Olivier Guibé
We prove a few existence results of a solution for a static system with a coupling of thermoviscoelastic type. As this system involves $L^1$ coupling terms we use the techniques of renormalized solutions for elliptic equations with $L^1$ data. We also prove partial uniqueness results.
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In this paper we establish the uniqueness of a solution to a stationary convection-diffusion equation in divergence form with an exponentially summable generalized divergence-free drift.
We prove the existence, uniqueness and non negativity of solutions for a nonlinear stationary Doi-Edwards equation. The existence is proved by a perturbation argument. We get the uniqueness and the non negativity by showing the convergence in time of the solution of the evolutionary Doi-Edwards equation towards any stationary solution.
This work is devoted to the study of a compressible viscoelastic fluids satisfying the Oldroyd-B model in a regular bounded domain. We prove the local existence of solutions and uniqueness of flows by a classical fixed point argument.
In this note we prove the existence and uniqueness of weak solutions for the boundary value problem modelling the stationary case of the bioconvective flow problem introduced by Tuval et. al. (2005, {\it PNAS} 102, 2277--2282). We derive some appropriate a priori estimates for the weak solution, which implies the existence, by application of Gossez theorem, and the uniqueness by standard methodology of comparison of two arbitrary solutions.
We consider a generalised Keller-Segel model with non-linear porous medium type diffusion and non-local attractive power law interaction, focusing on potentials that are more singular than Newtonian interaction. We show uniqueness of stationary states (if they exist) in any dimension both in the diffusion-dominated regime and in the fair-competition regime when attraction and repulsion are in balance. As stationary states are radially symmetric decreasing, the question of uniqueness reduces to the radial setting. Our key result is a sharp generalised Hardy-Littlewood-Sobolev type functional in…
Non-uniqueness of weak solutions to the 3-D stationary MHD equations in Besov space with negative regularity index
通过凸积分方法证明三维环面上分数阶稳态MHD方程存在无穷多个非平凡奇异解，从而在次临界Besov空间中建立弱解的非唯一性。
On non-uniqueness of mild solutions and stationary singular solutions to the Navier-Stokes equations
通过凸积分构造非平凡稳态奇异解，证明Navier-Stokes方程在负正则性Besov空间中温和解的非唯一性，并建立端点临界空间中稳态弱解的唯一性。
The role of the dimension in uniqueness results for the stationary quasi-geostrophic system
本文针对二维、三维和四维稳态分数阶准地转方程，在特定Lebesgue可积性条件下证明了弱解的唯一性，并阐明了维数与Laplace算子分数幂次对唯一性的影响。
A finite Larmor radius approximation is derived from the classical Vlasov equation, in the limit of large (and uniform) external magnetic field. We also provide an heuristic derivation of the electroneutrality equation in the finite Larmor radius setting. Existence and uniqueness of a solution is proven in the stationary frame for solutions depending only on the direction parallel to the magnetic field and factorizing in the velocity variables.
Cahn-Hilliard-Navier-Stokes system describes the evolution of two isothermal, incompressible, immiscible fluids in a bounded domain. In this work, we consider the stationary nonlocal Cahn-Hilliard-Navier-Stokes system in two and three dimensions with singular potential. We prove the existence of a weak solution for the system using pseudo-monotonicity arguments and Browder's theorem. Further we establish the uniqueness and regularity results for the weak solution of the stationary nonlocal Cahn-Hilliard-Navier-Stokes system for constant mobility parameter and viscosity. Finally, in two dimensi…
Some remarks about the stationary Micropolar fluid equations: existence, regularity and uniqueness
本文研究定常微极流体方程，推导速度场与微旋转场的耦合方程，探讨弱解存在性、正则性及唯一性，证明仅速度场满足无穷远衰减条件时平凡解唯一。
uniqueness and asymptotic behavior of the stationary Navier-Stokes equations in a slab
本文针对平板域带无滑移边界的定常Navier-Stokes方程，在小外力和小Poiseuille流假设下，证明存在H¹解，其远场围绕Poiseuille流逐点衰减，外力较小时解唯一，关键依赖Dirichlet积分与Stokes正则性估计。
We consider a class of stationary viscous Hamilton--Jacobi equations as $$ \left\{\begin{array}{l} \la u-{\rm div}(A(x) \nabla u)=H(x,\nabla u)\mbox{in }Ω, u=0{on}\partialΩ\end{array} \right. $$ where $\la\geq 0$, $A(x)$ is a bounded and uniformly elliptic matrix and $H(x,ξ)$ is convex in $ξ$ and grows at most like $|ξ|^q+f(x)$, with $1 < q < 2$ and $f \in \elle {\frac N{q'}}$. Under such growth conditions solutions are in general unbounded, and there is not uniqueness of usual weak solutions. We prove that uniqueness holds in the restricted class of solutions satisfying a suitable energy--typ…
Doubling variables and uniqueness of probability solutions to degenerate stationary Kolmogorov equations
针对具有退化扩散矩阵的平稳Kolmogorov方程，通过直接应用随机分析中的变量加倍方法，得到其概率解唯一性的充分条件。
We prove that a regular stationary black-hole solution of the Einstein vacuum equations which is "close" to some Kerr solution is, in fact, isometric to that Kerr solution.
