Mohammed El Ansari, Youssef Akdim, Soumia Lalaoui Rhali
In this paper, we interest on some class of Stefan type problems. We prove the existence and uniqueness of renormalized solution in anisotropic Sobolev spaces with data belongs to $L^1- data,$ based on the properties of the renormalized trunctions and the generalized monotonicity method in the functional spaces.
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We prove locally in time the existence of a smooth solution for multidimensional two-phase Stefan problem for degenerate parabolic equations of the porous medium type. We establish also natural Hölder class for the boundary conditions in the Cauchy-Dirichlet problem for a degenerate parabolic equation.
In this paper we are interested in a rigorous derivation of the Kuramoto-Sivashinsky equation (K--S) in a Free Boundary Problem. As a paradigm, we consider a two-dimensional Stefan problem in a strip, a simplified version of a solid-liquid interface model. Near the instability threshold, we introduce a small parameter $\varepsilon$ and define rescaled variables accordingly. At fixed $\varepsilon$, our method is based on: definition of a suitable linear 1D operator, projection with respect to the longitudinal coordinate only, Lyapunov-Schmidt method. As a solvability condition, we derive a self…
Existence of weak solutions to two-phase Stefan problems for parabolic partial differential equation system
本文针对面包烘焙过程的两相Stefan问题，在放宽初始函数条件后，证明了弱解的存在性，弥补了强解仅适用于严格条件下的局限。
Inverse Stefan problems of determining the time-dependent source coefficient and heat flux function
本文针对半无限长杆热传导的逆Stefan问题，提出确定时间相关源系数和热流函数的方法，并建立了弱解的存在性、唯一性及连续依赖性。
Two fractional Stefan problems are considered by using Riemann-Liouville and Caputo derivatives of order $α\in (0,1)$ such that in the limit case ($α=1$) both problems coincide with the same classical Stefan problem. For the one and the other problem, explicit solutions in terms of the Wright functions are presented. We prove that these solutions are different even though they converge, when $α\nearrow 1$, to the same classical solution. This result also shows that some limits are not commutative when fractional derivatives are used.
In this paper we analyze the singular set in the Stefan problem and prove the following results: - The singular set has parabolic Hausdorff dimension at most $n-1$. - The solution admits a $C^\infty$-expansion at all singular points, up to a set of parabolic Hausdorff dimension at most $n-2$. - In $\mathbb R^3$, the free boundary is smooth for almost every time $t$, and the set of singular times $\mathcal S\subset \mathbb R$ has Hausdorff dimension at most $1/2$. These results provide us with a refined understanding of the Stefan problem's singularities and answer some long-standing open quest…
From the one-dimensional consolidation of fine-grained soils with threshold gradient, it can be derived a special type of Stefan problems where the seepage front, due to the presence of this threshold gradient, exhibits the features of a moving boundary. In this kind of problems, in contrast with the classical Stefan problem, the latent heat is considered to depend inversely with the rate of change of the seepage front. In this paper a one-phase Stefan problem with a latent heat that not only depends on the rate of change of the free boundary but also on its position is studied. The aim of thi…
In this paper we consider a one-dimensional one-phase Stefan problem corresponding to the solidification process of a semi-infinite material with a convective boundary condition at the fixed face. The exact solution of this problem, available recently in the literature, enable us to test the accuracy of the approximate solutions obtained by applying the classical technique of the heat balance integral method and the refined integral method, assuming a quadratic temperature profile in space. We develop variations of these methods which turn out to be optimal in some cases. Throughout this paper…
We obtain solvability conditions for some elliptic equations involving non Fredholm operators with the methods of spectral theory and scattering theory for Schrodinger type operators. Though the Fredholm property is not satisfied, the solvability conditions are formulated in terms of orthogonality of the right-hand side to solutions of the homogeneous adjoint equation.
This paper has been withdrawn, and is replaced with paper "Solvability of elliptic systems with square integrable boundary data" by the same authors.
