Clément Denis
We give a description of a magnetohydrodynamical system in $n$ dimension using the exterior derivative. We then prove existence of global solutions for small initial data and local existence for arbitrary large data in two classes of critical spaces -- $L^q_tL^p_x$ and $\mathcal{C}_tL^p_x$, as well as uniqueness for solutions in $\mathcal{C}_tL^p_x$.
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We study an anisotropic system arising in magnetohydrodynamics (MHD) in the whole space R^3 , in the case where there are no diffusivity in the vertical direction and only a small diffusivity in the horizontal direction (of size $ε$$α$ with 0 \textless{} $α$ $\le$ $α$0, for some $α$0 \textgreater{} 0). We prove the local existence and uniqueness of a strong solution and then, using Strichartz-type estimates, we prove that this solution globally exists in time for large initial data, when the rotation is fast enough.
In this paper, we prove the uniqueness of solutions to the logarithmic Minkowski problem in $\mathbb{R}^3$ without symmetry condition, provided the density of the measure is close to $1$ in $C^α$ norm. This result also implies the uniqueness of self-similar solutions to the anisotropic Gauss curvature flow in $\mathbb{R}^3$ when the speed function is $C^α$ close to a positive constant.
We give a new proof of a classical uniqueness theorem of Alexandrov using the weak uniqueness continuation theorem of Bers-Nirenberg. We prove a version of this theorem with the minimal regularity assumption: the spherical hessians of the corresponding convex bodies as Radon measures are nonsingular.
By means of Banach fixed point theorem , the uniqueness of Boltzmann Equation generalizaled Solutions in Sobolev spaces in $L^1(\mathbb{R}^+\times \mathbb{R}^n\times \mathbb{R}^n)$, can be proved as well as Boltzmann Equation renormalized solutions uniqueness
We obtain the existence, regularity, uniqueness of the non-stationary problems of a class of non-Newtonian fluid is a power law fluid with $p>9/5$ in the half-space under slip boundary conditions.
In this short note, based on Carleman estimates and Holmgren's type theorems, we provide a converse theorem of the classical Huygens principle for free wave equations. Possible generalizations to other underlying space-times or other wave type equations are also discussed.
In this paper we show the uniqueness of the critical point for \emph{semi-stable} solutions of the problem $$\begin{cases} -Δu=f(u)&\text{in }Ω\\ u>0&\text{in }Ω\\ u=0&\text{on } \partialΩ,\end{cases}$$ where $Ω\subset\mathbb{R}^2$ is a smooth bounded domain whose boundary has \emph{nonnegative} curvature and $f(0)\ge0$. It extends a result by Cabré-Chanillo to the case where the curvature of $\partialΩ$ vanishes.
We construct a singular solution of a stationary nonlinear Schrödinger equation on $\mathbb{R}^2$ with square-exponential nonlinearity having linear behavior around zero. In view of Trudinger-Moser inequality, this type of nonlinearity has an energy-critical growth. We use this singular solution to prove non-uniqueness of strong solutions for the Cauchy problem of the corresponding semilinear heat equation. The proof relies on explicit computation showing a regularizing effect of the heat equation in an appropriate functional space.
In this paper, we present the result of maximum regularity of the mild solution of the fractional Cauchy problem. As our main result, we investigate the uniqueness of mild solutions for time-fractional Navier-Stokes equations in class $C\left([0,\infty);L^{N}\left( \mathbb{R}^{N}\right)^{N}\right)$ by means of the estimates $L^{p}-L^{q}$ of Giga-Shor inequality and the Gronwall inequality.
This note shows that under $(p,α, N)\in (1,\infty)\times(0,2)\times\mathbb Z_+$ the fractional order differential inequality $$ (\dagger)\quad u^p \le (-Δ)^{\fracα{2}} u\quad\hbox{in}\quad\mathbb R^{N} $$ has the property that if $N\leα$ then a nonnegative solution to $(\dagger)$ is unique, and if $N>α$ then the uniqueness of a nonnegative weak solution to $(\dagger)$ occurs when and only when $p\le N/(N-α)$, thereby innovatively generalizing Gidas-Spruck's result for $u^p+Δu\le 0$ in $\R^N$ discovered in \cite{GS}.