We consider an inviscid stochastically forced dyadic model, where the additive noise acts only on the first component. We prove that a strong solution for this problem exists and is unique by means of uniform energy estimates. Moreover, we exploit these results to establish strong existence and uniqueness of the stationary distribution.
On Stationary Gevrey Solutions to the Gravitational Boussinesq System and Applications to Uniqueness
本文研究了引力Boussinesq系统在一般重力加速度项下的平稳解，证明了弱解的存在性，并展示Gevrey类正则性可用于研究解的唯一性，同时获得Navier-Stokes方程的新正则性结果和Liouville型结论。
Weak-strong uniqueness for solutions to mean-field games
针对平稳一阶平均场博弈，本文提出线性化方法，在d维环面上证明弱-强唯一性，并给出具体成立条件。
We prove a conditional "no hair" theorem for smooth manifolds: if $E$ is the domain of outer communication of a smooth, regular, stationary Einstein vacuum, and if a technical condition relating the Ernst potential and Killing scalar is satisfied on the bifurcate sphere, then $E$ is locally isometric to the domain of outer communication of a Kerr space-time.
We study the existence and uniqueness for weak solutions to some classes of anisotropic elliptic Dirichlet problems with data belonging to the natural dual space.
We give a self-contained and short proof for the existence, uniqueness and measurability of so called $p$-harmonious functions. The proofs only use elementary analytic tools. As a consequence, we obtain existence, uniqueness and measurability of value functions for the tug-of-war game with noise.
In this article we find the equivalent conditions to assure the existence and uniqueness of positive solutions to semilinear elliptic equations wih double power nonlinearities. As a bonus, we give a simpler proof of our former result that the uniqueness condition comes from the existence condition.
We study the stationary Navier--Stokes equations in the whole plane with a compactly supported force term and with a prescribed constant spatial limit. Prior to this work, existence of solutions to this problem was only known under special symmetry and smallness assumptions. In the paper we solve the key difficulties in applying Leray's {\it invading domains method} and, as a consequence, prove the existence of $D$-solutions in the whole plane for arbitrary compactly supported force. The boundary condition at infinity are verified in two different scenarios: (I) the~limiting velocity is suffic…
In this paper, we mainly introduce a general method to study the existence and uniqueness of solution of free boundary problems with partially degenerate diffusion.
In this work we study the existence and uniqueness of (ω,c)-periodic solutions for semilinear evolution equations in complex Banach spaces.
A proof is given of the global existence and uniqueness of a weak solution to Navier-Stokes boundary problem. The proof is short and essentially self-contained.
We prove the existence of stationary solutions for the density of an infinitely extended plasma interacting with an arbitrary configuration of background charges. Furthermore, we show that the solution cannot be unique if the total charge of the background is attractive. In this case, infinitely many different stationary solutions exist. The non-uniqueness can be explained by the presence of trapped particles orbiting the attractive background charge.
In this paper, we provide a direct approach to the existence and uniqueness of strong (in the probabilistic sense) and weak (in the PDE sense) solutions to quasilinear stochastic partial differential equations, which are neither monotone nor locally monotone.