We consider the problem of recovering the initial condition in the one-dimensional one-phase Stefan problem for the heat equation from the knowledge of the position of the melting point. We first recall some properties of the free boundary solution. Then we study the uniqueness and stability of the inversion. The principal contribution of the paper is a new logarithmic type stability estimate that shows that the inversion may be severely ill-posed. The proof is based on integral equations representation techniques, and the unique continuation property for parabolic type solutions. We also pres…
We prove that the multidimensional dimensional initial value problem for the Navier-Stokes equations is globally well-posed in the so-called Moment and Grand Lebesgue Spaces (GLS), and give some a priory estimations for solution in this spaces.
In this paper we study a ratio-dependent predator-prey model with a free boundary causing by both prey and predator over a one dimensional habitat. We study the long time behaviors of the two species and prove a spreading-vanishing dichotomy, namely, as t goes to infinity, both prey and predator successfully spread to the whole space and survive in the new environment, or they spread within a bounded area and die out eventually. Then the criteria governing spreading and vanishing are obtained. Finally, when spreading occurs, we provide some estimates to the asymptotic spreading speed of h(t).
The work in this paper concerns the study of different approximations for one-dimensional one-phase Stefan-like problems with a space-dependent latent heat. It is considered two different problems, which differ from each other in their boundary condition imposed at the fixed face: Dirichlet and Robin conditions. The approximate solutions are obtained by applying the heat balance integral method (HBIM), a modified heat balance integral method, the refined integral method (RIM) . Taking advantage of the exact analytical solutions we compare and test the accuracy of the approximate solutions. The…
We will show a local solvability result for a class of degenerate second order linear partial differential operators with a complex subprincipal symbol. Due to the form of the operators in the class the subprincipal symbol is invariantly defined and we shall give sufficient conditions for the local solvability to hold involving the real and the imaginary part of the latter. Under suitable conditions we will prove that the class under consideration is L^2 to L^2 locally solvable.
On a multi-phase Stefan problem in the half-line with different boundary conditions at the fixed boundary
该研究针对半直线上带不同边界条件的多相Stefan问题，通过变分方法建立解的存在唯一性，确定了Dirichlet-to-Neumann映射的性质，并推广到Robin边界情形，还研究了负连接系数的不适定问题。
We consider the nonlinear Stefan problem $$ \left \{ \begin{array} {ll} -d Δu=a u-b u^2 \;\; & \mbox{for } x \in Ω(t), \; t>0, \\ u=0 \mbox{ and } u_t=μ|\nabla_x u |^2 \;\;&\mbox{for } x \in \partialΩ(t), \; t>0, \\ u(0,x)=u_0 (x) \;\; & \mbox{for } x \in Ω_0, \end{array}\right. $$ where $Ω(0)=Ω_0$ is an unbounded smooth domain in $\mathbb R^N$, $u_0>0$ in $Ω_0$ and $u_0$ vanishes on $\partialΩ_0$. When $Ω_0$ is bounded, the long-time behavior of this problem has been rather well-understood by \cite{DG1,DG2,DLZ, DMW}. Here we reveal some interesting different behavior for certain unbounded $Ω_…
On the solvability of some systems of quadratic integral equations in dimensions two and three
本文利用不动点技术，证明了二维和三维空间中某类二次积分方程组在H^2空间内扰动解的存在性。
Energy dissipation and free boundary stability of the Stefan problem
本文证明了任意维度下 Stefan 问题两相和单相弱解自由边界的稳定性与连续性，通过能量守恒原理和新的下密度界控制边界位移，并展示了弱解渐近行为的定性与定量结果。
This paper is concerned with the solvability of some abstract differential equation of type $$\dot u(t) + Au(t) + Bu(t) \ni f(t), t \in (0,T], u(0) = 0,$$ where $A$ is a linear selfadjoint operator and $B$ is a nonlinear(possibly multi-valued)maximal monotone operator in a (real) Hilbert space ${\mathbb H}$ with the normalization $0 \in B (0)$. We use the concept of variational sum introduced by H. Attouch, J.-B. Baillon, and M. Théra, to investigate solutions to the given abstract differential equation. Several applications will be discussed, among them the case where $B = \subdifferential ϕ$…
In this paper we will first present some results about the local solvability property of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration (which in turn is a generalization of the Kannai operator) exhibits a degeneracy due to the interplay between the singularity associated with the characteristic set of a system of vector fields and the vanishing of a function. Afterward we will also discuss some local solvability results for two classes of degenerate second order linear partial differential operators with non-smo…
