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.
4,252
questions
2
votes
1
answer
68
views
Estimates on divergence-type operator for the matrix
Is there any result (Schauder-like estimates, $L^2$ estimates or similar) to equations of the form
$$
{\rm div}(Av)=f
$$
where $A$ is the "unknown" (i.e. I would like estimates on $A$ depending on $f$,...
0
votes
1
answer
182
views
Existence of subsequences convergence with weak topology
Let $\left\{ {{\varphi _n}} \right\}$ is the sequence bounded in ${L^\infty }\left( {0,\infty ;H_0^1\left( {0,1} \right)} \right)$. Is there exists $\varphi \in {L^\infty }\left( {0,\infty ;H_0^1\...
8
votes
1
answer
252
views
Positive solutions for semilinear parabolic equations
Let $X$ be a Banach lattice. Consider the system
$$y'(t)=Ay(t)+f(t,y(t)) \qquad \text{in } (0,T) , \qquad y(0)=y_0, \qquad (*)$$
where $T>0$, $A$ generates an analytic positive semigroup $S(t)$ on $...
2
votes
2
answers
749
views
Laplace equation on the disk with Robin boundary condition
Consider the following two dimensional Laplace equation on the unit disk $D$ with homogeneous Robin boundary condition:
$$\Delta u = 0, ~~\frac{\partial u}{\partial n} = b(x) u(x)~~ \forall x \in \...
4
votes
0
answers
191
views
Compactness of multiplication operators
Let $n \ge 3$ and $0 \le V\in L^p(R^n)$ for some $p \ge n/2$. Then the multiplication operator $$Tu=V^{1/2}u$$ is compact from $H^1(R^n)$ to $L^2(R^n)$. If $p>n/2$, this follows from the ...
3
votes
1
answer
129
views
Smoothing-Strichartz estimates for the heat-Schrodinger evolution
Consider the heat-Schrodinger evolution $e^{(1+i)t\Delta}$, $t\geq 0$. For simplicity, suppose to work in dimension three. Due to the Strichartz estimates for the Schrodinger equation, and the ...
2
votes
0
answers
133
views
What is the motivation to define measure valued solutions to a PDE model?
Consider the model
$$\partial _{t}\mu + \partial_{x}(b(t,\mu)\mu)+c(t,\mu) \mu=0,~~~\mu \in \mathcal{M}^{+}(\mathbb{R}^{+}), t \in [0,T], x \in \mathbb{R}^{+} $$
$$ \mu(0)=\mu_{0} $$
where $ \mu (t)$...
1
vote
0
answers
142
views
Rigorous error estimate for semi-discrete heat equation in bounded domain
Let $\Omega$ be a bounded Lipschitz domain in $\mathbb R^N$ and $u_h$ be a solution of
$$
\begin{cases}
\partial_t u_h -\Delta_h u_h = f(x) & \text{ in } \Omega_h\\
u_h=0 &\text{ in } \...
1
vote
0
answers
164
views
A question about Stroock's notes on the Weyl lemma
On p.4 of these notes, D. Stroock gives a quick and efficient construction of the Markov transition functions of a certain diffusion. The idea of his construction (on page 4) is to 'freeze' the ...
1
vote
0
answers
70
views
Fixing constants of a series solution of a fourth-order PDE
The following is the PDE I want to solve,
$$\left(1+x^{2}\right)^{2}y_{xxxx}+8x\left(1+x^{2}\right)y_{xxx} + 4\left(1+3x^{2}\right)y_{xx} + K\left[2x yy_{xx}+\left(1+x^{2}\right)\left(yy_{xxx} + y_{x}...
5
votes
1
answer
589
views
Eigenvalue and eigenfunction convergence
Consider a bounded Euclidean domain $\Omega \subset \mathbb{R}^n$ (for simplicity, let's say, $\Omega$ has smooth boundary and is simply connected). Let $p \in \Omega$ be a point, and call $\Omega_n = ...
6
votes
1
answer
363
views
Interpolation for Sobolev spaces
How one can identify the following (complex) interpolation space
$$E_\theta :=[L^2(\Omega), H^2(\Omega)\cap H_0^1(\Omega)]_\theta,$$
where $\Omega$ is a regular domaine. After research, it seems that ...