Existence and non-uniqueness of classical solutions to the axially symmetric stationary Navier-Stokes equations in an exterior cylinder
研究外周期圆柱内轴对称稳态 Navier-Stokes 方程，构造满足边界与无穷远条件、具临界及超临界衰减的经典解，并证明径向汇足够强时解非唯一。
Existence and uniqueness results to a quasilinear singular Lane-Emden Neumann system
本文利用下上解方法，建立了一类受Neumann边界条件约束的拟线性奇异Lane-Emden型系统解的存在性与唯一性。
In this article we prove local-in-time existence and uniqueness of solution to a non-isothermal cross-diffusion system with Maxwell-Stefan structure.
In this paper we consider the linearized version of a system of partial differential equations arising from a fluid-structure interaction model. We prove the existence and the uniqueness of the solution under natural regularity assumptions.
We establish the existence and uniqueness of variational solution to the nonlinear Neumann boundary problem for the $p^{th}$-Sub-Laplacian associated to a system of Hörmander vector fields
Existence, Uniqueness, and Smoothing for Generalized EMHD
本文研究了广义电子磁流体力学的初值问题，证明了在临界Sobolev空间中局部解的存在唯一性，小初始数据下的全局解存在唯一性，并证明了解的瞬时平滑效应及全局解的时间衰减率。
Existence and uniqueness theorems for functional equations
本文针对受限加性及对数泛函方程，提出了扩展与唯一性定理，简化了相关理论。
Consider classical solutions to the parabolic reaction diffusion equation $$ &u_t =Lu+f(x,u), (x,t)\in R^n\times(0,\infty); &u(x,0) =g(x)\ge0, x\in R^n; &u\ge0, $$ where $$ L=\sum_{i,j=1}^na_{i,j}(x)\frac{\partial^2}{\partial x_i \partial x_j}+\sum_{i=1}^nb_i(x)\frac\partial{\partial x_i} $$ is a non-degenerate elliptic operator, $g\in C(R^n)$ and the reaction term $f$ converges to $-\infty$ at a super-linear rate as $u\to\infty$. We give a sharp minimal growth condition on $f$, independent of $L$, in order that there exist a universal, a priori upper bound for all solutions to the above Cauch…
In this paper, we prove existence and uniqueness of measure solutions for the Cauchy problem associated to the (vectorial) continuity equation with a non-local flow. We also give a stability result with respect to various parameters.
In this paper we investigate a class of elliptic problems involving a nonlocal Kirchhoff type operator with variable coefficients and data changing its sign. Under appropriated conditions on the coefficients, we have shown existence and uniqueness of solution.
This paper is concerned with the existence, uniqueness and nonlinear stability of stationary solutions to the Cauchy problem of the full compressible Navier-Stokes-Korteweg system effected by external force of general form in $\mathbb{R}^3$. Based on the weighted-$L^2$ method and some elaborate $L^\infty$ estimates of solutions to the linearized problem, the existence and uniqueness of stationary solution are obtained by the contraction mapping principle. The proof of the stability result is given by an elementary energy method and relies on some intrinsic properties of the full compressible N…
In this paper we extend results obtained in [3] and [5]. By considering a semi linear conservation law with velocity in $L^\infty$, we prove by fixed point arguments existence and uniqueness result and even in a penalized situation.
We obtain the existence, regularity, uniqueness of the non-stationary problems of a class of non-Newtonian fluid is a power law fluid with $p>9/5$ in the half-space under slip boundary conditions.
In this note we present some recent results for Kirchhoff equations in generalized Gevrey spaces. We show that these spaces are the natural framework where classical results can be unified and extended. In particular we focus on existence and uniqueness results for initial data whose regularity depends on the continuity modulus of the nonlinear term, both in the strictly hyperbolic case, and in the degenerate hyperbolic case.
Existence and uniqueness for renormalized solutions to a general noncoercive nonlinear parabolic equation
本文引入包含奇性与无界低阶项的一般非强制非线性抛物问题的重整化解概念，并证明其存在性与唯一性。
The integrable Novikov equation can be regarded as one of the Camassa-Holm-type equations with cubic nonlinearity. In this paper, we prove the global existence and uniqueness of the Hölder continuous energy conservative solutions for the Cauchy problem of the Novikov equation.
In this paper, we consider the existence and uniqueness of weak solutions of a nonlinear elliptic equation with a variable exponent, a monotonic type operator and a convection term. With the topological degree theory, we prove the existence of at least one weak solution under some Leray-Lions and growth conditions. Moreover, we obtain the uniqueness of the solution of the problem under some additional assumptions. Our results generalize and improve existing results with another approach.