On behavior of free boundaries to generalized two-phase Stefan problems for parabolic partial differential equation systems
本文研究了广义双相Stefan问题中自由边界的行为，通过提升解的正则性，克服了自由边界增长速率依赖于水分含量以及水分含量边界条件包含温度的困难，建立了局部时间内的解的存在性和唯一性，并在某些初始数据符号条件下推导了解存在最大区间的结果。
On self-similar solutions of a multi-phase Stefan problem in the half-line
本文研究半直线上带常数初值、Dirichlet或Neumann边界条件的多相Stefan问题自相似解，通过梯度系统与严格凸强制势函数证明解的存在唯一性，还推广到无穷相变情形并给出Neumann问题解的类型判定条件。
Solvability of some integro-differential equations with the logarithmic Laplacian
本文研究含对数拉普拉斯算子的积分微分方程序列解的存在性，基于不动点方法证明积分核的L^1收敛可推出解的存在性及L^2收敛。
On the solvability of some systems of integro-differential equations with and without a drift
本文利用不动点方法，研究含漂移与不含漂移项的积分–微分方程组在实轴及周期边界有限区间上的 H^2 可解性，并处理算子有无 Fredholm 性质的情形。
We review some fractional free boundary problems that were recently considered for modeling anomalous phase-transitions. All problems are of Stefan type and involve fractional derivatives in time according to Caputo's definition. We survey the assumptions from which they are obtained and observe that the problems are nonequivalent though all of them reduce to a classical Stefan problem when the order of the fractional derivatives is replaced by one. We further show that a simple heuristic approach built upon a fractional version of the energy balance and the classical Fourier's law leads to a …
Solvability of some integro-differential equations with the bi-Laplacian and transport
本文利用不动点技巧，证明了涉及漂移项和Laplace算子平方的积分微分方程在H^4空间中解的存在性与收敛性，适用于含或不含Fredholm性质的四阶微分算子。
We study the local solvability of a class of operators with multiple characteristics. The class considered here complements and extends the one studied in [9], in that in this paper we consider some cases of operators with complex coefficients that were not present in [9]. The class of operators considered here ideally encompasses classes of degenerate parabolic and Schrödinger type operators. We will give local solvability theorems. In general, one has $L^2$ local solvability, but also cases of local solvability with better Sobolev regularity will be presented.
We show that advection-diffusion equations with porous media type diffusion and integrable initial data are globally solvable under very mild conditions. Some generalizations and related results are also given.
In this paper, we consider the existence and non-existence of non-trivial solution to a Brezis-Nirenberg type problem with singular weights. First, we obtain a compact imbedding theorem which is an extension of the classical Rellich-Kondrachov compact imbedding theorem, and consider the corresponding eigenvalue problem. Secondly, we deduce a Pohozaev type identity and obtained a non-existence result. Thirdly, based on a generalized concentration compactness principle, we will give some abstract conditions when the functional satisfies the (PS)$_c$ condition. Finally, based on the explicit form…
In this article, we consider the solvability of two generalized div-curl systems. They are referred to as the equations of magnetostatics and electro- , resp.. Necessary compatibility conditions on the data for the existence of solutions are fully described, and existence of $W^{m, p}$-solutions is proved. Moreover, we give some description for the null spaces. As a corollary, we give the estimates of gradient via generalized ${\rm div}$ and ${\rm curl}$ in $L^p$-framework, which can be considered as Friedrichs inequalities. Furthermore, two decompositions of Sobolev spaces are given.
In the present paper, we discuss solvability questions of a non-local problem with integral form transmitting conditions for diffusion-wave equation with the Caputo fractional derivative in a domain bounded by smooth curves. The uniqueness of the solution of the formulated problem we prove using energy integral method with some modifications. The existence of solution will be proved by equivalent reduction of the studied problem into a system of second kind Fredholm integral equations.
This is an addendum to a previous article, which aims to provide the proofs of some results in that paper (Theorem 7.5 and Proposition 9.15) which were removed from its final version. The reason for such omission is that these proofs follow quite closely others already present in the literature, with minor modifications. I make them publicly available for the sake of completeness.
