We study the Muskat problem describing the vertical motion of two immiscible fluids in a two-dimensional homogeneous porous medium in an $L_p$-setting with $p\in(1,\infty)$. The Sobolev space $W^s_p(\mathbb{R})$ with $s=1+1/p$ is a critical space for this problem. We prove, for $s\in (1+1/p,2),$ that the Rayleigh-Taylor condition identifies an open subset of $W^s_p(\mathbb{R})$ within which the Muskat problem is of parabolic type. This enables us to establish the local well-posedness of the problem in all these subcritical spaces together with a parabolic smoothing property.
This paper is devoted to the study of solutions with critical regularity for the two-dimensional Muskat equation. We prove that the Cauchy problem is well-posed on the endpoint Sobolev space of $L^2$ functions with three-half derivative in $L^2$. This result is optimal with respect to the scaling of the equation. One well-known difficulty is that one cannot define a flow map such that the lifespan is bounded from below on bounded subsets of this critical Sobolev space. To overcome this, we estimate the solutions for a norm which depends on the initial data themselves, using the weighted fracti…
We consider the Muskat problem with surface tension for one fluid or two fluids, with or without viscosity jump, with infinite depth or Lipschitz rigid boundaries, and in arbitrary dimension $d$ of the interface. The problem is nonlocal, quasilinear, and to leading order, is scaling invariant in the Sobolev space $H^{s_c}(\mathbb{R}^d)$ with $s_c=1+\frac d2$. We prove local well-posedness for large data in all subcritical Sobolev spaces $H^s(\mathbb{R}^d)$, $s>s_c$, allowing for initial interfaces whose curvatures are unbounded and, furthermore when $d=1$, not square integrable. To the best of…
The present paper is concerned with the analysis of two strongly coupled systems of degenerate parabolic partial differential equations arising in multiphase thin film flows. In particular, we consider the two-phase thin film Muskat problem and the two-phase thin film approximation of the Stokes flow under the influence of both, capillary and gravitational forces. The existence of global weak solutions for medium size initial data in large function spaces is proved. Moreover, exponential decay results towards the equilibrium state are established, where the decay rate can be estimated by expli…
The Muskat problem involves filtration of two incompressible fluids throughout a porous medium. In this paper we shall discuss in 3-D the relevance of the Rayleigh-Taylor condition, and the topology of the initial interface, in order to prove its local existence in Sobolev spaces.
In this work we study the evolution of the interface between two different fluids in a porous media with two different permeabilities. We prove local existence in Sobolev spaces, when the free boundary is given by the discontinuity among the densities and viscosities of the fluids.
This article is devoted to the study of the Cauchy problem for the Muskat equation. We consider initial data belonging to the critical Sobolev space of functions with three-half derivative in $L^2$, up to a fractional logarithmic correction. As a corollary, we obtain the first local and global well-posedness results for initial free surface which are not Lipschitz.
We paralinearize the Muskat equation to extract an explicit parabolic evolution equation having a compact form. This result is applied to give a simple proof of the local well-posedness of the Cauchy problem for rough initial data, in homogeneous Sobolev spaces $\dot{H}^1(\mathbb{R})\cap \dot{H}^s(\mathbb{R})$ with $s>3/2$. This paper is essentially self-contained and does not rely on general results from paradifferential calculus.