2
votes
0
answers
62
views
Proving that $(f,g)$ are Cauchy data for the Schrödinger equation iff $(f,g)$ satisfies an equation
I have to prove that if $f\in H^{1/2}(\partial\Omega)$ then $(f,g)$ are Cauchy
data for the Schrödinger equation if and only if
$$g=\gamma^{-1/2} \Lambda_{\gamma}(\gamma^{-1/2} f)+1/2 \gamma^{-1}\...
1
vote
0
answers
34
views
Reference request- Shock development problem for the compressible Euler equation in 1D
I was wondering if there is any good reference discussing the shock development problem for Euler in 1D? Something in the spirit of Christodoulou's work on the same for higher dimension.
I am ...
2
votes
0
answers
65
views
Properties of solution to Burger's equation using Cole-Hopf transformation
I am currently looking at a $1$D-Burger's equation defined by
\begin{equation} \label{ex burgers}
\left\{
\begin{array}{ll}
{} & \frac{\partial V_m}{\partial t} (t,x) = \frac{\sigma^2}{2} \...
3
votes
2
answers
876
views
Gradient $L^p$ estimates for heat equation
I'm looking for a proof of the gradient estimate associated to the heat equation with Dirichlet boundary conditions, to see if I can express the constant $\color{red}{C}$.
$$\|e^{t\Delta_d}f\|_{W^{1,...
2
votes
0
answers
42
views
Commonly used metrics to compare two Young measures
Let $\Omega\subset \mathbb{R}^n$ be a bounded open set, $K\subset \mathbb{R}^d$ be a compact set, and $M_1(K)$ be the set of probability measures on $K$. Then a Young measure is defined as a $\textrm{...
4
votes
0
answers
162
views
Inequality between Dirichlet and Neumann eigenvalue for Sturm Liouville problem
Consider the following Sturm Liouville problem on an interval $[a,b]$
$$\frac{\mathrm{d}}{\mathrm{d} x}\left[p(x) \frac{\mathrm{d} y}{\mathrm{d} x}\right]+q(x) y=-\lambda w(x) y$$
for given ...
2
votes
1
answer
357
views
Regularity on the boundary for the heat equation with linear source
This is probably a known problem but I was not able to find exactly what I am looking for.
I have the following linear heat equation with zero-flux boundary conditions:
\begin{equation}
\begin{cases}...
1
vote
1
answer
242
views
Metric and particular system of PDE
I have a big problem to solve this system:
$\Delta f−hf^2=0$
$p|\nabla f|^2+hf^3=0$
where $h$ and $p$ are constants (with $h \neq 0$ and $p \neq 0$, $p \neq -1$), $f$ is a scalar function defined ...
1
vote
0
answers
79
views
Reference for the following flow equation
I'm looking for reference on the following partial differential equation. $\partial_tF(t,x) = G(t)((\partial_xF(t,x))^2 + \partial_x^2F(t,x))$, where G(t) is a fixed Schwartz function. If possible, I ...
4
votes
0
answers
154
views
What's the essential definition of resonance of Schrodinger operator?
Rencently, I am reading some articles about time decay estimates or Strichartz estimates for Schrodinger equations with potential. When considering Strichartz estimates for potential $V$ with decay $|...
1
vote
2
answers
699
views
Proof of the du Bois-Reymond lemma "by approximation" [closed]
I'm curious about the following argument in Morrey ("Multiple integrals in the calculus of variations", Lemma 2.3.1). Suppose $f\in L^1[0,1]$ and $$\int_0^1 fg\,dx=0$$ for every test function $g\in C^\...
2
votes
1
answer
153
views
Regularity of Neumann eigenfunctions at vertices of polygons
Given a bounded polygonal domain $D$ in $\mathbb{R}^2$, the Neumann eigenfunctions have continuous version on $\overline{D}$. The eigenfunctions also have critical points at vertices of $D$ (I have ...
2
votes
1
answer
168
views
References for Neumann eigenfunctions
I am looking for references on eigenfunctions with Neumann boundary condition.
In an article, the author wrote in introduction that when a domain is planar polygon, the second eigenfunctions on it ...