We address the deterministic homogenization, in the general context of ergodic algebras, of a doubly nonlinear problem which generalizes the well known Stefan model, and includes the classical porous medium equation. It may be represented by the differential inclusion, for a real-valued function $u(x,t)$, $$ \frac{\partial}{\partial t}\partial_u Ψ(x/\ve,x,u)-\nabla_x\cdot \nabla_ηψ( x/\ve,x,t,u,\nabla u) \ni f(x/\ve,x,t, u), $$ on a bounded domain $\Om\subset \R^n$, $t\in(0,T)$, together with initial-boundary conditions, where $Ψ(z,x,\cdot)$ is strictly convex and $ψ(z,x,t,u,\cdot)$ is a $C^1$…
Solvability conditions for some non-Fredholm operators with shifted arguments
本文研究了具有移位参数的非Fredholm算子的可解条件，通过序列意义下的存在性和收敛性分析，探讨了微分方程和积分微分方程在不同函数空间中的解的性质。
The Riccati equation method is used to establish some global solvability criteria for some classes of second order nonlinear ordinary differential equations. Two oscillation theorems are proved. The results are applied to the Emden - Fowler equation and to the Van der Pol type equation.
By using some recent results for divergence form equations, we study the $L_p$-solvability of second-order elliptic and parabolic equations in nondivergence form for any $p\in (1,\infty)$. The leading coefficients are assumed to be in locally BMO spaces with suitably small BMO seminorms. We not only extend several previous results by Krylov and Kim [14]-[18] to the full range of $p$, but also deal with equations with more general coefficients.
In this paper, we study the Dirichlet problem for the implicit degen- erate nonlinear elliptic equation with variable exponent in a bounded domain. We obtain sufficient conditions for the existence of a solution with- out regularization and any restriction between the exponents. Furthermore, we define the domain of the operator generated by posed problem and investigate its some properties and also its relations with known spaces that enable us to prove existence theorem.
Let $Ω$ be a Lipschitz domain in $\mathbb R^n,n\geq 3,$ and $L=\divt A\nabla$ be a second order elliptic operator in divergence form. We will establish that the solvability of the Dirichlet regularity problem for boundary data in Hardy-Sobolev space $\HS$ is equivalent to the solvability of the Dirichlet regularity problem for boundary data in $H^{1,p}$ for some $1<p<\infty$. This is a "dual result" to a theorem in \cite{DKP09}, where it has been shown that the solvability of the Dirichlet problem with boundary data in $\text{BMO}$ is equivalent to the solvability for boundary data in $L^p(\pa…
We study a model elliptic pseudo-differential equation and simplest boundary value problems for a half-space and a special cone in Sobolev--Slobodetskii spaces which have different smoothness with respect to separate variables. Sufficient conditions for a unique solvability for such boundary value problems are described.
The present paper deals with the nonhomogeneous incompressible asymmetric fluids equations in dimension $d= 2,3$. The aim is to prove the unique global solvability of the system with only bounded nonnegative initial density and $H^{1}$ initial velocities. We first construct the global existence of the solution with large data in 2-D. Next, we establish the existence of local in time solution for arbitrary large data and global in time for some smallness conditions in 3-D. Finally, the uniqueness of the solution is proved under quite soft assumptions about its regularity through a Lagrangian ap…
We study a Hele-Shaw problem with a mushy region obtained as a Mesa type limit of one phase Stefan problems in exterior domains. We study the convergence, determine some of the qualitative properties and regularity of the unique limiting solution, and prove regularity of the free boundary of this limit under very general conditions on the initial data. Indeed, our results handle changes in topology and multiple injection slots.
We consider the Cauchy problem for the Hardy parabolic equation $\partial_t u-Δu=|x|^{-γ}u^p$ with initial data $u_0$ singular at some point $z$. Our main results show that, if $z\neq 0$, then the optimal strength of the singularity of $u_0$ at $z$ for the solvability of the equation is the same as that of the Fujita equation $\partial_t u-Δu=u^p$. Moreover, if $z=0$, then the optimal singularity for the Hardy parabolic equation is weaker than that of the Fujita equation. We also obtain analogous results for a fractional case $\partial_t u+(-Δ)^{θ/2} u=|x|^{-γ}u^p$ with $0<θ<2$.
