**1**

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113 views

### Bound for a certain integral expression

I am working to establish an estimate in $X^{s,b}$ spaces to prove local well-posedness of a certain equation, and I need to consider some sub-cases. In particular, I wish to show that the following ...

**1**

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**0**answers

110 views

### A parabolic PDE with Lipschitz nonlinearity, how to obtain well-posedness?

Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^n$ (or more generally a compact manifold). I'm interested in well-posedness (existence most importantly) of equations of the form
$$u_t(t) - ...

**1**

vote

**0**answers

93 views

### How to use, $(|u|^{2}u - |v|^{2}v)(s)= (u-v)|u|^{2}(s)+ v(|u|^{2}-|v|^{2}) (s)$; to prove contraction in a Banach space $C([0,T]; M^{p,1})$?

(May be this is very basic question for MO)
(For details or this question you may see the paper page no. 9, MR2506839, Local well-posedness of nonlinear dispersive equations on modulation spaces; ...

**1**

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**0**answers

78 views

### Regularity of weak solutions for a quasilinear problem

Theorem 6.2.7 in the book Nonlinear Analisis of Gasisnski and Papageorgiou states: If $u\in W^{1,p}(\Omega)\cap L^\infty(\Omega)$ and $\Delta_pu \in L^\infty(\Omega)$ then we have $u\in ...

**1**

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**0**answers

101 views

### Weak periodic solution of parabolic PDE

Take
$$
u_t(t) + A(t)u(t) = f(t),
$$
$$
u(0) = u(T),
$$
where $A$ is an linear elliptic operator and the first equation is an equality in $L^2(0,T;V^*)$ for $V \subset H \subset V^*$ Hilbert triple. ...

**1**

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**0**answers

118 views

### Half-wave group $e^{it\sqrt{-\Delta_g}}$ for large $t$

Consider the Laplace-Beltrami operator $\Delta_g$ on compact Riemannian manifold $(M,g)$, then $e^{it\sqrt{-\Delta_g}}f$ is the solution of the following Cauchy problem.
$$
...

**1**

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**0**answers

160 views

### Reverse Holder Inequality and the higher integrability of the gradient of a solution to Euler's equation for a certain functional

In Giaquinta-Giusti's (1978) paper "Nonlinear Elliptic Systems with Elliptic Growth" (thm 1.1) they consider the following system:
\begin{equation}
\sum_{i, j=1}^{n}\sum_{\alpha, ...

**1**

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**0**answers

133 views

### Comparison principle for partial differential equation with singular coefficients

How (or if) a comparison principle works in the case of equations
singular at some point? For example, I am analyzing a partial
differential equation
$$
...

**1**

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**0**answers

43 views

### Rational homogenous functions

I'm interested in the set $\mathcal{S}$ of rational functions $F \colon \mathbb{R}^3 \to \mathbb{R}$ verifying:
\begin{align}
\Delta F=0 \quad \text{et} \quad F(\lambda x)= \lambda^d F(x) \quad d \in ...

**1**

vote

**0**answers

59 views

### Existence of harmonic maps between loops

Given a Riemannian manifold $M$ and two smooth loops $\gamma_0, \gamma_1: S^1 \longrightarrow M$ in it, I am looking for maps $\phi: [0, T] \times S^1 \longrightarrow M$ which minimize the energy
...

**1**

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**0**answers

72 views

### Analyticity of one-dimensional PDE solutions with respect to the space variable

Let $n>1$ and $u$ be a solution of a linear PDE with constant coefficients
$$
u_t-\sum_{k=0}^n a_k \partial_x^k u=0,\quad a_k\in \mathbb C,\quad a_n\ne0,
$$
in some neighborhood of a point ...

**1**

vote

**0**answers

117 views

### Focusing NLS: $L^2$ convergence of a solution as $t\rightarrow +\infty$

Consider the cubic focusing non linear Schrodinger equation in dimension $n\geq 2$:
$$(iu_t+\Delta)=-|u|^2u\qquad u(0,x)=u_0(x)\in L^2(\mathbb{R}^n)$$
Can we find an initial data $u_0\neq 0$ such ...

**1**

vote

**0**answers

103 views

### localization of the $L^p$ variation for heat equation

I'm struggling with yet another question for the classical heat equation in the whole space $R^d$. This question seems basic at first sight, but I think it is nontrivial in the end so here it is.
The ...

**1**

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**0**answers

130 views

### Extension of solutions of PDE

Let $\Omega \subset \mathbb{R}^{2}$ be an open set such that $\mathbf{0} \in \Omega$. Let $A := \Omega \setminus (\{0\}\times \mathbb{R})$, that is, $A$ is $\Omega$ with the $y$-axis removed.
Let ...