In this paper, we are concerned with the uniqueness result for non-negative solutions of the higher-order Lane-Emden equations involving the GJMS operators on $\mathbb{S}^n$. Since the classical moving-plane method based on the Kelvin transform and maximum principle fails in dealing with the high-order elliptic equations in $\mathbb{S}^n$, we first employ the Mobius transform between $\mathbb{S}^n$ and $\mathbb{R}^n$, poly-harmonic average and iteration arguments to show that the higher-order Lane-Emden equation on $\mathbb{S}^n$ is equivalent to some integral equation in $\mathbb{R}^n$. Then …
This paper concerns the uniqueness of weak solutions to the Cauchy problem to the Ericksen-Leslie system of liquid crystal models in $\mathbb R^2$, with both general Leslie stress tensors and general Oseen-Frank density. It is shown here that such a system admits a unique weak solution provided that the Frank coefficients are close to some positive constant, which solves an interesting open problem. One of the main ideas of our proof is to perform suitable energy estimates at level one order lower than the natural basic energy estimates for the Ericksen-Leslie system.
In this paper, we establish a type of uniqueness and nondegeneracy results for positive solutions to the following nonlocal Kirchhoff equations \begin{eqnarray*} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\text{d} x\right)Δu+u=|u|^{p-1}u & & \text{in }\mathbb{R}^{3}, \end{eqnarray*} where $a,b$ are positive constants and $1<p<5$. Before this paper, it seems that there have no this type of results even on positive ground states solutions to Kirchhoff type equations, much less on general positive solutions. To overcome the difficulty brought by the nonlocality, some new observation on Kirchhof…
We consider the hydrodynamics for biaxial nematic phases described by a field of orthonormal frame, which can be derived from a molecular-theory-based tensor model. We prove the uniqueness of global weak solutions to the Cauchy problem of the frame hydrodynamics in dimensional two. The proof is mainly based on the suitable weaker energy estimates within the Littlewood--Paley analysis. We take full advantage of the estimates of nonlinear terms with rotational derivatives on $SO(3)$, together with cancellation relations and dissipative structures of the biaxial frame system.
Existence and uniqueness of local regular solution to the Schrödinger flow from a bounded domain in $\mathbb{R}^3$ into $\mathbb{S}^2$
本文利用抛物扰动法、内蕴几何能量论证及边界几何量分析，证明了$\mathbb{R}^3$中有界区域到$\mathbb{S}^2$的Schrödinger流初-Neumann边值问题局部正则解的存在性与唯一性。
Existence and Uniqueness for the Fractional Gelfand Equation in $\mathbb{R}$
本文利用基于v=√(e^u)的不动点格式和非局部打靶法，证明了一维分数阶Gelfand方程在s∈(1/2,1)时解的存在、对称、唯一、有限Morse指标及非退化性，并推广到一般正偶递减K。
Existence, uniqueness, and decay results for singular $Φ$-Laplacian systems in $\mathbb{R}^N$
本文利用移位法、变分法及多种分析技术，研究了奇异 $\Phi$-Laplacian 系统解的存在性、先验估计、衰减性与唯一性，并给出了 Diaz-Saa 型结果。
In this paper, we prove that the existence and uniqueness of globally weak solutions to the Cauchy problem for the weakly dissipative Camassa-Holm equation in time weighted $H^1$ space. First, we derive an equivalent semi-linear system by introducing some new variables, and present the globally conservative solutions of this equation in time weighted $H^1$ space. Second, we show that the peakon solutions are conservative weak solutions in $H^1.$ Finally, given a conservative solution, we introduce a set of auxiliary variables tailored to this particular solution, and prove that these variables…