Solvability in the sense of sequences for certain non-Fredholm operators with a drift and Laplace and bi-Laplace operators
研究某些线性非齐次椭圆问题的可解性，证明在特定技术假设下，右端项在L^2中收敛可导致解的存在性及在H^4中收敛，涉及含或不含Fredholm性质的四阶微分算子，并指出输运项提供正则化。
Existence of stationary solutions for some systems of integro-differential equations with Laplace and bi-Laplace operators
本文研究了含标准拉普拉斯与双拉普拉斯算子的积分微分方程组解的存在性，采用不动点技术证明，基于无界域中椭圆算子的可解性条件。
Solvability of a transmission problem in $L^p$-spaces with generalized diffusion equation
针对相邻种群栖息地间含 Laplace 算子与双调和项的广义扩散方程透射问题，本文利用半群理论和泛函演算给出系数关系，证明 $L^p$ 空间中经典解的存在唯一性。
A Characterization of Solvability of the Parabolic $L^p$ Dirichlet Problem on Lipschitz Graph Domains Via Carleson Measure Estimates of Bounded Solutions
该研究通过有界解的Carleson测度估计，刻画了Lipschitz图区域上抛物型$L^p$Dirichlet问题的可解性，改进了需附加抛物一致可求长性等额外假设的现有结果。
Different one-phase Stefan problems for a semi-infinite slab are considered, involving a moving phase change material as well as temperature dependent thermal coefficients. Existence of at least one similarity solution is proved imposing a Dirichlet, Neumann, Robin or radiative-convective boundary condition at the fixed face. The velocity that arises in the convective term of the diffusion-convection equation is assumed to depend on temperature and time. In each case, an equivalent ordinary differential problem is obtained giving rise to a system of an integral equation coupled with a conditio…
We prove a global existence result for weak solutions to a one-dimensional free boundary problem with flux boundary conditions describing swelling along a halfline. Additionally, we show that solutions are not only unique but also depend continuously on data and parameters. The key observation is that the structure of our system of partial differential equations allows us to show that the moving a priori unknown interface never disappears. As main ingredients of the global existence proof, we rely on a local weak solvability result for our problem, uniform estimates of the solution, integral e…
We address the existence and uniqueness of the so-called modified error function that arises in the study of phase-change problems with specific heat and thermal conductivity given by linear functions of the material temperature. This function is defined from a differential problem that depends on two parameters which are closely related with the slopes of the specific heat and the thermal conductivity. We identify conditions on these parameters which allow us to prove the existence of the modified error function. In addition, we show its uniqueness in the space of non-negative bounded analyti…
In the first part of this article, we complete the program announced in the preliminary note [8] by proving a conjecture presented in [9] that states the equivalence of contractibility and p_{1}-stability for generalized spaces of formal barycenters and hence we get purely algebraic conditions for the solvability of the singular Liouville equation on Riemann surfaces. This relies on a structure decomposition theorem for these model spaces in terms of maximal strata, and on elementary combinatorial arguments based on the selection rules that define such spaces. Moreover, we also show that these…
In this article, we consider the following problem: $$ \quad \left\{ \begin{array}{lr} \quad (-Δ)^s u = αu^+ -βu^{-} + f(u) + h \; \text{in}\;Ω\quad \quad \quad \quad u =0 \; \text{on}\; \mathbb{R}^n\setminus Ω, \end{array} \right. $$ where $Ω\subset \mathbb{R}^n$ is a bounded domain with Lipschitz boundary, $n> 2s$, $0<s<1$, $(α, β) \in \mathbb{R}^2$, $f: \mathbb{R}\to \mathbb{R}$ is a bounded and continuous function and $h\in L^2(Ω)$. We prove the existence results in two cases: First, the nonresonance case, where $(α,β)$ is not an element of the Fučik spectrum. Second, the resonance case, w…
Solvability of a class of fully nonlinear elliptic equations on tori