3
votes
0
answers
77
views
Logarithmic Sobolev growth of time-space-periodic Schrödinger solutions
Consider the following Schrödinger equation
$$i\partial_t \psi (t,x) + \Delta \psi - V(t,x) \psi = 0 \, ,$$
where $x\in \mathbb{T}^d$ and $V(t,\cdot)$ is real, smooth, and periodic (with a diophantine ...
0
votes
1
answer
419
views
How can I prove this Weitzenbock formula
Update: It is almost sure that the expression of $\kappa$ in coordinates given in the book, namely
$$\kappa(u_t)=h_{\alpha\beta}\frac{\partial u_t^\alpha}{\partial x^i}\frac{\partial u_t^\beta}{\...
11
votes
0
answers
2k
views
A question on trig series
Assume $\{a_k\}_{k\ge1}$ is a real sequence such that $u(x) = \sum_{k\ge 1}a_k\sin(kx)$ is a smooth function, and for every $x \in [-\pi, \pi]$
$$\left(\sum_{k\ge 1}\frac{a_k}{k}\sin(kx)\right)\left(\...
5
votes
1
answer
182
views
References for systems of elliptic PDEs
I was wondering if there were any recent references dealing with the theory of systems of elliptic PDEs: in particular, someone was telling me about something which sounded like 'Schur complementarity'...
2
votes
0
answers
54
views
What's the state of the art on stochastic representations of hyperbolic PDE?
I saw this paper: https://arxiv.org/abs/1306.2382
Chatterjee gives a representation of many solutions of the wave equation in terms of Brownian motion. I haven't seen much other than this. Is there ...
1
vote
0
answers
42
views
Estimates on density for Stokes equation
Consider a bounded, smooth domain $\Omega\subset \mathbb{R}^3$ and in there the Stokes equations
$\nabla p(\rho)-\Delta u=\rho f\\
\operatorname{div}(\rho u)=0\\
u\restriction_{\partial \Omega}=0$
...
1
vote
0
answers
128
views
Topology on function spaces for pointwise convergence
Suppose I have some collection of maps $T_\lambda: C^\infty(\Omega)\rightarrow C^\infty(\Omega)$ which are linear and parameterized by a parameter $\lambda >0$. (Perhaps more generally take $C^\...
3
votes
2
answers
868
views
Representing a nonlinear elliptic PDE as an energy minimization problem
I need to solve a PDE in 2D representing a (time-independent) nonlinear diffusion process. The unknown function is $\phi(x,y)$ and its gradients create fluxes $\vec J$ through a nonlinear relation:
$$\...
2
votes
1
answer
215
views
Continuity of solution of a parabolic PDE w.r.t. system parameters
If we have a system of PDE of the form:
$$\begin{cases} \dfrac{\partial y}{\partial t}(t,x)=D\Delta y+F(t,x,f(x),y) ,\ (t,x)\in (0,T)\times\Omega\\ \dfrac{\partial y}{\partial \nu}(t,x)=0,\ (t,x)\in (...
3
votes
0
answers
75
views
solutions of a pde smooth with respect to a parameter
I have a generic type question, but I will pose it with a specific example. Suppose $1<p<\frac{N+2}{N-2}$ and $B_1$ is the unit ball in $ R^N$ for $N \ge 3$. Consider the pde
$$-\Delta u(x) ...
14
votes
1
answer
372
views
Regularity of conformal maps
In order to define what it means for a map $f \colon \Omega \subseteq \mathbb R^n \to \mathbb R^n$ to be conformal, it is sufficient to require that $f$ is everywhere differentiable. Does conformality ...
3
votes
1
answer
140
views
Non-trivial examples of regular Lagrangian flow in BV case
What is a concrete example of BV vector field $v$ with $\mathrm{div}\, v = 0$ that makes Ambrosio's theory of regular Lagrangian flow relevant?
With concrete I mean that we can compute the flow ...
2
votes
1
answer
200
views
Finite energy solution for Allen -Cahn equation
I am interested in the Allen-Cahn equation in $ R^N$ and one can consider the related energy functional
$$ E(u):= \frac{1}{2}\int_{R^N}| \nabla u(x)|^2 dx + \frac{1}{4} \int_{R^N} (u^2-1)^2dx.$$ ...
