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200 | Some non-analytic-hypoelliptic sums of squares of vector fields | math.AP | Certain second-order partial differential operators, which are expressed as
sums of squares of real-analytic vector fields in $\Bbb R^3$ and which are well
known to be $C^\infty$ hypoelliptic, fail to be analytic hypoelliptic. | math |
201 | A steepest descent method for oscillatory Riemann-Hilbert problems | math.AP | In this announcement we present a general and new approach to analyzing the
asymptotics of oscillatory Riemann-Hilbert problems. Such problems arise, in
particular, in evaluating the long-time behavior of nonlinear wave equations
solvable by the inverse scattering method. We will restrict ourselves here
exclusively to ... | math |
202 | Semilinear wave equations | math.AP | We survey existence and regularity results for semi-linear wave equations. In
particular, we review the recent regularity results for the $u^5$-Klein Gordon
equation by Grillakis and this author and give a self-contained, slightly
simplified proof. | math |
203 | A sharp pointwise bound for functions with $L^2$-Laplacians on arbitrary domains and its applications | math.AP | For all functions on an arbitrary open set $\Omega\subset\R^3$ with zero
boundary values, we prove the optimal bound \[ \sup_{\Omega}|u| \leq
(2\pi)^{-1/2} \left(\int_{\Omega}|\nabla u|^2 \,dx\, \int_{\Omega}|\Delta u|^2
\,dx\right)^{1/4}. \] The method of proof is elementary and admits
generalizations. The inequality ... | math |
204 | user's guide to viscosity solutions of second order partial differential equations | math.AP | The notion of viscosity solutions of scalar fully nonlinear partial
differential equations of second order provides a framework in which startling
comparison and uniqueness theorems, existence theorems, and theorems about
continuous dependence may now be proved by very efficient and striking
arguments. The range of imp... | math |
205 | Smooth static solutions of the Einstein-Yang/Mills equation | math.AP | We consider the Einstein/Yang-Mills equations in $3+1$ space time dimensions
with $\SU(2)$ gauge group and prove rigorously the existence of a globally
defined smooth static solution. We show that the associated Einstein metric is
asymptotically flat and the total mass is finite. Thus, for non-abelian gauge
fields the ... | math |
206 | A new result for the porous medium equation derived from the Ricci flow | math.AP | Given $\Bbb R^2, $ with a ``good'' complete metric, we show that the unique
solution of the Ricci flow approaches a soliton at time infinity. Solitons are
solutions of the Ricci flow, which move only by diffeomorphism. The Ricci flow
on $\Bbb R^2$ is the limiting case of the porous medium equation when $m$ is
zero. The... | math |
207 | A Formula for Finding a Potential from Nodal Lines | math.AP | In this announcement we consider an eigenvalue problem which arises in the
study of rectangular membranes. The mathematical model is an elliptic equation,
in potential form, with Dirichlet boundary conditions. We have shown that the
potential is uniquely determined, up to an additive constant, by a subset of
the nodal ... | math |
208 | Local Solvability For a Class of Partial Differential Operators With Double Characteristics | math.AP | A necessary and sufficient condition for local solvability is presented for
the linear partial differential operators $-X^2-Y^2+ia(x)[X,Y]$ in $\bold
R^3=\{(x,y,t)\}$, where $X=\partial_x,\; Y=\partial_y+x^k\partial_t$, and $a\in
C^{\infty}(\bold R^1)$ is real valued, for each positive integer $k$. | math |
209 | Global Irregularity For Degenerate Elliptic Operators | math.AP | Examples are given of degenerate elliptic operators on smooth, compact
manifolds that are not globally regular in $C^\infty$. These operators
degenerate only in a rather mild fashion. Certain weak regularity results are
proved, and an interpretation of global irregularity in terms of the associated
heat semigroup is gi... | math |
210 | Infinite dimensional families of locally nonsolvable partial differential operators | math.AP | Local solvability is analyzed for natural families of partial differential
operators having double characteristics. In some families the set of all
operators that are not locally solvable is shown to have both infinite
dimension and infinite codimension. | math |
211 | The monodromy matrix for a family of almost periodic Schrödinger equations in the adiabatic case | math.AP | This work is devoted to the study of a family of almost periodic
one-dimensional Schr\"odinger equations. We define a monodromy matrix for this
family. We study the asymptotic behavior of this matrix in the adiabatic case.