We investigate a stochastic transport equation driven by a multiplicative noise. For $L^q(0,T;W^{1,p}({\mathbb R}^d;{\mathbb R}^d))$ drift coefficient and $W^{1,r}({\mathbb R}^d)$ initial data, we obtain the existence and uniqueness of stochastic strong solutions (in $W^{1,r}_{loc}({\mathbb R}^d))$.In particular, when $r=\infty$, we establish a Lipschitz estimate for solutions and this question is opened by Fedrizzi and Flandoli in case of $L^q(0,T;L^p({\mathbb R}^d;{\mathbb R}^d))$ drift coefficient. Moreover, opposite to the deterministic case where $L^q(0,T;W^{1,p}({\mathbb R}^d;{\mathbb R}…
We study the equation \begin{equation*}\label{P0} (-Δ)^s u = |x|^α u^{\frac{N+2s+2α}{N-2s}}\mbox{ in }\mathbb{R}^N,\tag{P} \end{equation*} where $(-Δ)^s$ is the fractional Laplacian operator with $0 < s < 1$, $α>-2s$ and $N>2s$. We prove the linear non-degeneracy of positive radially symmetric solutions of the equation (\ref{P0}) and, as a consequence, a uniqueness result of those solutions with Morse index equal to one. In particular, the ground state solution is unique. Our non-degeneracy result extends in the radial setting some known theorems done by Dávila, Del Pino and Sire (see \cite[Th…
The goal of this paper is to study uniqueness of a one-dimensional Hamilton-Jacobi equation \begin{equation*} \begin{cases} u_t=|u_x|^2+R(x,I(t)) &\text{in }\mathbb{R} \times (0,\infty), \max_{\mathbb{R}} u(\cdot,t)=0 &\text{on }[0,\infty), \end{cases} \end{equation*} with an initial condition $u_0(x,0)=u_0(x)$ on $\mathbb{R}$. A reaction term $R(x,I(t))$ is given while $I(t)$ is an unknown constraint (Lagrange multiplier) that forces maximum of $u$ to be always zero. In the paper, we prove uniqueness of a pair of unknowns (u,I) using dynamic programming principle in one dimensional space for …
In the present paper, we establish the uniqueness and nondegeneracy of positive energy solutions to the Kirchhoff equation \begin{eqnarray*} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\right)Δu+u=|u|^{p-1}u & & \text{in }\mathbb{R}^{3}, \end{eqnarray*} where $a,b>0$, $1<p<5$ are constants. Then, as applications, we derive the existence and local uniqueness of solutions to the perturbed Kirchhoff problem \begin{eqnarray*} -\left(ε^2a+εb\int_{\mathbb{R}^{3}}|\nabla u|^{2}\right)Δu+V(x)u=|u|^{p-1}u & & \text{in }\mathbb{R}^{3} \end{eqnarray*} for $ε>0$ sufficiently small, under some mild assump…
Non-uniqueness and failure of Calderón-Zygmund estimates below the critical exponent for non-monotone PDE with linear growth
该研究通过凸积分方法，展示了非单调PDE在低于临界指数下解的非唯一性及Calderón-Zygmund估计的失效
On the uniqueness and the regularity of solutions to the Navier Stokes in the Lorentz space $L^{3,\infty}(\mathbb{R}^3)$
本文证明Navier-Stokes方程在Lorentz空间$L^{3,\infty}$相关函数类中解的唯一性，并证明此类解在正时间光滑，同时给出$t\to0^+$时解的精确估计。
On uniqueness for hyperbolic half-wave maps in dimension $d \geq 3$
本文研究了维度 $d \geq 3$ 下 Half-Wave Maps 方程的唯一性问题，通过时间求导、Nash 嵌入定理及几何性质，结合分数阶 Leibniz 规则与交换子估计，利用 Grönwall 不等式在自然能量类中证明了唯一性。
We prove some uniqueness results for positive harmonic functions on the unit ball satisfying a nonlinear boundary condition
Existence and uniqueness of global large-data solutions for the Chemotaxis-Navier-Stokes system in $\mathbb{R}^2$
本文研究了二维Chemotaxis-Navier-Stokes系统的Cauchy问题，通过利用系统内禀熵结构推导熵-能量估计并结合bootstrap论证，建立了大初值条件下强解、经典解及任意光滑解的全局存在性与唯一性。
We prove existence and uniqueness of Radon measure-valued solutions of the Cauchy problem $$ \begin{cases} u_t+[φ(u)]_x=0 & \text{in } \mathbb{R}\times (0,T) \\ u=u_0\ge 0 &\text{in } \mathbb{R}\times \{0\}, \end{cases} $$ where $u_0$ a positive Radon measure whose singular part is a finite superposition of Dirac masses, and $φ\in C^2([0,\infty))$ is bounded. The novelty of the paper is the introduction of a compatibility condition which, combined with standard entropy conditions, guarantees uniqueness.
