We study properties of positive functions satisfying (E) --$Δ$u+m|$\nabla$u| q -- u p = 0 is a domain $Ω$ or in R N + when p > 1 and 1 < q < 2. We give sufficient conditions for the existence of a solution to (E) with a nonnegative measure $μ$ as boundary data, and these conditions are expressed in terms of Bessel capacities on the boundary. We also study removable boundary singularities and solutions with an isolated singularity on $\partial$$Ω$. The different results depends on two critical exponents for p = p c := N +1 N --1 and for q = q c := N +1 N and on the position of q with respect to 2p p+1. Contents
Asymptotic solutions of a quasilinear parabolic equation with a small parameter at the higher derivative are constructed near large-gradient and Lagrange singularities of A-type, which represent interest for studying processes of shock waves formation in physical media with a small nonzero viscosity.
We consider the blow-up of solutions to the following parameterized nonlinear wave equation: $ u_{tt} = c(u)^{2} u_{xx} + λc(u)c'(u)( u_x)^2$ with the real parameter $λ$. In previous works, it was reported that there exist finite time blow-up solutions with $λ=1$ and $2$. However, the construction of a blow-up solution depends on the symmetric structure of the equation (e.g., the energy conservation law). In the present paper, we extend the blow-up result with $λ=1$ to the case with $λ\in (0,1]$ by using a new $L^{2/λ}$ estimate. Moreover, some properties for the blow-up solution including the…
We study the boundary behaviour of the solutions of (E) $-Δ_p u+|\nabla u|^q=0$ in a domain $Ω\subset \mathbb{R}^N$, when $N\geq p > q >p-1$. We show the existence of a critical exponent $q_* < p$ such that if $p-1 < q < q_*$ there exist positive solutions of (E) with an isolated singularity on $\partialΩ$ and that these solutions belong to two different classes of singular solutions. If $q_*\leq q < p$ no such solution exists and actually any boundary isolated singularity of a positive solution of (E) is removable. We prove that all the singular positive solutions are classified according the…
We study some properties of the solutions of (E) $\;-\Gd_p u+|\nabla u|^q=0$ in a domain $\Gw \sbs \BBR^N$, mostly when $p\geq q>p-1$. We give a universal priori estimate of the gradient of the solutions with respect to the distance to the boundary. We give a full classification of the isolated singularities of the positive solutions of (E), a partial classification of isolated singularities of the negative solutions. We prove a general removability result in expressed in terms of some Bessel capacity of the removable set. We extend our estimates to equations on complete non compact manifolds …
We study a boundary value elliptic problem having a lower order nonlinear term with subquadratic growth in the gradient of the solution and possibly singular when the solution vanishes. If the singularity is mild enough (and even in the absence of the singularity), we prove an existence and multiplicity result. On the contrary, we prove an existence and uniqueness result for strong singularities.
This work deals with the existence of at least two positive solutions for the class of quasilinear elliptic equations with cylindrical singularities and multiple critical nonlinearities that can be written in the form \begin{align*} -\operatorname{div}\left[\frac{|\nabla u|^{p-2}}{|y|^{ap}}\nabla u\right] -μ\,\frac{u^{p-1}}{|y|^{p(a+1)}} = h\,\frac{u^{p^*(a,b)-1}}{|y|^{bp^*(a,b)}} +λg\,\frac{u^{q-1}}{|y|^{cp^*(a,c)}}, \qquad (x,y) \in \mathbb{R}^{N-k}\times\mathbb{R}^k. \end{align*} We consider $N \geqslant 3$, $λ>0$, $p < k \leqslant N$, $1<p<N$, $0 \leqslant μ<\barμ \equiv \left\{[k-p(a+1)]/…
This work deals with existence of solutions for the class of quasilinear elliptic problems with cylindrical singularities and multiple critical nonlinearities that can be written in the form \begin{align*} -\operatorname{div}\left[\frac{|\nabla u|^{p-2}}{|y|^{ap}}\nabla u\right] -μ\,\frac{u^{p-1}}{|y|^{p(a+1)}} = \frac{u^{p^*(a,b)-1}}{|y|^{bp^*(a,b)}} + \frac{u^{p^*(a,c)-1}}{|y|^{cp^*(a,c)}}, \qquad (x,y) \in \mathbb{R}^{N-k}\times\mathbb{R}^k. \end{align*} The existence of a positive, weak solution $u \in \mathcal{D}_a^{1,p}(\mathbb{R}^N\backslash\{|y|=0\})$ is proved with the help of the mou…
In this paper, we prove the global existence for some 4-D quasilinear wave equations with small, radial data in $H^{3}\times H^{2}$. The main idea is to exploit local energy estimates with variable coefficients, together with the trace estimates.