**1**

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**0**answers

128 views

### How to pick out harmonics based on boundary conditions?

(..this is almost a continuation of my last question (which got closed!)...) Let me first rewrite one of the main results of this paper, http://calvino.polito.it/~camporesi/JMP94.pdf in a coordinate ...

**1**

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**0**answers

44 views

### 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 ...

**1**

vote

**0**answers

87 views

### Elliptic problem on half space; infinite boundary values; Liouville theorem

In a the study of a boundary value problem the following problem is arising:
$-\Delta v(x)= e^{v(x)}$ in $ R^N_+$
$v= - \infty$ $\qquad $ on $ \partial R^N_+$ $ \qquad $ $ v \le 0$ in $ R^N_+$.
...

**1**

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**0**answers

177 views

### local existence for a singular quasilinear parabolic equation

I'm considering the following type of PDE:
$u_{t}=u_{xx}+u_{x}+u_{x}^2+u_{x}^3+\frac{u_{x}}{x(1-x)}+\left(\frac{u_{x}}{x(1-x)}\right)^3$
with periodic boundary conditions $u_{x}(0,t)=u_{x}(1,t)=0$, ...

**1**

vote

**0**answers

62 views

### Kernel of perturbation of biharmonic operator

Suppose we have a linear fourth order operator defined on $\mathbb{R}^{2n}$ with $n\geq2$ of the form:
$$\mathcal{L}(f)=\Delta^{2}f+\sum_{i,j=1}^{2n}P_{ij}(x)\partial_{i}\partial_{j}f$$
with ...

**1**

vote

**0**answers

281 views

### A variation of Poisson's equation in cylindrical coordinates

Our team of undergraduate physicists are familiar with finding numerical approximations to the following Poisson-like PDE central to our plasma research in a torus:
$\nabla^2 V = \frac{f(V)}{R^2}$
...

**1**

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**0**answers

53 views

### Checking initial condition of PDE is satisfied in Galerkin method

I asked this question here: http://math.stackexchange.com/questions/416885/checking-initial-condition-of-pde-is-satisfied-in-galerkin-method
But I did not receive the solution so I post it here.
The ...

**1**

vote

**0**answers

80 views

### 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}} ...

**1**

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**0**answers

69 views

### 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 ...

**1**

vote

**0**answers

164 views

### Laplacian type operator on compact Lie group

Consider the operator $S = \sum_{i,j} X_{ij}^2$ on $L^2(SO(n+1))$, where $X_{ij}$ generates the rotation of the sphere $S^n$ in the $ij$-plane keeping the $(n - 2)$ complementary directions fixed. ...

**1**

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**0**answers

105 views

### optimal regularity for Laplace equation with inhomogeneous L^p Robin boundary condition

Consider the problem
$$-\Delta u = 0 \mbox{ in }\Omega,\qquad \partial_\nu u+\tau u=g\mbox{ on }\partial\Omega,$$
where $\Omega\subset R^n$ is a bounded $C^2$-domain, $\tau>0$ is a constant, and ...

**1**

vote

**0**answers

104 views

### (localized) L^2 norm of quasimode for Laplacian

Lately I've been thinking about the following distribution on the flat torus $\mathbb{T}^2$:
$u_k=\frac{1}{\sqrt{2\lfloor k^{0.99}\rfloor+1}}\sum_{|l|\leq ...

**1**

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**0**answers

89 views

### null controllability of linear wave equation

Consider the linear wave equation :
$$z_{tt}=\Delta z + k(x) z + h(t) , \; in \; \Omega\times (0,T)$$
Are there sufficient conditions on the functions $k(x)$ and $h(t)$ for which $(z,z_t)$ vanish ...

**1**

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**0**answers

111 views

### nodal lines in the dirichlet problem

In the Dirichlet problem if nodal lines do not touch $\partial\Omega$ (unit disk), what happens to the eigenvalues?
Thanks for help.

**1**

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**0**answers

87 views

### About definition of weak derivative in abstract PDE problems

I'm confused about weak derivative definition.
$u \in L^2(0,T;V)$ has weak derivative $u'\in L^2(0,T;V')$ iff
$$\int_0^T u(t)\varphi'(t) = -\int_0^T u'(t)\varphi(t)$$
holds for all $\varphi \in ...

**1**

vote

**0**answers

54 views

### 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 $t\to+\infty$ ...

**1**

vote

**0**answers

213 views

### Solving a PDE involving a mixed derivative for a partial derivative

Consider a PDE of the form
\begin{equation}
\frac{\partial^2u}{\partial p\partial t}=F\left(\frac{\partial u}{\partial p},u,p\right)
\end{equation}
or
\begin{equation}
\frac{\partial^2u}{\partial ...