Therefore, w develop a complex WKB method for adiabatic perturbations of
periodic Schr\"odinger e... | math |
212 | Low regularity semi-linear wave equations | math.AP | We prove local well-posedness results for the semi-linear wave equation for
data in $H^\gamma$, $0 < \gamma < \frac{n-3}{2(n-1)}$, extending the previously
known results for this problem. The improvement comes from an introduction of a
two-scale Lebesgue space $X^{r,p}_k$. | math |
213 | Variational evolution problems and nonlocal geometric motion | math.AP | We consider two variational evolution problems related to Monge-Kantorovich
mass transfer. These problems provide models for collapsing sandpiles and for
compression molding. We prove the following connection between these problems
and nonlocal geometric curvature motion: The distance functions to surfaces
moving accor... | math |
214 | Complete Integrability of Completely Integrable Systems | math.AP | The question of complete integrability of evolution equations associated to
$n\times n$ first order isospectral operators is investigated using the inverse
scattering method. It is shown that for $n>2$, e.g. for the three-wave
interaction, additional (nonlinear) pointwise flows are necessary for the
assertion of comple... | math |
215 | Action-Angle variables for the Gel'fand-Dikii flows | math.AP | Using the scattering transform for $n^{th}$ order linear scalar operators,
the Poisson bracket found by Gel'fand and Dikii, which generalizes the Gardner
Poisson bracket for the KdV hierarchy, is computed on the scattering side.
Action-angle variables are then constructed. Using this, complete integrability
is demonstr... | math |
216 | Transition operators of diffusions reduce zero-crossing | math.AP | If $u(t,x)$ is a solution of a one--dimensional, parabolic, second--order,
linear partial differential equation (PDE), then it is known that, under
suitable conditions, the number of zero--crossings of the function $u(t,\cdot)$
decreases (that is, does not increase) as time $t$ increases. Such theorems
have application... | math |
217 | On a singular limit problem for nonlinear Maxwell's equations | math.AP | In this paper we study the following nonlinear Maxwell's equations \\
$\varepsilon \E_{t}+\sigma(x,|\E|)\E= \g \vh +\F,\, \vh_{t}+\g \E=0$, where
$\sigma(x,s)$ is a monotone graph of $s$. It is shown that the system has a
unique weak solution. Moreover, the limit of the solution as
$\varepsilon\rightarrow 0$ converges ... | math |
218 | Forced symmetry-breaking via boundary conditions | math.AP | We study impact of a forced symmetry-breaking in boundary conditions on the
bifurcation scenario of a semilinear elliptic partial differential equation. We
show that for the square domain the orthogonality of eigenfunctions of the
Laplacian may compensate partially the loss of symmetries in the boundary
conditions and ... | math |
219 | On a class of linearizable Monge-Ampère equations | math.AP | Monge-Amp\`ere equations of the form, $u_{xx}u_{yy}-u_{xy}^2=F(u,u_x,u_y)$
arise in many areas of fluid and solid mechanics. Here it is shown that in the
special case $F=u_y^4f(u, u_x/u_y)$, where $f$ denotes an arbitrary function,
the Monge-Amp\`ere equation can be linearized by using a sequence of Amp\`ere,
point, Le... | math |
220 | A nonlinear transformation of the dispersive long wave equations in (2+1) dimensions and its applications | math.AP | A nonlinear transformation of the dispersive long wave equations in (2+1)
dimensions is derived by using the homogeneous balance method. With the aid of
the transformation given here, exact solutions of the equations are obtained. | math |
221 | On symmetries of KdV-like evolution equations | math.AP | The $x$-dependence of the symmetries of (1+1)-dimensional scalar
translationally invariant evolution equations is described. The sufficient
condition of (quasi)polynomiality in time $t$ of the symmetries of evolution
equations with constant separant is found. The general form of time dependence
of the symmetries of KdV... | math |
222 | Axisymmetric Solutions of the Euler Equations for Sub-Square Polytropic Gases | math.AP | We establish rigorously the existence of a three-parameter family of
self-similar,globally bounded, and continuous weak solutions in two space
dimensions to the compressible Euler equations with axisymmetry for gamma-law
polytropic gases with gamma between 1 and 2, including 1. The initial data of
these solutions have ... | math |
223 | Local and global well-posedness of wave maps on $\R^{1+1}$ for rough data | math.AP | We prove local and global existence from large, rough initial data for a wave
map between 1+1 dimensional Minkowski space and an analytic manifold. Included
here is global existence for large data in the scale-invariant norm $\dot
L^{1,1}$, and in the Sobolev spaces $H^s$ for $s > 3/4$. This builds on