We study generalized solutions of multidimensional transport equation with bounded measurable solenoidal field of coefficients $a(x)$. It is shown that any generalized solution satisfies the renormalization property if and only if the operator $a\cdot\nabla u$, $u\in C_0^1(\mathbb{R}^n)$ in the Hilbert space $L^2(\mathbb{R}^n)$ is an essentially skew-adjoint operator, and this is equivalent to the uniqueness of generalized solutions. We also establish existence of a contractive semigroup, which provides generalized solutions, and give a criterion of its uniqueness.
We consider the problem of existence and uniqueness of strong a.e. solutions $u: \mathbb{R}^n \longrightarrow \mathbb{R}^N$ to the fully nonlinear PDE system \[\label{1} \tag{1} F(\cdot,D^2u ) \,=\, f, \ \ \text{ a.e. on }\mathbb{R}^n, \] when $ f\in L^2(\mathbb{R}^n)^N$ and $F$ is a Carathéodory map. \eqref{1} has not been considered before. The case of bounded domains has been studied by several authors, firstly by Campanato and under Campanato's ellipticity condition on $F$. By introducing a new much weaker notion of ellipticity, we prove solvability of \eqref{1} in a tailored Sobolev "ener…
Results of existence and uniqueness for the Cauchy problem of semilinear heat equations on stratified Lie groups
本文研究分层李群上半线性热方程 Cauchy 问题，利用广义分数阶链式法则估计非线性项，证明 Sobolev 初值下解的存在唯一性，并推广 Ribaud 在欧氏空间的结果。
Uniqueness of bound states to $Δu-u+|u|^{p-1}u= 0$ in $\mathbb{R}^n$, $n\ge 3$
本文证明了 $n\ge 3$ 时非线性 Schrödinger/Klein-Gordon 模型方程中，具有任意给定零点数的径向束缚态在平移和反射下唯一，解决了 Berestycki–Lions 猜想。
Uniqueness of solutions for the logarithmic Schrödinger equation
本文研究对数薛定谔方程柯西问题，通过积分方程、局部光滑估计和次线性效应定量估计，将弱解唯一性从 $H^1$ 推广到 $H^s$（$s\in(0,1)$），并探讨了环面及幂非线性扰动情形。
Uniqueness for phaseless inverse elastic scattering problem for periodic structures
本文针对二维和三维周期结构中的无相位逆弹性散射问题，利用双点源叠加入射场和近场测量数据，通过推导准周期/双周期 Green 函数、互易关系及 Rayleigh 展开，建立了唯一性结果。
We prove the uniqueness property for a class of entire solutions to the equation \begin{equation*} \left\{ \begin{array}{ll} -{\rm div}\, \mathcal{A}(x,\nabla u) = σ, \quad u\geq 0 \quad \text{in } \mathbb{R}^n, \\ \displaystyle{\liminf_{|x|\rightarrow \infty}}\, u = 0, \end{array} \right. \end{equation*} where $σ$ is a nonnegative locally finite measure in $\mathbb{R}^n$, absolutely continuous with respect to the $p$-capacity, and ${\rm div}\, \mathcal{A}(x,\nabla u)$ is the $\mathcal{A}$-Laplace operator, under standard growth and monotonicity assumptions of order $p$ ($1<p<\infty$) on $\mat…
This paper studies the existence and uniqueness of local weak solutions to the d-dimensional tropical climate model without thermal diffusion. We establish that, when $α=β\geq1$, $η=0$, any initial data $(u_{0},v_{0})\in B_{2,1}^{1+\frac{d}{2}-2α}(\mathbb{R}^{d})$ and $θ_{0}\in B_{2,1}^{1+\frac{d}{2}-α}(\mathbb{R}^{d})$ yields a unique weak solution.