In this paper we study the two-dimensional multiphase Muskat problem describing the motion of three immiscible fluids with general viscosities in a vertical homogeneous porous medium under the influence of gravity. Employing Rellich type identities in the regime where the fluids are ordered according to their viscosities, respectively a Neumann series argument when the fluids are not ordered by viscosity, we may recast the governing equations as a strongly coupled nonlinear and nonlocal evolution problem for the functions that parameterize the sharp interfaces that separate the fluids. This pr…
The Muskat problem, in its general setting, concerns the interface evolution between two incompressible fluids of different densities and viscosities in porous media. The interface motion is driven by gravity and capillarity forces, where the latter is due to surface tension. To leading order, both the Muskat problems with and without surface tension effect are scaling invariant in the Sobolev space $H^{1+\frac{d}{2}}(\mathbb{R}^d)$, where $d$ is the dimension of the interface. We prove that for any subcritical data satisfying the Rayleigh-Taylor condition, solutions of the Muskat problem with…
We study the Muskat problem for one fluid or two fluids, with or without viscosity jump, with or without rigid boundaries, and in arbitrary space dimension $d$ of the interface. The Muskat problem is scaling invariant in the Sobolev space $H^{s_c}(\mathbb{R}^d)$ where $s_c=1+\frac{d}{2}$. Employing a paradifferential approach, we prove local well-posedness for large data in any subcritical Sobolev spaces $H^s(\mathbb{R}^d)$, $s>s_c$. Moreover, the rigid boundaries are only required to be Lipschitz and can have arbitrarily large variation. The Rayleigh-Taylor stability condition is assumed for …
We consider the Dirichlet-to-Neumann operator in strip-like and half-space domains with Lipschitz boundary. It is shown that the quadratic form generated by the Dirichlet-to-Neumann operator controls some sharp homogeneous fractional Sobolev norm. As an application, we prove that the global Lipschitz solutions constructed in \cite{DGN} for the one-phase Muskat problem decays exponentially in time in any Hölder norm $C^α$, $α\in (0, 1)$.
We study the dynamics of the interface between two incompressible 2-D flows where the evolution equation is obtained from Darcy's law. The free boundary is given by the discontinuity among the densities and viscosities of the fluids. This physical scenario is known as the two dimensional Muskat problem or the two-phase Hele-Shaw flow. We prove local-existence in Sobolev spaces when, initially, the difference of the gradients of the pressure in the normal direction has the proper sign, an assumption which is also known as the Rayleigh-Taylor condition.
We study the two-dimensional Muskat problem in a horizontally periodic setting and for fluids with arbitrary densities and viscosities. We show that in the presence of surface tension effects the Muskat problem is a quasilinear parabolic problem which is well-posed in the Sobolev space $H^r(\mathbb{S})$ for each $r\in(2,3)$. When neglecting surface tension effects, the Muskat problem is a fully nonlinear evolution equation and of parabolic type in the regime where the Rayleigh-Taylor condition is satisfied. We then establish the well-posedness of the Muskat problem in the open subset of $H^2(\…
In this work we study the evolution of the free boundary between two different fluids in a porous medium where the permeability is a two dimensional step function. The medium can fill the whole plane $\mathbb{R}^2$ or a bounded strip $S=\mathbb{R}\times(-π/2,π/2)$. The system is in the stable regime if the denser fluid is below the lighter one. First, we show local existence in Sobolev spaces by means of energy method when the system is in the stable regime. Then we prove the existence of curves such that they start in the stable regime and in finite time they reach the unstable one. This chan…
In this paper, we prove the existence of solutions to quasilinear elliptic equations on a bounded domain of $\R^N$ under subcritical Musielak-Orlicz-Sobolev growth. Our proofs rely essentially on Mountain Pass Theorem with corresponding variational techniques. Furthermore, we establish the sharpness of our central assumptions. As far as we know, our approach is new, even for the Orlicz case.
We consider the mass-subcritical NLS in dimensions $d\geq 3$ with radial initial data. In the defocusing case, we prove that any solution that remains bounded in the critical Sobolev space throughout its lifespan must be global and scatter. In the focusing case, we prove the existence of a threshold solution that has a compact flow.
We consider the subcritical SQG equation in its natural scale invariant Sobolev space and prove the existence of a global attractor of optimal regularity. The proof is based on a new energy estimate in Sobolev spaces to bootstrap the regularity to the optimal level, derived by means of nonlinear lower bounds on the fractional laplacian. This estimate appears to be new in the literature, and allows a sharp use of the subcritical nature of the $L^\infty$ bounds for this problem. As a byproduct, we obtain attractors for weak solutions as well. Moreover, we study the critical limit of the attracto…
We investigate a natural approximation by subcritical Sobolev embeddings of the Sobolev quotient for the fractional Sobolev spaces $H^s$ for any $0<s<N/2$, using $Γ$-convergence techniques. We show that, for such approximations, optimal functions always exist and exhibit a concentration effect of the $H^s$ energy at one point.