previous work in ... | math |
224 | Application of the group-theoretical method to physical problems | math.AP | The concept of the theory of continuous groups of transformations has
attracted the attention of applied mathematicians and engineers to solve many
physical problems in the engineering sciences. Three applications are presented
in this paper. The first one is the problem of time-dependent vertical
temperature distribut... | math |
225 | Some homogenization and corrector results for nonlinear monotone operators | math.AP | This paper deals with the limit behaviour of the solutions of quasi-linear
equations of the form \ $\ds -\limfunc{div}\left(a\left(x, x/{\varepsilon
_h},Du_h\right)\right)=f_h$ on $\Omega $ with Dirichlet boundary conditions.
The sequence $(\varepsilon _h)$ tends to $0$ and the map $a(x,y,\xi )$ is
periodic in $y$, mon... | math |
226 | Similarity reductions for a nonlinear diffusion equation | math.AP | Similarity reductions and new exact solutions are obtained for a nonlinear
diffusion equation. These are obtained by using the classical symmetry group
and reducing the partial differential equation to various ordinary differential
equations. For the equations so obtained, first integrals are deduced which
consequently... | math |
227 | The Dirichlet problem for superdegenerate differential operators | math.AP | Let $L$ be an infinitely degenerate second-order linear operator defined on a
bounded smooth Euclidean domain. Under weaker conditions than those of
H\"ormander, we show that the Dirichlet problem associated with $L$ has a
unique smooth classical solution. The proof uses the Malliavin calculus. At
present, there appear... | math |
228 | From the Polya-Szego symmetrization inequality for Dirichlet integrals to comparison theorems for p.d.e.'s on manifolds | math.AP | A method for proving symmetrization inequalities for some elliptic p.d.e.'s
on manifolds equipped with appropriate isoperimetric inequalities is outlined.
The method is based on a modification of an approach of Baernstein. The
question of what is the most general result that can be proved in this way is
still open, and... | math |
229 | On the analytical approach to the N-fold Bäcklund transformation of Davey-Stewartson equation | math.AP | N-fold B\"acklund transformation for the Davey-Stewartson equation is
constructed by using the analytic structure of the Lax eigenfunction in the
complex eigenvalue plane. Explicit formulae can be obtained for a specified
value of N. Lastly it is shown how generalized soliton solutions are generated
from the trivial on... | math |
230 | Fundamental solution of the Volkov problem (characteristic representation) | math.AP | The characteristic representation, or Goursat problem, for the
Klein-Fock-Gordon equation with Volkov interaction [1] is regarded. It is shown
that in this representation the explicit form of the Volkov propagator can be
obtained. Using the characteristic representation technique, the Schwinger
integral [2] in the Volk... | math |
231 | Differential constraints compatible with linearized equations | math.AP | Differential constraints compatible with the linearized equations of partial
differential equations are examined. Recursion operators are obtained by
integrating the differential constraints. | math |
232 | Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation | math.AP | The endpoint Strichartz estimates for the Schr\"odinger equation are known to
be false in two dimensions. However, if one averages the solution in $L^2$ in
the angular variable, we show that the homogeneous endpoint and the retarded
half-endpoint estimates hold, but the full retarded endpoint fails. In
particular, the ... | math |
233 | Ill-posedness for one-dimensional wave maps at the critical regularity | math.AP | We show that the wave map equation in $\R^{1+1}$ is in general ill-posed in
the critical space $\dot H^{1/2}$, and the Besov space $\dot B^{1/2,1}_2$. The
problem is attributed to the bad behaviour of the one-dimensional bilinear
expression $D^{-1}(f Dg)$ in these spaces. | math |
234 | Interactions of Andronov-Hopf and Bogdamov-Takens bifurcations | math.AP | A codimension-three bifurcation, characterized by a pair of purely imaginary
eigenvalues and a nonsemisimple double zero eigenvalue, arises in the study of
a pair of weakly coupled nonlinear oscillators with Z_2 + Z_2 symmetry. The
methodology is based on Arnold's ideas of versal deformations of matrices for
the linear... | math |
235 | Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation | math.AP | The problem we are concerned with is whether singularities form in finite
time in incompressible fluid flows. It is well known that the answer is ``no''
in the case of Euler and Navier-Stokes equations in dimension two. In dimension
three it is still an open problem for these equations.