We classify positive solutions to a class of quasilinear equations with Neumann or Robin boundary conditions in convex domains. Our main tool is an integral formula involving the trace of some relevant quantities for the problem. Under a suitable condition on the nonlinearity, a relevant consequence of our results is that we can extend to weak solutions a celebrated result obtained for stable solutions by Casten and Holland and by Matano.
In this article we provide a local wellposedness theory for quasilinear Maxwell equations with absorbing boundary conditions in $\mathcal{H}^m$ for $m \geq 3$. The Maxwell equations are equipped with instantaneous nonlinear material laws leading to a quasilinear symmetric hyperbolic first order system. We consider both linear and nonlinear absorbing boundary conditions. We show existence and uniqueness of a local solution, provide a blow-up criterion in the Lipschitz norm, and prove the continuous dependence on the data. In the case of nonlinear boundary conditions we need a smallness assumpti…
In this paper we give necessary and sufficient conditions for the existence of solutions to quasilinear equations of Lane--Emden type with measure data on a Carnot group $\mathbb G$ of arbitrary step. The quasilinear part involves operators of the $p$-Laplacian type $Δ_{\mathbb G,\,p}\,$, $1<p<\infty$. These results are based on new a priori estimates of solutions in terms of nonlinear potentials of Th. Wolff's type. As a consequence, we characterize completely removable singularities, and prove a Liouville type theorem for supersolutions of quasilinear equations with source terms which has be…
In this paper, we investigate the formation and propagation of singularities for the system for one-dimensional Chaplygin gas, which is described by a quasilinear hyperbolic system with linearly degenerate characteristic fields. The phenomena of concentration and the formation of ``$δ$-shock'' waves are identified and analyzed systematically for this system under suitably large initial data. In contrast to the Rankine-Hogoniot conditions for classical shock, the generalized Rankine-Hogoniot conditions for ``$δ$-shock'' waves are established. Finally, it is shown that the total mass and momentu…
Let $Ω\subset \mathbb{R}^{N}$ be a smooth bounded domain, $H$ a Caratheodory function defined in $Ω\times \mathbb{R\times R}^{N},$ and $μ$ a bounded Radon measure in $Ω.$ We study the problem% \begin{equation*} -Δ_{p}u+H(x,u,\nabla u)=μ\quad \text{in}Ω,\qquad u=0\quad \text{on}\partial Ω, \end{equation*} where $Δ_{p}$ is the $p$-Laplacian ($p>1$)$,$ and we emphasize the case $H(x,u,\nabla u)=\pm \left\| \nabla u\right\| ^{q}$ ($q>0$). We obtain an existence result under subcritical growth assumptions on $H,$ we give necessary conditions of existence in terms of capacity properties, and we prov…
The existence problem is solved, and global pointwise estimates of solutions are obtained for quasilinear and Hessian equations of Lane-Emden type, including the following two model problems: $ -Δ_p u = u^q + μ$ and $F_k[-u] = u^q + μ$, $u > 0$, on $R^n$, or on a bounded domain $Ω$. Here $Δ_p$ is the p-Laplacian, and $F_k[u]$ is the $k$-Hessian defined as the sum of $k\times k$ principal minors of the Hessian matrix $D^2 u$; $μ$ is a nonnegative measurable function (or measure) on $Ω$. The solvability of these classes of equations in the renormalized (entropy) or viscosity sense has been an op…
In a recent preprint, we showed that for the Dirichlet Laplacian $Δ$ on the unit disk, the wave trace ${Tr}(e^{it\sqrtΔ})$, which has complicated singularities on $2π- ε< t < 2π$, is, on the interval $2π< t < 2π+ ε$, the restriction to this interval of a $C^\infty$ function on its closure. In this paper we prove the analogue of this somewhat counter-intuitive result for the Friedlander model. The proof for the Friedlander model is simpler and more transparent than in the case of the unit disk.