In this paper we establish the reversed sharp Hardy-Littlewood-Sobolev (HLS for short) inequality on the upper half space and obtain a new HLS type integral inequality on the upper half space (extending an inequality found by Hang, Wang and Yan in \cite{HWY2008}) by introducing a uniform approach. The extremal functions are classified via the method of moving spheres, and the best constants are computed. The new approach can also be applied to obtain the classical HLS inequality and other similar inequalities.
This work concerns the semilinear wave equation in three space dimensions with a power-like nonlinearity which is greater than cubic, and not quintic (i.e. not energy-critical). We prove that a scale-invariant Sobolev norm of any non-scattering solution goes to infinity at the maximal time of existence. This gives a refinement on known results on energy-subcritical and energy-supercritical wave equation, with a unified proof. The proof relies on the channel of energy method, as in arXiv:1204.0031, in weighted scale-invariant Sobolev spaces which were introduced in arXiv:1506.00788. These space…
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 …
Let $N\geq 2$ and $1 < p < (N+2)/(N-2)_{+}$. Consider the Lane-Emden equation $Δu + u^p = 0$ in $\mathbb{R}^N$ and recall the classical Liouville type theorem: if $u$ is a non-negative classical solution of the Lane-Emden equation, then $u \equiv 0$. The method of moving planes combined with the Kelvin transform provides an elegant proof of this theorem. A classical approach of J. Serrin and H. Zou, originally used for the Lane-Emden system, yields another proof but only in lower dimensions. Motivated by this, we further refine this approach to find an alternative proof of the Liouville type t…
This paper is devoted to optimal functional inequalities for fractional Laplace operators on the sphere. Based on spectral properties, subcritical inequalities are established. Their consequences for fractional heat flows are considered. These subcritical inequalities interpolate between fractional Sobolev and subcritical fractional logarithmic Sobolev inequalities. Their optimal constants are determined by a spectral gap. In the subcritical range, the method also provides us with remainder terms which can be considered as an improved version of the optimal inequalities. We also consider inequ…
In this paper, we are concerned with the critical and subcritical Trudinger-Moser type inequalities for functions in a fractional Sobolev space $H^{1/2,2}$ on the whole real line. We prove the relation between two inequalities and discuss the attainability of the suprema.
In this paper, we consider minimization problems related to the combined power-type nonlinear scalar field equations involving the Sobolev critical exponent in three space dimensions. In four and higher space dimensions, it is known that for any frequency and any power of the subcritical nonlinearity, there exists a ground state. In contrast to those cases, when the space dimension is three and the subcritical power is three or less, we can show that there exists a threshold frequency, above which no ground state exists, and below which the ground state exists. Furthermore, we prove the differ…
In this note, we give a new proof of subcritical Trudinger-Moser inequality on $\mathbb{R}^n$. All the existing proofs on this inequality are based on the rearrangement argument with respect to functions in the Sobolev space $W^{1,n}(\mathbb{R}^n)$. Our method avoids this technique and thus can be used in the Riemannian manifold case and in the entire Heisenberg group.
Sharp Trudinger-Moser inequalities on the first order Sobolev spaces and their analogous Adams inequalities on high order Sobolev spaces play an important role in geometric analysis, partial differential equations and other branches of modern mathematics. Such geometric inequalities have been studied extensively by many authors in recent years and there is a vast literature. There are two types of such optimal inequalities: critical and subcritical sharp inequalities, both are with best constants. Critical sharp inequalities are under the restriction of the full Sobolev norms for the functions…
We consider the Cauchy problem for the defocusing power type nonlinear wave equation in $(1+3)$-dimensions for energy subcritical powers $p$ in the range $3 < p< 5$. We prove that any solution is global-in-time and scatters to free waves in both time directions as long as its critical Sobolev norm stays bounded on the maximal interval of existence.
