Questions tagged [ap.analysis-of-pdes]
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
4,466 questions
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Compactness of solutions of elliptic equation
Consider the following nonlinear elliptic equation
$$
-\triangle u + u + u^3 = g, \quad x \in R^3.
$$
If $g \in L^2(R^3)$, then the set $Q$ of solutions of above equation is bounded in $H^2(R^3)$, and ...
- 425
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Compatibility conditions for parabolic regularity
I'm trying to understand the compatibility conditions for regularity of second order parabolic equations.
Let's consider the equation $u_t - Lu = f$ with $u(0)=g$ on $\Omega \times [0,T)$ with $u = 0$ ...
- 2,641
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Parabolic PDE; uniform bound on approximations $u'_n$ in $L^2(0,T;V^*)$ without using orthogonal basis?
Let $V \subset H \subset V^*$ be a Gelfand triple, all Hilbert and separable spaces.
I consider the PDE with weak form: find $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V')$ such that
$$\langle u'(t), v(...
- 405
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Dirichlet-to-Neumann map on $C^{k,1}$ domains
I am interested in the mapping properties of the Dirichlet-to-Neumann map (also called the Poincare-Steklov operator) for $C^{k,1}$ domains, between Sobolev spaces on the boundary. What I know is in ...
- 3,322
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Test function .
For a smooth test function \eta and some constant C is it possible to have an estimate like the following?
|grad \eta|^2 < C {\eta}^2 ?
Thanks.
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Core of divergence form operator with unbounded coefficient
Consider the unbounded operator $L$ on $L^2(\mathbb{R^d})$ to be the self-adjoint extension of
$$Lf := \nabla \cdot \left(a(x) \nabla f(x) \right)$$ on $C^2_c(\mathbb{R^d})$.
I also assume that $a(x)...
- 617
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Lagrangean uniqueness versus Eulerian uniqueness
(1) Lagrangean description. Let us consider a $N\times N$ system of autonomous ODE:
$$
\dot x=a(x),\quad \mathbb R\ni t\mapsto x(t)\in \mathbb R^N,\quad a:\mathbb R^N\rightarrow \mathbb R^N.
$$
...
- 16.2k
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General solutions for HJB equations in a special case.
I am reading the book of Wendell Flemming in control theorem to learn the HJB equation
Here is the setting that interests me: Let $g_i: i=1,2$ be $C^2 =C^{2}(-\infty,\infty)$ functions such that $0\...
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Vanishing solution of the Poisson equation at infinity
Hi, I am interested in finding some vanish bahavior at infinity of the solutions of this kind of equations:
$-\Delta\phi+a(x)\phi=b(x)$
where $a(x), b(x)\in L^{p}$ with $1\leq p\leq 3$. Besides $\...
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Interpolation with time continuity
If $u(x,t)$ is a function depends on $x\in\Omega$ and $t\in[0,T]$. The following result could be found in L.C. Evans's book "PDE".
Suppose $u\in L^2(0,T;H_0^1(\Omega))$, with $u_t\in L^2(0,T;H^{-1}(\...
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Generalized bilinear estimates
Hello. Let $ X^{s,b} $ be the Bourgain space generated by $ \tau - \xi^3 $. It is proved that, for $ s\in (-\frac{1}{2}, 0] $, we have
$$
\|(u^2)_x\| _{X^{s,b'-1}} \leq c \|u\|_{X^{s,b}} \|u\|_{X^{-...
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Is it classical that the solution to an hyperbolic equation equation is Lipschitz -continuous $[0,\infty)\to L^1(\mathbb{R})$?
Recently I am reading a paper "Global solution and smoothing effect for a non-local regularization of a hyperbolic equation" published on J.E.E, 2004. In the proof, the authors write "It is classical ...
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A critical elliptic PDE
I am considering the problem $-\Delta u=|u|^4u$, $x\in \Omega\subset \mathbb{R}^3$, $u|_{\partial \Omega}=0$. Where $\Omega$ is a unbounded domain. Some special case like $\Omega=\mathbb{R}^3-B_1(0)$, ...
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$\mathcal{D}(0,T;V)$ is dense in $W(0,T)$
Where can I find a proof that $\mathcal{D}(0,T;V)$ (the space of $V$-valued compactly supported functions on $[0,T]$) is dense in the space $W(0,T)$, where $$W(0,T) := \{ u \in L^2(0,T;V) : u' \in L^2(...
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Is this integral operator about Stokes' Flow compact?
Consider the following integral operator $\mathcal{A}$ on [EDITED: continuous vector function $f=(f_i):\partial S\to{\mathbb R}^3$]:
$$
({\mathcal A}f)_j(x_0):=\int_{\partial S}\sum_{i=1}^3 f_i(x)G_{...
user14319
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Weak solution of a certain pde with integral term
Let us consider the following pde on the domain $(0,T)\times(0,1)$
$
\dot{p}(t,x)+v(t)p_{x}(t,x)+v'(t)\int_{0}^{1} \rho(t,s)p_{s}(t,s)\ ds=0
$
with initial data $p(0,x)=p_{0}(x)$ and boundary data $...