**1**

vote

**0**answers

116 views

### Trace Inequality question

There is a result in a paper I am reading :
Let $\Omega$ be a bounded domain. For any $\epsilon > 0$, there is a constant $C(\epsilon)$ such that
$$\lVert n \times u\rVert_{H^{-1/2}(\partial ...

**1**

vote

**0**answers

164 views

### Galerkin method for existence for PDE with nonsymmetric bilinear form

Suppose we have a PDE
$$\langle u', v \rangle + a(u,v) = 0$$
where $a:V\times V \to \mathbb{R}$ is a bounded symmetric bilinear form, then if $u_0 \in V$ then $u \in L^2(0,T; V)$ with $u' \in ...

**1**

vote

**0**answers

176 views

### Stationary Phase and Propagation Speed (Reference)

I'm trying to understand how one can make precise statements about propagation speed for various (linear and nonlinear) PDEs (in particular, ones with infinite propagation speed) and what, if ...

**1**

vote

**0**answers

70 views

### 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( ...

**1**

vote

**0**answers

157 views

### multivalued solution of a equation

Definition: A scalar k-th order differential equation on a smooth manifold $M$ ,
is $F(x,v,\frac{\partial {^\left | \sigma \right |}v}{\partial x^\sigma })=0 $
for $\left | \sigma \right |\leqslant ...

**1**

vote

**0**answers

115 views

### Spectrum of Combinatorial Laplacian

The spectrum of the combinatorial laplacian is well understood for a square lattice. What about for other lattices?
In particular:
Let $ f: \mathbb{Z}^2 \rightarrow \mathbb{R} $. The usual ...

**1**

vote

**0**answers

162 views

### Reference request: Anisotropic Sobolev spaces

Hello,
I am interested in what is known about anisotropic Sobolev spaces, by which I mean spaces of functions satisfying
$ \| f \|_p < \infty, \|Df \|_q < \infty, $
where $p \ne q$ (as ...

**1**

vote

**0**answers

91 views

### 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 $\Omega=\mathbb{R}^3-B_1(0)$, ...

**1**

vote

**0**answers

149 views

### two polynomial equations

Let $f:\mathbb R^2\rightarrow\mathbb R$ be a smooth function such that for every point $(x,y)\in\mathbb R^2$ the system
$$f_{11}+2tf_{12}+t^2f_{22}=0$$
$$f_{111}+3tf_{112}+3t^2f_{122}+t^3f_{222}=0$$
...

**1**

vote

**0**answers

132 views

### Weyl quantization and convexity

Let $C$ be a convex subset of $\mathbb R^{2n}$ and $\mathbf 1_C$ be the characteristic function of $C$. Is it true that
$$\forall u\in\mathscr S(\mathbb R^n),\quad
\langle\mathbf ...

**1**

vote

**0**answers

126 views

### 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 ...

**1**

vote

**0**answers

276 views

### Relation between interpolation spaces and besov spaces

Consider the following two norms:
The interpolation norm:
1) $\|u; [L_2,\dot H_1^{\infty}]_{1/3,\infty}\| := \sup_{s > 0} \inf_{u=u_0+u_1} \frac{\|u_0\|_{L^2}}{s^{1/2}} + s \|\partial_x ...

**1**

vote

**0**answers

57 views

### When can a perturbation be treated as a regular perturbation?

I am working with cauchy problem of the form
$$ ( - \partial_t + A^\delta) u^\delta = 0 , \qquad u^\delta(0,x) = h(x), $$
where the domain of $u^\delta$ is $[0,\infty) \times \mathbb{R}$. The ...

**1**

vote

**0**answers

226 views

### Convolution Estimates on a Smooth Manifold

Suppose $f,g$ are $a$-Hoelder continuous real-valued functions on some domain $\Omega \subset \mathbb{R}^n$ satisfying
$$
\|f\|_{C^{0,a}(\bar\Omega)},\|g\|_{C^{0,a}(\bar\Omega)}<\infty.
$$
Then ...

**1**

vote

**0**answers

101 views

### 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 ...

**1**

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**0**answers

169 views

### Pitfalls when generalizing the heat kernel of a Riemannian metric

Suppose $M$ is a Riemannian manifold with some compact quotient under isometries.
Associated with the Riemannian metric one has the Laplace-Beltrami operator $\Delta$ and the heat kernel $p(t,x,y)$ ...

**1**

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**0**answers

52 views

### elliptic system; bounds on $v$ when $u$ is small

I am interested in the following system
$-\Delta u = f(u,v) $
$-\Delta v = g(u,v)$ in $ \Omega$ a bounded domain in $ R^N$ with $ u=v=0$ on the boundary.
The solutions are smooth and positive. ...

**1**

vote

**0**answers

115 views

### 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:
...

**1**

vote

**0**answers

172 views

### 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 ...