In this paper we focus on a tw... | math |
236 | Large amplitude gravitational waves | math.AP | We derive an asymptotic solution of the Einstein field equations which
describes the propagation of a thin, large amplitude gravitational wave into a
curved space-time. The resulting equations have the same form as the colliding
plane wave equations without one of the usual constraint equations. | math |
237 | The structure of the solutions to semilinear equations at a critical exponent | math.AP | This paper is concerned with the structure of the solutions to subcritical
elliptic equations related to the Matukuma equation. In certain cases the
complete structure of the solution set is known, and is comparable to that of
the original Matukuma equation. Here we derive sufficient conditions for a more
complicated s... | math |
238 | Blowup of small data solutions for a quasilinear wave equation in two space dimensions | math.AP | For the quasilinear wave equation
\partial_t^2u - \Delta u = u_t u_{tt},
we analyze the long-time behavior of classical solutions with small (not
rotationally invariant) data. We give a complete asymptotic expansion of the
lifespan and describe the solution close to the blowup point. It turns out that
this solution... | math |
239 | The instability of naked singularities in the gravitational collapse of a scalar field | math.AP | One of the fundamental unanswered questions in the general theory of
relativity is whether ``naked'' singularities, that is singular events which
are visible from infinity, may form with positive probability in the process of
gravitational collapse. The conjecture that the answer to this question is in
the negative has... | math |
240 | Variational methods for solving nonlinear boundary problems of statics of hyper-elastic membranes | math.AP | A number of important results of studying large deformations of hyper-elastic
shells are obtained using discrete methods of mathematical physics. In the
present paper, using the variational method for solving nonlinear boundary
problems of statics of hyper-elastic membranes under the regular hydrostatic
load, we invest... | math |
241 | Symmetries of a class of nonlinear fourth order partial differential equations | math.AP | In this paper we study symmetry reductions of a class of nonlinear fourth
order partial differential equations \be u_{tt} = \left(\kappa u + \gamma
u^2\right)_{xx} + u u_{xxxx} +\mu u_{xxtt}+\alpha u_x u_{xxx} + \beta u_{xx}^2,
\ee where $\alpha$, $\beta$, $\gamma$, $\kappa$ and $\mu$ are constants. This
equation may b... | math |
242 | On the Grushin operator and hyperbolic symmetry | math.AP | Complexity of geometric symmetry for differential operators with mixed
homogeniety is examined here. Sharp Sobolev estimates are calculated for the
Grushin operator in low dimensions using hyperbolic symmetry and conformal
geometry. | math |
243 | Weak Convergence and Deterministic Approach to Turbulent Diffusion | math.AP | The purpose of this contribution is to show that some of the basic ideas of
turbulence can be addressed in a deterministic setting instead of introducing
random realizations of the fluid. Weak limits of oscillating sequences of
solutions are considered and along the same line the Wigner transform replaces
the Kolmogoro... | math |
244 | Explode-decay dromions in the non-isospectral Davey-Stewartson I (DSI) equation | math.AP | In this letter, we report the existence of a novel type of explode-decay
dromions, which are exponentially localized coherent structures whose amplitude
varies with time, through Hirota method for a nonisospectral Davey-Stewartson
equation I discussed recently by Jiang. Using suitable transformations, we also
point out... | math |
245 | Semiclassical solutions of the nonlinear Schrödinger equation | math.AP | A concept of semiclassically concentrated solutions is formulated for the
multidimensional nonlinear Schr\"odinger equation (NLSE) with an external
field. These solutions are considered as multidimensional solitary waves. The
center of mass of such a solution is shown to move along with the
bicharacteristics of the bas... | math |
246 | Lectures on Pseudo-differential Operators | math.AP | This lecture notes cover a Part III (first year graduate) course that was
given at Cambridge University over several years on pseudo-differential
operators. The calculus on manifolds is developed and applied to prove
propagation of singularities and the Hodge decomposition theorem. Problems are
included. | math |
247 | Doubling properties for second order parabolic equations | math.AP | We prove the doubling property of L-caloric measure corresponding to the
second order parabolic equation in the whole space and in Lipschitz domains.