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partial maximum principle for elliptic differential operators
Let $(M,g)$ be a closed, smooth Riemannian manifold. Let $P$ be a self-adjoint, elliptic differential operator defined on $C^\infty(M)$ with smooth coefficients. Suppose as well that the lowest ...
- 1,701
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A special Integral Kernel
Does there exist either one / general class of non-negative definite , symmetric Integral Kernel map satisfying the following properties ??
$f(x)=(Kg)(x)=\int_{\Omega}K(x,y)g(y)dy$
$K:L^2(\...
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Does the following measurable Halmilton-Jacobian equation admit a Lipschitz solution?
I have the following question:
Let $F:\Omega\times \mathbb{R}^n\to [0,\infty)$ be a convex Finsler norm, which means that
$F(x,\cdot)$ is convex with respect to the second variable.
$F(\cdot,v)$ is ...
- 1,881
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separation of variables in differential euqtions and compact self-adjoint operators [closed]
For some partial differential equations in physics, people may separate the variables and get some eigenfunctions. And then for any solutions for that equation, people often suppose them to be a ...
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References for weak ellipticity
There are good books (like Evans) for strongly elliptic second order linear PDE. I want to learn about weakly elliptic PDE (of any order). Are there any good books for the same? I am very curious as ...
- 3,383
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Finding singular "solutions" to the Dirichlet problem for Schrödinger operators that do not admit smooth solutions
Suppose that $\Omega\subset \mathbb{R}^n$ is a smooth open region and that $V:\Omega\to \mathbb{R}^+$ be a positive smooth function.
Then we have a family of operators
$$L_\epsilon =-\Delta -\epsilon ...
- 2,299
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Inclusions between $L^p$ continuous functions and Triebel-Lizorkin spaces
Working in $\mathbb{R}^{d}$, consider on the one hand the space of continuous $L^{p}$ functions (let's use $V$ to denote this space), and on the other the family $\{ F_ {\alpha}^{p, q} \}_{\alpha, q}$ ...
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Integrability of ground state solution for elliptic equation
For the solution of semi-linear elliptic equation, for example I'm considering the 2D cubic nonlinear Schroedinger equation, the correspongding elliptic equation is $\Delta u+u^3=u$, with $u>0$. By ...
- 331
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Strichartz estimates over cones
I'm trying to understand Sogge's book Lectures on Non-Linear Wave Equations, the part where he proves global existence for semilinear equations. There is one part he uses the following inequality:
$\|...
- 11
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285
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Sobolev imbedding failure due to a kink in the domain
I'm looking for a simple example where an inequality of the form $||u||_{L^q} \leq C||u||_{W^{1,p}}$ fails for some $1 \leq q \leq p^*$ (ie. within the acceptable range for which the bound should ...
- 2,641
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Geometric description of Jacobi's theorem on complete integrals of HJ eqn.
I am not sure if this question is adapted to this site, if it is not, then I will delete it.
The Hamilton--Jacobi theory is about the connection between:
the solutions of an Hamilton--Jacobi ...
- 4,306
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Orthogonal projection of discontinuous piecewise polynomial space in energy scalar product
Let $I = [0,1]$ be the unit interval Let $I$ be partioned into $n$ closed subintervals $(I_j)_J$, each of length $1/n$.
Let $X_{DC} = \{ v \in L^2[0,1] | 1 \leq j \leq n : v_{|I_j} \in \mathcal P_1( ...
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Partial $L^2$ control on (part of) the Hessian of a harmonic function.
I have a simple little analysis question that I'm hoping is well known.
Suppose $D=\lbrace(x,y): x^2+y^2<1\rbrace$ is the unit disk and that $u$ is a harmonic function on $D$. Suppose in addition ...
- 2,299
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A non-elliptic PDE
I wish to know if this PDE can be solved (for a real smooth function $\rho$) on a compact complex surface X :
$\bar{\partial}\partial \rho \wedge \bar{\partial}\partial \rho + \bar{\partial}\partial \...
- 3,383
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Splitting wave equation for application of CPML
A recent paper (Roden and Gedney, 2000) proposed the application of a Convolutional Perfectly Matched Layer (CPML) to approximate free-field conditions for Finite-Difference Time-Domain (FDTD) ...
- 157
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A limit involving a regularizing kernel
I am studying the following article by Benoit Perthame: http://www.mendeley.com/research/uniqueness-error-estimates-first-order-quasilinear-conservation-laws-via-kinetic-entropy-defect-measure/#
...
- 2,222
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Can the solution manifold for an exterior differential system be represented using alternating multivectors?
Differential equations can be written as an ideal of n-forms. Solutions are manifolds where the forms pull back to zero. Is it possible, or useful, to represent the solution by multivectors? For ...
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Asymptotic behaviour near the boundary in the Dirichlet problem for the Laplacian.
Perturbative behaviour of solutions of the solutions of the Dirichlet problem for the Laplacian:
Lets consider $ B = B(0, 1) \in \mathbb{R}^2$ be the unit circle with center at $0\in\mathbb{R}^2$. ...