该研究聚焦平坦环面上一类完全非线性椭圆方程的可解性，这类方程源自环面丛相关Calabi-Yau型问题的研究，为相关几何分析问题提供支撑。
This is a the first in a series of two articles devoted to the question of local solvability of doubly characteristic differential operators $L,$ defined, say, in an open set $\Om\subset \RR^n.$ Suppose the principal symbol $p_k$ of $L$ vanishes to second order at $(x_0,ξ_0)\in T^*\Om\setminus 0,$ and denote by $Q_\H$ the Hessian form associated to $p_k$ on $T_{(x_0,ξ_0)}T^*\Om.$ As the main result of this paper, we show (under some rank conditions and some mild additional conditions) that a necessary condition for local solvability of $L$ at $x_0$ is the existence of some $θ\in\RR$ such that …
Exact controllability to the trajectories of the one-phase Stefan problem
本文针对一维单相Stefan自由边界问题，通过将其转化为耦合PDE-ODE系统的零能控问题，结合新的Carleman不等式与Liusternik-Graves定理，证明了光滑轨迹的局部精确能控性，并附带得到一类带边界记忆项抛物方程的零能控性结果。
Let $Ω\subset \rr^2$ be a bounded simply connected domain. We show that, for a fixed (every) $p\in (1,\fz),$ the divergence equation $\mathrm{div}\,\mathbf{v}=f$ is solvable in $W^{1,p}_0(Ω)^2$ for every $f\in L^p_0(Ω)$, if and only if $Ω$ is a John domain, if and only if the weighted Poincaré inequality $$\int_Ω|u(x)-u_Ω|^q\,dx\le C\int_Ω|\nabla u(x)|^q\dist(x,\partial Ω)^q\,dx$$ holds for some (every) $q\in [1,\fz)$. In higher dimensions similar results are proved under some additional assumptions on the domain in question.
In this paper we formulate the equilibrium equation of a beam made of graphene material subjected to some boundary conditions and acted upon by axial compression and nonlinear lateral constrains as a fourth-order nonlinear boundary value problem. We also formulate the nonlinear eigenvalue for buckling analysis of the beam. We verify the solvability of the buckling problem as an asymptotic expansion in a ratio of the elastoplastic parameters, that the spectrum is bounded away from zero and contains a discrete infinite sequence of eigenvalues.We also verify, for certain ranges of the lateral for…
We study the existence and properties of solutions and free boundaries of the one-phase Stefan problem with fractional diffusion posed in $\mathbb{R}^N$. In terms of the enthalpy $h(x,t)$, the evolution equation reads $\partial_t h+(-Δ)^sΦ(h) =0$, while the temperature is defined as $u:=Φ(h):=\max\{h-L,0\}$ for some constant $L>0$ called the latent heat, and $(-Δ)^s$ stands for the fractional Laplacian with exponent $s\in(0,1)$. We prove the existence of a continuous and bounded selfsimilar solution of the form $h(x,t)=H(x\,t^{-1/(2s)})$ which exhibits a free boundary at the change-of-phase le…
We address some global solvability issues for classes of smooth nonsingular vector fields $L$ in the plane related to cohomological equations $Lu=f$ in geometry and dynamical systems. The first main result is that $L$ is not surjective in $C^\infty(\R^2)$ iff the geometrical condition -- the existence of separatrix strips -- holds. Next, for nonsurjective vector fields, we demonstrate that if the RHS $f$ has at most infra-exponential growth in the separatrix strips we can find a global weak solution $L^1_{loc}$ near the boundaries of the separatrix strips. Finally we investigate the global sol…
It is well-known that quantitative, scale invariant absolute continuity (more precisely, the weak-$A_\infty$ property) of harmonic measure with respect to surface measure, on the boundary of an open set $ Ω\subset \mathbb{R}^{n+1}$ with Ahlfors-David regular boundary, is equivalent to the solvability of the Dirichlet problem in $Ω$, with data in $L^p(\partialΩ)$ for some $p<\infty$. In this paper, we give a geometric characterization of the weak-$A_\infty$ property, of harmonic measure, and hence of solvability of the $L^p$ Dirichlet problem for some finite $p$. This characterization is obtain…
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