1
vote
1
answer
177
views
Log-concavity of function
Consider the function
$$f_{n}(x)=e^{-x^2}x^n.$$
My goal is to show that
$$ G(y):=\frac{(f_2*f_0)(y)}{(f_0*f_0)(y)}- \left(\frac{(f_1*f_0)(y) }{(f_0*f_0)(y)}\right)^2$$
is log-concave.
Let us ...
21
votes
1
answer
684
views
Non real eigenvalues for elliptic equations
I am looking for an example of a pure second order uniformly elliptic operator
$L=\sum_{i,j=1}^da_{ij}(x)D_{ij}$ in a bounded domain $\Omega$ (with Dirichlet boundary conditions, for example) having a ...
2
votes
1
answer
449
views
Some properties of fractional Dirichlet heat kernel
Let $\Omega\subset \mathbb{R}^n (n\geq 2)$ be a bounded open domain with smooth boundary $\partial\Omega$. Consider the fractional heat equation with Dirichlet boundary condition:
\begin{equation}
\...
1
vote
0
answers
78
views
Reference (foundamental sol. and grad estimate, etc.): a particular elliptic PDE
In $\mathbb{R}^d$, consider the following equation
$$\Delta u -x\cdot \nabla u = f $$
where $f$ can be $C^\infty$ and decay like $e^{-\frac{c|x|^2}{2}}$.
I would like to know fundamental sol. to this ...
1
vote
1
answer
158
views
Vlasov Poisson: linear momentum conservation [closed]
The 3-dimensional Vlassov -Poisson equation I am studying at university is
$$ \partial_t f (t,x,v) + v\cdot \nabla_x f (t,x,v) - \nabla_x \phi (t,x) \nabla_v f (t,x,v) =0,$$
where $$\Delta \phi = 4\pi\...
2
votes
1
answer
101
views
WKB expansion for NLS
We consider the equation (NLS)
\begin{eqnarray}\label{gnls}
i \epsilon\partial_t u^{\epsilon} + \frac{\epsilon^2}{2}\Delta_{\eta}u^{\epsilon} = \epsilon |u^{\epsilon}|^{2}u^{\epsilon}, \quad x \in \...
3
votes
1
answer
394
views
Courant nodal domain theorem for fractional Laplacian
Let $\lambda_k$ and $\varphi_k$ be the $k$-th eigenvalue and a corresponding eigenfunction of the fractional Laplacian in a bounded domain $\Omega \subset \mathbb{R}^N$, $N \geq 2$.
That is, $\...
4
votes
2
answers
357
views
Angle of analyticity of semigroup
Is there any known parabolic PDEs in the literature where the angle of analyticity of the associated semigroup is $<\pi/2$ ?
For example, the angle of heat semigroup in $L^2$ is exactly $=\pi/2$. ...
0
votes
0
answers
162
views
Help to understand a limit $\varepsilon\rightarrow 0$ computation on a fluid mechanic paper
In Córdoba and Gancedo - Contour dynamics of incompressible 3-D fluids in a porous medium with different densities (page 4) I read that if
$$ v (x_1,x_2,x_3,t)=-\frac{\rho_2-\rho_1}{4\pi} \...
1
vote
0
answers
87
views
Explanation for the energy method used here
I am reading a paper where the authors prove
$$ \frac{d}{dt} \Vert f \Vert_{H^4} (t)\leq C\Vert f \Vert^5_{H^4}(t) $$
Where $f=f(x,t)$ and $H^4$ stands for the usual Sobolev space. Using Gronwall's ...
2
votes
1
answer
420
views
Existence of weak solutions of a parabolic PDE
Assume that $\Omega\subset\mathbb{R}^n(n\geq3)$ is a bounded open set with smooth boundary, $\varphi\in H^1_0(\Omega)$, $F(t)$ is differentiable in $\mathbb{R}$ and $F'$ is bounded. Given a PDE
$$
\...
0
votes
1
answer
119
views
finding Morse index for the following functional
not sure if this meets the standards here in this forum. I was dealing with the following functional $I(u)=\frac{1}{p}\int_{\Omega}|\nabla u|^{p}dx-\frac{1}{q}\lambda\int_{\Omega}|u|^qdx$ for $p \geq ...