In this paper we give a necessary and suffcient conditions for the existence and uniqueness of periodic solutions of functional differential equations with n delay d dt x(t) = Ax(t) + n j=1 Bx(t -- r j) + f (t). The conditions are obtained in terms of R-boundedness of operator valued Fourier multipliers.
Existence and uniqueness of solutions to some anisotropic elliptic equations with a singular convection term
研究具有奇异对流项的各类各向异性椭圆方程，证明了弱解的存在唯一性，并推导了先验估计与正则性结果。
The paper deals with the existence and uniqueness of a non-trivial solution to non-homogeneous $ p ( x ) -$laplacian equations, managed by non polynomial growth operator in the framework of variable exponent Sobolev spaces on Riemannian manifolds. The mountain pass Theorem is used.
We prove a global existence, uniqueness and regularity result for a two-species reaction-diffusion volume-surface system that includes nonlinear bulk diffusion and nonlinear (weak) cross diffusion on the active surface. A key feature is a proof of upper $L^{\infty}$-bounds that exploits the entropic gradient structure of the system.
We study a model due to J.M. Lasry and P.L. Lions, describing the evolution of a scalar price which is realized as a free boundary in a 1-D diffusion equation with dynamically evolving, non-standard sources. We establish global existence and uniqueness.
Motivated mainly by the localization over an open bounded set $Ω$ of $\mathbb R^n$ of solutions of the Schrödinger equations, we consider the Schrödinger equation over $Ω$ with a very singular potential $V(x) \ge C d (x, \partial Ω)^{-r}$ with $r\ge 2$ and a convective flow $\vec U$. We prove the existence and uniqueness of a very weak solution of the equation, when the right hand side datum $f(x)$ is in $L^1 (Ω, d(\cdot, \partial Ω))$, even if no boundary condition is a priori prescribed. We prove that, in fact, the solution necessarily satisfies (in a suitable way) the Dirichlet condition $u…
In this paper, we study the time-fractional diffusion equation on a metric star graph. The existence and uniqueness of the weak solution are investigated and the proof is based on eigenfunction expansions. Some priori estimates and regularity results of the solution are proved.
In this paper a nonlinear coupled Schrodinger system in the presence of mixed cubic and superlinear power laws is considered. Focus are made on the steady state solutions of the continuous system for existence and uniqueness by minimizing problem on some associated Nehari manifold.
We study in this paper the axisymmetric 3-D Navier-Stokes system where the horizontal viscosity is zero. We prove the existence of a unique global solution to the system with initial data of Yudovitch type
An existence and uniqueness result, up to fattening, for a class of crystalline mean curvature flows with natural mobility is proved. The results are valid in any dimension and for arbitrary, possibly unbounded, initial closed sets. The comparison principle is obtained by means of a suitable weak formulation of the flow, while the existence of a global-in-time solution follows via a minimizing movements approach.
We study the $p$-Choquard equation in 3-dimensional case and establish existence and uniqueness of ground states for the corresponding Weinstein functional. For proving the uniqueness of ground states, we use the radial symmetry to transform the equation into an ordinary differential system, and applying the Pohozaev identities and Gronwall lemma we show that any two Weinstein minimizers coincide.
Existence and uniqueness of solutions to Bogomol'nyi-Prased-Sommerfeld equations on graphs
本文研究连通有限图上的两个BPS方程，给出了解存在且唯一的充要条件。
This paper deals with existence and uniqueness results for a transient nonlinear radiative-conductive system in three-dimensional case. This system describes the heat transfer for a grey, semi-transparent and non-scattering medium with general boundary conditions. We reformulate the full transient state system as a fixed-point problem. The existence and uniqueness proof is based on Banach fixed point theorem.
A standard assumption in mean-field game (MFG) theory is that the coupling between the Hamilton-Jacobi equation and the transport equation is monotonically non-decreasing in the density of the population. In many cases, this assumption implies the existence and uniqueness of solutions. Here, we drop that assumption and construct explicit solutions for one-dimensional MFGs. These solutions exhibit phenomena not present in monotonically increasing MFGs: low-regularity, non-uniqueness, and the formation of regions with no agents.
Existence and uniqueness of weak solutions to singular anisotropic elliptic problems
本文研究了一类带p次次线扰动的非线性各向异性椭圆问题，证明了弱解的唯一性，并给出了存在性的必要充分条件。
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