We prove that for large enough data, the life span of smooth solutions to the Cauchy problem for the following two quasilinear hyperbolic systems is finite: (1) equations of relativistic compressible fluid dynamics, (2) equations of plasma dynamics (Euler-Maxwell equations with background charge) in the spherically symmetric case.
Let $-λ_j$ be the eigenvalues of the Laplace operator on the unit disk with Dirichlet conditions. The distribution $h(t) = \sum_j e^{i\sqrtλ_j t}$ is the trace of the solution operator of the wave equation on the disk. It is well known that $h$ has isolated singularities at the lengths of the reflecting geodesics. In particular, $h$ is singular at $t_k$, the perimeter of the regular inscribed polygon with $k$ sides. Evidently, $t_k < 2π$, the perimeter of the circle, and $t_k$ tends to $2π$. In this paper, we show that $h(t)$ is infinitely differentiable as $t$ tends to $2π$ from the right.
Singular solutions of the harmonic Einstein evolution equation are constructed which are related to spatially global and time-local solutions for a certain class of quasilinear hyperbolic systems of second order. The constructed singularities of curvature invariants occur generically and are accessible by g.a.p. curves. The singularities are not strongly censored, and for strongly asymptotically predictable space-times, they are located in the causal past of the future null infinity, and are, hence, not shielded by a black hole. This is an alternative construction of singularities, which may b…
We consider the inviscid unsteady Prandtl system in two dimensions, motivated by the fact that it should model to leading order separation and singularity formation for the original viscous system. We give a sharp expression for the maximal time of existence of regular solutions, showing that singularities only happen at the boundary or on the set of zero vorticity, and that they correspond to boundary layer separation. We then exhibit new Lagrangian formulae for backward self-similar profiles, and study them also with a different approach that was initiated by Elliott-Smith-Cowley and Cassel-…
The last decades saw growing interest across multiple disciplines in nonlinear phenomena described by partial differential equations (PDE). Integrability of such equations is tightly related with the Painleve property - solutions being free from moveable critical singularities. The problem of Painleve classification of ordinary and partial nonlinear differential equations lasting since the end of XIX century saw significant advances for the equation of lower (mainly up to fourth with rare exceptions) order, however not that much for the equations of higher orders. Recent works of the author ha…
In this paper, we prove a sharp local well-posedness result for spherically symmetric solutions to quasilinear wave equations with rough initial data, when the spatial dimension is three or higher. Our approach is based on Morawetz type local energy estimates with fractional regularity for linear wave equations with variable $C^1$ coefficients, which rely on multiplier method, weighted Littlewood-Paley theory, duality and interpolation. Together with weighted linear and nonlinear estimates (including weighted trace estimates, Hardy's inequality, fractional chain rule and fractional Leibniz rul…
Let $(X,g)$ be a compact manifold with conic singularities. Taking $Δ_g$ to be the Friedrichs extension of the Laplace-Beltrami operator, we examine the singularities of the trace of the half-wave group $e^{- i t \sqrt{ \smash[b]{Δ_g}}}$ arising from strictly diffractive closed geodesics. Under a generic nonconjugacy assumption, we compute the principal amplitude of these singularities in terms of invariants associated to the geodesic and data from the cone point. This generalizes the classical theorem of Duistermaat-Guillemin on smooth manifolds and a theorem of Hillairet on flat surfaces wit…
We derive a detailed asymptotic expansion of the heat trace for the Laplace-Beltrami operator on functions on manifolds with conic singularities, using the Singular Asymptotics Lemma of Jochen Bruening and Robert T. Seeley [BS]. In the subsequent paper we investigate how the terms in the expansion reflect the geometry of the manifold.