This paper deals with nonlinear singular partial differential equations of the form $t \partial u/\partial t=F(t,x,u,\partial u/\partial x)$ with independent variables $(t,x) \in \mathbb{R} \times \mathbb{C}$, where $F(t,x,u,v)$ is a function continuous in $t$ and holomorphic in the other variables. Under a very weak assumption we show the uniqueness of the solution of this equation. The results are applied to the problem of analytic continuation of local holomorphic solutions of equations of this type.
We consider ground states of $L^2$-subcritical nonlinear Schrödinger equation (1.1), which can be described equivalently by minimizers of the following constraint minimization problem $$ e(ρ):=\inf\{E_ρ(u):u\in \mathcal{H}(\mathbb{R}^d),\|u\|_2^2=1\}.$$ The energy functional $E_ρ(u)$ is defined by $$ E_ρ(u):=\frac{1}{2}\int_{\mathbb{R}^d}|\nabla u|^2dx +\frac{1}{2}\int_{\mathbb{R}^d}V(x)|u|^2dx-\frac{ρ^{p-1}}{p+1}\int_{\mathbb{R}^d}|u|^{p+1}dx,$$ where $d\geq1$, $ρ>0$, $p\in\big(1, 1+\frac{4}{d}\big)$ and $0\leq V(x)\to\infty$ as $|x| \to\infty$. We present a detailed analysis on the concentra…
$l^{2}$-decoupling and the unconditional uniqueness for the Boltzmann equation
本文基于$l^2$分解理论，通过Strichartz估计和时空双线性估计，证明了Boltzmann方程在不同空间中的解的无条件唯一性，方法源于非线性Schrödinger方程的统一层级方案。
We prove existence and uniqueness of distributional, bounded, nonnegative solutions to a fractional filtration equation in ${\mathbb R}^d$. With regards to uniqueness, it was shown even for more general equations in [19] that if two bounded solutions $u,w$ of (1.1) satisfy $u-w\in L^1({\mathbb R}^d\times(0,T))$, then $u=w$. We obtain here that this extra assumption can in fact be removed and establish uniqueness in the class of merely bounded solutions, provided they are nonnegative. Indeed, we show that a minimal solution exists and that any other solution must coincide with it. As a conseque…
Uniqueness and nonuniqueness for mean field equations with multiple singularities on flat tori
本文研究平坦环面上带多奇点的平均场方程，建立了基于总质量和谱量的唯一性判据，并证明最优系数 8/π²，通过对称奇点例子展示非唯一性，并在两奇点情形下给出 ρ 接近 4π 时的唯一性与三解结果。
In this paper, we consider the following critical fractional Kirchhoff equation \begin{equation*} \Big(a+b{\int_{\mathbb{R}^{N}}}|(-Δ)^{\frac{s}{2}}u|^2dx\Big)(-Δ)^su=|u|^{2^*_s-2}u,\quad \text{in}\ \mathbb{R}^{N}, \end{equation*} where $a,b>0$, $\frac{N}{4}<s<1$, $2^*_s=\frac{2N}{N-2s}$ and $(-Δ)^s$ is the fractional Laplacian. We prove the uniqueness and nondegeneracy of positive solutions to the problem, which can be used to study the singular perturbation problems concerning fractional Kirchhoff equations.