Two Positive Normalized Solutions and Phase Separation for Coupled Schrödinger Equations on Bounded Domain with L2-Supercritical and Sobolev Critical or Subcritical Exponent
This paper is devoted to study a fractional Choquard problem with slightly subcritical exponents on bounded domains. When the exponent of the convolution type nonlinearity tends to the fractional critical one in the sense of Hardy-Littlewood-Sobolev inequality, we obtain the existence of multiple positive solutions via Lusternik-Schnirelmann category and nonlocal global compactness. Moreover, we prove that the topology of the domain furnishes a lower bound for the number of positive solutions.
In this paper, we consider the wave equation in 3-dimensional space with an energy-subcritical nonlinearity, either in the focusing or defocusing case. We show that any radial solution of the equation which is bounded in the critical Sobolev space is globally defined in time and scatters. The proof depends on the compactness/rigidity argument, decay estimates for radial, "compact" solutions, gain of regularity arguments and the "channel of energy" method.
In this paper, we study the critical Sobolev embeddings $W^{1,p(x)}(Ω)\subset L^{p^*(x)}(Ω)$ for variable exponent Sobolev spaces from the point of view of the $Γ$-convergence. More precisely we determine the $Γ$-limit of subcritical approximation of the best constant associated with this embedding. As an application we provide a sufficient condition for the existence of extremals for the best constant.
We study the scattering theory for the Schrödinger and wave equations with rough potentials in a scale of homogeneous Sobolev spaces. The first half of the paper concerns with an inverse-square potential in both of subcritical and critical constant cases, which is a particular model of scaling-critical singular perturbations. In the subcritical case, the existence of the wave and inverse wave operators defined on a range of homogeneous Sobolev spaces is obtained. In particular, we have the scattering to a free solution in the homogeneous energy space for both of the Schrödinger and wave equati…
The inverse scattering method for the Novikov-Veselov equation is studied for a larger class of Schrödinger potentials than could be handled previously. Previous work concerns so-called conductivity type potentials, which have a bounded positive solution at zero energy and are a nowhere dense set of potentials. We relax this assumption to include logarithmically growing positive solutions at zero energy. These potentials are stable under perturbations. Assuming only that the potential is subcritical and has two weak derivatives in a weighted Sobolev space, we prove that the associated scatteri…
This paper is mainly devoted to the study of the reversed Hardy-Littlewood-Sobolev (HLS) inequality on Heisenberg group $\mathbb{H}^n$ and CR sphere $\mathbb{S}^{2n+1}$. First, we establish the roughly reversed HLS inequality and give a explicitly lower bound for the sharp constant. Then, the existence of the extremal functions with sharp constant is proved by subcritical approach and some compactness techniques. Our method is rearrangement free and can be applied to study the classical HLS inequality and other similar inequalities.
In this paper, we study the following singular nonlinear elliptic problem \begin{equation}\label{eq:1} \left\{ \begin{array}{ll} \displaystyle (-Δ)^{\frac α2} u=λ|u|^{r-2}u+μ\frac{|u|^{q-2}u}{|x|^{s}}\quad &{\rm in }\quad Ω, \\ \\ u=0 &{\rm on }\quad \partialΩ, \end{array} \right. \end{equation} where $Ω$ is a smooth bounded domain in $\mathbb R^N$ with $0\in Ω$, $λ,μ>0,0<s\leqα$, $(-Δ)^{\frac α2}$ is the fractional Laplacian operator with $0<α<2$. We establish existence results of problem \eqref{eq:1} for subcritical, Sobolev critical and Hardy-Sobolev critical cases.
We study the following boundary value problem (P)\ \ \ \ \ {-\mathrm{div}(a(|\nabla u|)\nabla u)=f(x,u),\ & in $Ω$, u=0, & on $\partialΩ$} with nonhomogeneous principal part. By assuming the nonlinearity $f(x, t)$ being subcritical growth, some abstract results of problem (P) are obtained: (1) Regularity; (2) Orlicz-Sobolev versus Hölder local minimizer; (3) Strong comparison principle. Applying these abstract results and critical point theory, we prove the existence of multiple solutions of problem (P) in an Orlicz-Sobolev space.