Motivated by a new kind of initial boundary value problem (IBVP) with a free boundary arising in wave-structure interaction, we propose here a general approach to one-dimensional IBVP as well as transmission problems. For general strictly hyperbolic 2x2 quasilinear hyperbolic systems, we derive new sharp linear estimates with refined dependence on the source term and control on the traces of the solution at the boundary. These new estimates are used to obtain sharp results for quasilinear IBVP and transmission problems, and for fixed, moving, and free boundaries. In the latter case, two kinds …
In a previous work [I. Rodnianski and Y. Shlapentokh-Rothman, Naked Singularities for the Einstein Vacuum Equations: The Exterior Solution, arXiv:1912.08478] we constructed solutions to the Einstein vacuum equations in 3+1 dimensions which corresponded to the exterior region of a naked singularity. In this work we construct solutions which correspond to the interior region and show that the two solutions may be glued together to produce a naked singularity. Fundamental to our construction is the novel type of self-similarity for the Einstein vacuum equations that we introduced in our previous …
We construct positive weak solutions of a class of semilinear elliptic equation which vanish in suitable trace sense on the boundary of a given smooth bounded N-dimensional domain, but which are singular at prescribed isolated points of the boundary. Similar constructions are carried out for solutions which are singular on any given embedded submanifold of the boundary.
In his 2007 monograph, D. Christodoulou proved a breakthrough result giving a detailed description of the formation of shocks in solutions to the relativistic Euler equations in three spatial dimensions. He assumed that the data have small $H^N$ norm, where $N$ is a sufficiently large integer. To deduce the shock formation, he also assumed that the data verify a signed integral inequality. In the present monograph, we extend Christodoulou's framework and use it to prove that shock singularities often develop in initially small, regular solutions to two important classes of quasilinear wave equ…
We investigate the singularities of the trace of the half-wave group, $\mathrm{Tr} \, e^{-it\sqrtΔ}$, on Euclidean surfaces with conical singularities $(X,g)$. We compute the leading-order singularity associated to periodic orbits with successive degenerate diffractions. This result extends the previous work of the third author \cite{Hil} and the two-dimensional case of the work of the first author and Wunsch \cite{ForWun} as well as the seminal result of Duistermaat and Guillemin \cite{DuiGui} in the smooth setting. As an intermediate step, we identify the wave propagators on $X$ as singular …
Let $n\geq2$ and $ Ω\subset \mathbb{R}^{n+1}$ be a Lipschitz wedge- like domain . We construct positive weak solutions of the problem $$Δu + u^p = 0 \quad\hbox{in}\, Ω,$$ which vanish in a suitable trace sense on $\partialΩ,$ but which are singular at prescribed "edge" of $Ω$ if $p$ is equal or slightly above a certain exponent $p_0>1$ which depends on $Ω.$ Moreover, in the case which $Ω$ is unbounded, the solutions have fast decay at infinity.
We consider the case of scattering of several obstacles in $\mathbb{R}^d$ for $d \geq 2$ for the Laplace operator $Δ$ with Dirichlet boundary conditions imposed on the obstacles. In the case of two obstacles, we have the Laplace operators $Δ_1$ and $Δ_2$ obtained by imposing Dirichlet boundary conditions only on one of the objects. The relative trace operator $g(Δ) - g(Δ_1) - g(Δ_2) + g(Δ_0)$ was introduced in [18] and shown to be trace-class for a large class of functions $g$, including certrain functions of polynomial growth. When $g$ is sufficiently regular at zero and fast decaying at infi…
We prove the existence of a solution of (--$Δ$) s u + f (u) = 0 in a smooth bounded domain $Ω$ with a prescribed boundary value $μ$ in the class of positive Radon measures for a large class of continuous functions f satisfying a weak singularity condition expressed under an integral form. We study the existence of a boundary trace for positive moderate solutions. In the particular case where f (u) = u p and $μ$ is a Dirac mass, we prove the existence of several critical exponents p.
We show the existence and the uniqueness of initial traces of nonnegative solutions to a semilinear heat equation on a half space of ${\mathbb R}^N$ under the zero Dirichlet boundary condition. Furthermore, we obtain necessary conditions and sufficient conditions on the initial data for the solvability of the corresponding Cauchy--Dirichlet problem. Our necessary conditions and sufficient conditions are sharp and enable us to find optimal singularities of initial data for the solvability of the Cauchy--Dirichlet problem.