We study nonlocal conservation laws with a discontinuous flux function of regularity $\mathsf{L}^{\infty}(\mathbb{R})$ in the spatial variable and show existence and uniqueness of weak solutions in $\mathsf{C}\big([0,T];\mathsf{L}^{1}_{\text{loc}}(\mathbb{R})\big)$, as well as related maximum principles. We achieve this well-posedness by a proper reformulation in terms of a fixed-point problem. This fixed-point problem itself necessitates the study of existence, uniqueness and stability of a class of discontinuous ordinary differential equations. On the ODE level, we compare the solution type …
We establish the uniqueness of a saddle-shaped solution to the diffusion equation $-Δu = f(u)$ in all of $\mathbb{R}^{2m}$, where $f$ is of bistable type, in every even dimension $2m \geq 2$. In addition, we prove its stability whenever $2m \geq 14$. Saddle-shaped solutions are odd with respect to the Simons cone ${\mathcal C} = \{(x^1,x^2) \in \mathbb{R}^m \times \mathbb{R}^m : |x^1|=|x^2| \}$ and exist in all even dimensions. Their uniqueness was only known when $2m=2$. On the other hand, they are known to be unstable in dimensions 2, 4, and 6. Their stability in dimensions 8, 10, and 12 rem…
Uniqueness of Cylindrical Tangent Cones $C_{p,q} \times \mathbb{R}$
研究面积最小超曲面在 $\mathbb{R}^9$ 中柱形切锥 $C(\mathbb{S}^2 \times \mathbb{S}^4) \times \mathbb{R}$ 的唯一性，完善了 Simon 和 Székelyhidi 对高维切锥唯一性的证明。
Uniqueness and nondegeneracy of ground states for $(-Δ)^su+u=2(I_2\star u^2)u$ in $\mathbb{R}^N$ when $s$ is close to 1
本文研究分数阶 Choquard 方程在 $s$ 接近 1 时基态的唯一性与非退化性，核心方法是对幂次 $s$ 进行连续性论证。
The moving plane method and the uniqueness of high order elliptic equation with GJMS operator
本文利用移动平面法研究含GJMS算子的高阶椭圆方程，证明在特定参数条件下解必为零，并由此给出经典Beckner不等式的新证明。
Uniqueness in Lorentz Spaces of the 2d Navier-Stokes equation
本文证明了在特定光滑性假设下，二维纳维-斯托克斯方程在洛伦茨空间中的解的唯一性。
We obtain uniqueness and nondegeneracy results for ground states of Choquard equations $-Δu+u=\left(|x|^{-1}\ast|u|^{p}\right)|u|^{p-2}u$ in $\mathbb{R}^{3}$, provided that $p>2$ and $p$ is sufficiently close to 2.
We consider the Navier-Stokes equation on $\mathbb{H}^{2}(-a^{2})$, the two dimensional hyperbolic space with constant sectional curvature $-a^{2}$. We prove an ill-posedness result in the sense that the uniqueness of the Leray-Hopf weak solutions to the Navier-Stokes equation breaks down on $\mathbb{H}^{2}(-a^{2})$. We also obtain a corresponding result on a more general negatively curved manifold for a modified geometric version of the Navier-Stokes equation. Finally, as a corollary we also show a lack of the Liouville theorem in the hyperbolic setting both in two and three dimensions.
Non-uniqueness and energy dissipation for 2D Euler equations with vorticity in Hardy spaces
本文通过凸积分方法构造了在 $\mathbb{R}^2$ 上具有 $H^p(\mathbb{R}^2)$ 空间涡度的 2D Euler 方程能量耗散解，证明了其非唯一性。
We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic fourth order nonlinear Schrödinger equation with initial data $u_{0}\in X$, where $X\in\{M_{2,q}^{s}(\mathbb R), H^σ(\mathbb T), H^{s_{1}}(\mathbb R)+H^{s_{2}}(\mathbb T)\}$ and $q\in[1,2]$, $s\geq0$, or $σ\geq0$, or $s_{2}\geq s_{1}\geq0$. Moreover, if $M_{2,q}^{s}(\mathbb R)\hookrightarrow L^{3}(\mathbb R)$, or if $σ\geq\frac16$ or if $s_{1}\geq\frac16$ and $s_{2}>\frac12$ we show that the Cauchy problem is unconditionally wellposed in $X$. Similar results hold true for all higher order nonlinea…
In this paper, we investigate the critical points of solutions to the prescribed constant mean curvature equation with Neumann and Robin boundary conditions respectively in a bounded smooth convex domain $Ω$ of $\mathbb{R}^{n}(n\geq2)$. Firstly, we show the non-degeneracy and uniqueness of the critical points of solutions in a planar domain by using the local Chen & Huang's comparison technique and the geometric properties of approximate surfaces at the non-degenerate critical points. Secondly, we deduce the uniqueness and non-degeneracy of the critical points of solutions in a rotationally sy…
It is shown that a function $u$ satisfying $|\partial_tu+\sum_{i,j}\partial_i(a^{ij}\partial_ju)|\leq N(|u|+|\nabla u|)$, $|u(x,t)|\leq Ne^{N|x|^2}$ in $\mathbb{R}^n_+\times[0,T]$ and $u(x,0)=0$ in $\mathbb{R}^n_+$ under certain conditions on $\{a^{ij}\}$ must vanish identically in $\mathbb{R}^n_+\times[0,T]$. The main point of the result is that the conditions imposed on $\{a^{ij}\}$ are of the type: $\{a^{ij}\}$ are Lipschitz and $|\nabla_xa^{ij}(x,t)|\leq \frac{E}{|x|}$, where $E$ is less than a given number, and the conditions are in some sense optimal.