We prove sharp inequalities of Hardy type for functions in the Sobolev space $W^{1,p}$ on the unit sphere $\mathbb{S}^{n-1}$ in $\mathbb{R}^{n}$. We achieve this in both the subcritical and critical cases. The method we use to show optimality takes into account all the constants involved in our inequalities.
We consider the long time well-posedness of the Cauchy problem with large Sobolev data for a class of nonlinear Schrödinger equations (NLS) on $\mathbb{R}^2$ with power nonlinearities of arbitrary odd degree. Specifically, the method in this paper applies to those NLS equations having either elliptic signature with a defocusing nonlinearity, or else having an indefinite signature. By rigorously justifying that these equations govern the modulation of wave packet-like solutions to an artificially constructed equation with an advantageous structure, we show that a priori every subcritical inhomo…
In this paper we consider lower order perturbations of the critical Lane-Emden system posed on a bounded smooth domain $Ω\subset \mathbb{R}^N$, with $N \geq3$, inspired by the classical results of Brezis and Nirenberg \cite{BrezisNirenberg1983}. We solve the problem of finding a positive solution for all dimensions $N \geq 4$. For the critical dimension $N=3$ we show a new phenomenon, not observed for scalar problems. Namely, there are parts on the critical hyperbola where solutions exist for all $1$-homogeneous or subcritical superlinear perturbations and parts where there are no solutions fo…
In this paper, we look for solutions to the following coupled Schrödinger system \begin{equation*} \begin{cases} -Δu+λ_{1}u=α_{1}|u|^{p-2}u+μ_{1}u^{3}+ρv^{2}u & \text{in} \ \ \mathbb{R}^{N}, -Δv+λ_{2}v=α_{2}|v|^{p-2}v+μ_{2}v^{3}+ρu^{2}v& \text{in} \ \ \mathbb{R}^{N}, \end{cases} \end{equation*} with the additional conditions $\int_{\mathbb{R}^{N}}u^{2}dx=b^{2}_{1}$ and $\int_{\mathbb{R}^{N}}v^{2}dx=b^{2}_{2}.$ Here $b_1, b_2>0$ are prescribed, $N\leq3$, $μ_{1}, μ_{2}, α_{1},α_{2},ρ>0$, $p\in (2,4)$ and the frequencies $λ_{1},λ_{2}$ are unknown and will appear as Lagrange multipliers. In the on…
This paper deals with the qualitative analysis of solutions to the following $(p,q)$-fractional equation: \begin{equation*} \begin{array}{rllll} (-Δ)^{s_1}_{p}u+(-Δ)^{s_2}_{q}u+V(x) \big(|u|^{p-2}u+|u|^{q-2}u\big) = K(x)\frac{f(u)}{|x|^\ba} \; \text{ in } \mb R^N, \end{array} \end{equation*} \noi where $1< q< p$, $0<s_2\leq s_1<1$, $ps_1=N$, $\ba\in[0,N)$, and $V,K:\mb R^N\to\mb R$, $f:\mb R\to \mb R$ are continuous functions satisfying some natural hypotheses. We are concerned both with the case when $f$ has a subcritical growth and with the critical framework with respect to the exponential …
We study semilinear damped wave equations with power nonlinearity $|u|^p$ and initial data belonging to Sobolev spaces of negative order $\dot{H}^{-γ}$. In the present paper, we obtain a new critical exponent $p=p_{\mathrm{crit}}(n,γ):=1+\frac{4}{n+2γ}$ for some $γ\in(0,\frac{n}{2})$ and low dimensions in the framework of Soblev spaces of negative order. Precisely, global (in time) existence of small data Sobolev solutions of lower regularity is proved for $p>p_{\mathrm{crit}}(n,γ)$, and blow-up of weak solutions in finite time even for small data if $1<p<p_{\mathrm{crit}}(n,γ)$. Furthermore, …