We study the boundary value problem with measures for (E1) $-\Gd u+g(|\nabla u|)=0$ in a bounded domain $\Gw$ in $\BBR^N$, satisfying (E2) $ u=\gm$ on $\prt\Gw$ and prove that if $g\in L^1(1,\infty;t^{-(2N+1)/N}dt)$ is nondecreasing (E1)-(E2) can be solved with any positive bounded measure. When $g(r)\geq r^q$ with $q>1$ we prove that any positive function satisfying (E1) admits a boundary trace which is an outer regular Borel measure, not necessarily bounded. When $g(r)=r^q$ with $1
We study the boundary behaviour of the of (E) $-\Gd u-\myfrac{\xk }{d^2(x)}u+g(u)=0$, where $0<\xk <\frac{1}{4}$ and $g$ is a continuous nonndecreasing function in a bounded convex domain of $\BBR^N$. We first construct the Martin kernel associated to the the linear operator $\CL_{\xk }=-\Gd-\frac{\xk }{d^2(x)}$ and give a general condition for solving equation (E) with any Radon measure $\gm$ for boundary data. When $g(u)=|u|^{q-1}u$ we show the existence of a critical exponent $q_c=q_c(N,\xk )>1$: when $0<q<q_c$ any measure is eligible for solving (E) with $\gm$ for boundary data; if $q\geq …
Motivated by the study of the corner singularities in the so-called cavity flow, we establish in this article, the existence and uniqueness of solutions in $L^2(Ω)^2$ for the Stokes problem in a domain $Ω,$ when $Ω$ is a smooth domain or a convex polygon. We establish also a trace theorem and show that the trace of $u$ can be arbitrary in $L^2(\partialΩ)^2.$ The results are also extended to the linear evolution Stokes problem.
In this article we study the positive solutions of the parabolic semilinear system of competitive type \[ \left\{\begin{array} [c]{c}% u_{t}-Δu+v^{p}=0, v_{t}-Δv+u^{q}=0, \end{array} \right. \] in $Ω\times\left(0,T\right) $, where $Ω$ is a domain of $\mathbb{R}^{N},$ and $p,q>0,$ $pq\neq1.$ Despite of the lack of comparison principles, we prove local upper estimates in the superlinear case $pq>1$ of the form \[ u(x,t)\leqq Ct^{-(p+1)/(pq-1)},\qquad v(x,t)\leqq Ct^{-(q+1)/(pq-1)}% \] in $ω\times\left(0,T_{1}\right) ,$ for any domain $ω\subset\subsetΩ$ and $T_{1}\in\left(0,T\right) ,$ and $C=C(N…
We study leading order singularities of the wave trace of the Aharonov--Bohm Hamiltonian on $\mathbf{R}^2$ with multiple solenoids under a generic assumption that no three solenoids are collinear. Then we apply our formula to get a lower bound of scattering resonances in a logarithmic neighborhood near the real axis.
We study the Robin boundary-value problem for bounded domains with isolated singularities. Because for such domains trace spaces of space $H^1(D)$ on its boundaries are weighted Sobolev spaces $L^{2, ξ}(\partial D)$ existence and uniqueness of corresponding Robin boundary-value problems depends on properties of embedding operators $I_1: H^{1}(D)\to L^{2}(D)$ and $I_{2}:H^{1}(D)\to L^{2,ξ}(\partial D)$ i.e. on type of singularities. We obtain an exact description of the weights $ξ$ for bounded domains with 'outside peaks' on its boundaries. This result allows us to formulate correctly the corre…
Based on Guillemin's work on gauged Lagrangian distributions, we will introduce the notion of a poly-$\log$-homogeneous distribution as an approach to $ζ$-functions for a class of Fourier Integral Operators which includes cases of amplitudes with asymptotic expansion $\sum_{k\in\mathbb{N}}a_{m_k}$ where each $a_{m_k}$ is $\log$-homogeneous with degree of homogeneity $m_k$ but violating $\Re(m_k)\to-\infty$. We will calculate the Laurent expansion for the $ζ$-function and give formulae for the coefficients in terms of the phase function and amplitude as well as investigate generalizations to th…
