation

We study the problem of unconditional uniqueness of solutions to the cubic nonlinear Schrödinger equation (NLS). We introduce a new strategy to approach this problem on bounded domains, in particular on rectangular tori. It is a known fact that solutions to the cubic NLS give rise to solutions of the Gross–Pitaevskii (GP) hierarchy, which is an infinite-dimensional system of linear equations. By using the uniqueness analysis of the GP hierarchy, we obtain new unconditional uniqueness results for the cubic NLS on rectangular tori, which cover the full scaling-subcritical regime in high dimensions. In fact, we prove a more general result which is conditional on the domain. In addition, we observe that well-posedness of the cubic NLS in Fourier–Lebesgue spaces implies unconditional uniqueness.

Unconditional uniqueness for the derivative nonlinear Schrödinger equation on the real line

We prove the unconditional uniqueness of solutions to the derivative nonlinear Schrodinger equation (DNLS) in an almost end-point regularity. To this purpose, we employ the normal form method and we transform (a gauge-equivalent) DNLS into a new equation (the so-called normal form equation) for which nonlinear estimates can be easily established in \begin{document}$ H^s({\mathbb{R}}) $\end{document} , \begin{document}$ s>\frac12 $\end{document} , without appealing to an auxiliary function space. Also, we prove that low-regularity solutions of DNLS satisfy the normal form equation and this is done by means of estimates in the \begin{document}$ H^{s-1}({\mathbb{R}}) $\end{document} -norm.

Well-posedness issues on the periodic modified Kawahara equation

This paper is concerned with the Cauchy problem of the modified Kawahara equation (posed on T), which is well-known as a model of capillary-gravity waves in an infinitely long canal over a flat bottom in a long wave regime [26]. We show in this paper some well-posedness results, mainly the global well-posedness in L2(T). The proof basically relies on the idea introduced in Takaoka-Tsutsumi's works [60,69], which weakens the non-trivial resonance in the cubic interactions (a kind of smoothing effect) for the local result, and the global well-posedness result immediately follows from L2 conservation law. An immediate application of Takaoka-Tsutsumi's idea is available only in Hs(T), s>0, due to the lack of L4-Strichartz estimate for arbitrary L2 data, a slight modification, thus, is needed to attain the local well-posedness in L2(T). This is the first low regularity (global) well-posedness result for the periodic modified Kwahara equation, as far as we know. A direct interpolation argument ensures the unconditional uniqueness in Hs(T), s>12, and as a byproduct, we show the weak ill-posedness below H12(T), in the sense that the flow map fails to be uniformly continuous.

Regularity and uniqueness for the rough solutions of the derivative nonlinear Schrödinger equation

In this paper, we obtain the unconditional uniqueness for the rough solutions of the derivative nonlinear Schrödinger equation $$ i\partial_t u + \partial^2_{x} u =i\partial_{x}(|u|^2u) $$ in $C([0,T];H^s(\mathbb R))$, $s\in(\frac{2}{3},1]$. The arguments used here are the normal form argument, resonant decomposition and the Bourgain argument. The main ingredient in the proof is to improve the regularity of the solution by iteration method and finally show that the solution belongs to some Bourgain space.