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Poisson problem with a "scaled" Laplacian.
Let $d_1$ and $d_2$ be positive constants. I'm considering a 2D Poisson-like problem of the form
$$ d_1\frac{\partial^2 u}{\partial x_1^2} + d_2\frac{\partial^2 u}{\partial x_2^2} = f$$ in the ...
- 617
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question about mixed spectrum of a linear operator $\mathcal{L}$
Suppose $\mathcal{L}$ is a bounded linear operator and I have the solution to Eigenvalue problem
$\mathcal{L} \phi + \lambda \phi = 0$
wish to solve the following PDE
$\left(-\partial_t + \mathcal{...
- 351
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optimality of energy estimates for non smooth metric
Consider the linear (geometric) wave equation in dimension (3+1) with non smooth background metric $g$ say $g \in L^\infty_t H^3_x$ and $\partial_t g \in L^\infty H^2_x$, then energy estimates enable ...
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Controlling the Second Eigenvalue of a Schrödinger Operator
Consider a bounded domain $\Omega$ (with smooth boundary) in some Riemannian $n$-manifold $M^n$.
Let $L$ be the operator
$$
L=\Delta+V
$$
where $\Delta$ is the Laplace-beltrami operator on $M$ (so is ...
- 2,299
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Quantitative Weierstrass Approximation and Paley-Wiener for the Laplace Transform II
This is a modification of a previous question.
Question: Suppose $a(s)\in C^\infty([0,1])$ and $H(s,x)\in C^\infty([0,1]\times [0,1])$ with $H(s,x)>0$, $\forall s,x\in [0,1]$. Suppose,
$$\sup_{\...
- 468
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h-oscillating function
I need help understanding the following condition:
$u_h\in L^2(\mathbb{T}^d)$, $\|u_h\|_{L^2(\mathbb{T}^d)}=1$, where $h$ is the semiclassical parameter and $\mathbb{T}^d$ is the flat torus, is ...
- 11
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base change for distributions
For distributions on smooth manifolds one can consider the push-forward which is defined for proper maps, and the pull-back which is defined under certain condition on the wave front set see ...
- 2,649
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Numerical solution
Last time, I asked this question
but after discussing with some friends, I have given up finding closed-form solutions. Now I have a simpler question.Let $g_i: i=1,2$ be $C^2 =C^{2}(-\infty,\infty)$ ...
- 281
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Neumann problem in case f=1
Is there a solution to the following problem?
$-\Delta u = 1$ in $\Omega$ and $\frac{\partial u}{\partial \nu} = 0$ on $\partial \Omega$.
where $\Omega$ is bounded.
- 11
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parabolic PDE with pseudomonotone operators
I am looking for a reference where well-posedness of problems
$$u_t + A(t)u = f$$
is addressed via the Galerkin method where $A$ is a pseudomonotone operator. I am aware that Roubicek's book ...
- 131
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Does there exist a base $\{e_j\}_{j\geq 1}$ of $H(\Omega)$ such that $\{e_j\}_{j\geq 1}$ is linearly independent in $L^2(\omega)^d$?
Does there exist a base $\{e_j\}_{j\geq 1}$ of $H(\Omega)$ such that $\{e_j\}_{j\geq 1}$ is linearly independent in $L^2(\omega)^d$?
Where $\omega\subset\subset \Omega$ with $\Omega$ is a $C^2$ ...
- 11
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0
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74
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strong stability for the wave equation
Consider the $n-$dimensional wave equation
$$z_{tt}=\Delta z + k(x) z - \epsilon {1}_\omega z_t, \; in \; \Omega\times (0,T)$$
where $\omega\subset \Omega.$ Can I have $z(t) \to 0,$ as $t\to+\infty$ ...
- 51
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86
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About the boundedness of the derivative of a function which is in a special function space.
If $f \in C^1 ([0,T] , L^2) \cap C^0 ([0,T] , W^{1,2} )$, $f (t,x) : bounded\; on \; [0,T] \times \Bbb R^n $ then how can I conclude that
$$ \left \| \frac{\partial f}{\partial t} \right \|_{L^\...
- 43
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0
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123
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A slightly subcritical elliptic equation on the ball; blow-up behavior near zero
I am interested in positive ground state solutions of the following elliptic pde:
$-\Delta u(x) = u(x)^{p-\epsilon} $ in the unit ball $B$ in $ R^N$ with $ u=0$ on $ \partial B$. Here $ p:=\...
- 1
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292
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Why is a smooth weak solution strong for stationary linear Stokes problem with zero-traction boundary condition?
Can anyone provide me with a reference giving details on how smooth generalized solutions of the stationary linear Stokes problem can be shown to be classical solutions when a zero-traction boundary ...
2
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164
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Regularity properties of the derivatives of a particular function on $D \times D\to \bar{D} $ ?
This question might sound a little less rigorously formulated, but I hope the question still makes sense.
Let $h: S^1 \to S^1$ be an oriention-preserving homeomorphism and let $p(z,t) = \frac{1}{2\...
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