In this paper, we focus on the two-dimensional surface quasi-geostrophic equation with fractional horizontal dissipation and vertical thermal diffusion which represents a general case of the classical surface quasi-geostrophic equation. On the one hand, we will show the local existence and uniqueness of the solution in Sobolev space $H^{2-2α}(\mathbb{R}^2)\cap H^{2-2β}(\mathbb{R}^2)$, which is the critical space in the classical case. Furthermore, we will demonstrate that the solution is global even when the initial data is very small. Finally, we will study the asymptotic representation of ou…
In this paper, we study the global well-posedness for the Camassa-Holm(C-H) equation with a forcing in $H^1(\mathbb{R})$ by the characteristic method. Due to the forcing, many important properties to study the well posedness of weak solutions do not inherit from the C-H equation without a forcing, such as conservation laws, integrability. By exploiting the balance law and some new estimates, we prove the existence and uniqueness of global weak solutions for the C-H equation with a forcing in $H^1(\mathbb{R})$.
We are concerned with Dirichlet problems of the form $${\mathop{\rm div}\nolimits} (|D u|^{p-2}Du)+f(u)=0\ \mbox{ in }Ω,\qquad u=0\ \mbox{ on }\partialΩ, $$ where $Ω$ is a bounded domain of $\mathbb{R}^n$, $n\ge 2$, $1<p<n$ and $f$ is a continuous function with supercritical growth from the viewpoint of the Sobolev embedding. In particular, if $n=2$ and $γ:[a,b]\to\mathbb{R}^2$ is a smooth curve such that $γ(t_1)\neqγ(t_2)$ for $t_1\neq t_2$, we prove that, for $\varepsilon>0$ small enough, there exists a unique solution of the Dirichlet problem in the domain $Ω=Ω^Γ_\varepsilon=\{(x_1,x_2)\in\…
A current based approach for the uniqueness of the continuity equation
本文研究连续性方程解的唯一性问题，在不使用重整化解理论的情况下，提出了一种基于电流理论和交换子强 $L^1$ 估计的几何方法。
Let $X(Γ)$ be the space of all finite Borel measure $μ$ in $\mathbb R^2$ which is supported on the curve $Γ$ and absolutely continuous with respect to the arc length of $Γ$. For $Λ\subset\mathbb R^2,$ the pair $\left(Γ, Λ\right)$ is called a Heisenberg uniqueness pair for $X(Γ)$ if any $μ\in X(Γ)$ satisfies $\hatμ\vert_Λ=0,$ implies $μ=0.$ We explore the Heisenberg uniqueness pairs corresponding to the cross, exponential curves, and surfaces. Then, we prove a characterization of the Heisenberg uniqueness pairs corresponding to finitely many parallel lines. We observe that the size of the deter…
Uniqueness of the blow-down limit for triple junction problem
本文证明具有三井势的平面 Allen-Cahn 系统整体极小解在无穷远处具有唯一的 $L^1$ blow-down 极限，并可通过三重点映射逼近，证明基于能量上下界的精细分析。
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