For parabolic equations in the divergence form, a weaker form of the doubling
property follows easily from a recent result, the backward Harnack inequality,
and known est... | math |
248 | Versal deformations of a Dirac type differential operator | math.AP | If we are given a smooth differential operator in the variable $x\in {\mathbb
R}/2\pi {\mathbb Z},$ its normal form, as is well known, is the simplest form
obtainable by means of the $\mbox{Diff}(S^1)$-group action on the space of all
such operators. A versal deformation of this operator is a normal form for some
param... | math |
249 | The Thual-Fauve pulse: skew stabilization | math.AP | It is possible to choose the parameters of a real quintic Ginzburg-Landau
equation so that it possesses localized pulse-like solutions; Thual and Fauve
have observed numerically that these pulses are stabilized by perturbations
destroying the gradient structure of the real equation. For parameters such
that the real pa... | math |
250 | Semiclassical estimates in asymptotically Euclidean scattering | math.AP | In this note we obtain semiclassical resolvent estimates for non-trapping
long range perturbations of the Laplacian on asymptotically Euclidean
manifolds. Our proof is based on a positive commutator argument which differs
from Mourre-type estimates by making the commutant also positive. The resolvent
estimates, includi... | math |
251 | On elliptic operator pencils with general boundary conditions | math.AP | In this paper operator pencils $A(x,D,\lambda)$ are investigated which depend
polynomially on the parameter $\lambda$ and act on a manifold with boundary.
The operator A is assumed to satisfy the condition of N-ellipticity with
parameter which is an ellipticity condition formulated with the use of the
Newton polygon. W... | math |
252 | Continuous and discrete transformations of a one-dimensional porous medium equation | math.AP | We consider the one-dimensional porous medium equation $u_t=\left (u^nu_x
\right )_x+\frac{\mu}{x}u^nu_x$. We derive point transformations of a general
class that map this equation into itself or into equations of a similar class.
In some cases this porous medium equation is connected with well known
equations. With th... | math |
253 | Geometry of Stationary Sets for the Wave Equation in R^n, The Case of Finitely Supported Initial Data, An Announcement | math.AP | We consider the Cauchy problem for the wave equation in the whole space, R^n,
with initial data which are distributions supported on finite sets. The main
result is a precise description of the geometry of the sets of stationary
points of the solutions to the wave equation. | math |
254 | An inverse boundary value problem for harmonic differential forms | math.AP | We show that the full symbol of the Dirichlet to Neumann map of the k-form
Laplace's equation on a Riemannian manifold (of dimension greater than 2) with
boundary determines the full Taylor series, at the boundary, of the metric.
This extends the result of Lee and Uhlmann for the case $k=0$. The proof avoids
the comput... | math |
255 | Global Strichartz estimates for nontrapping perturbations of the Laplacian | math.AP | The authors prove global Strichartz estimates for compact perturbations of
the wave operator in odd dimensions when a non-trapping assumption is
satisfied. | math |
256 | Weighted Strichartz estimates and global existence for semilinear wave equations | math.AP | We prove certain weighted Strichartz estimates and use these to prove a sharp
theorem for global existence of small amplitude solutions of $\square u=
|u|^p$, thus verifying the so-called "Strauss conjecture". | math |
257 | Null form estimates for (1/2,1/2) symbols and local existence for a quasilinear Dirichlet-wave equation | math.AP | The authors show that bilinear estimates for null forms hold for
Dirichlet-wave equations outside of convex obstacle. This generalizes results
for the Euclidean case of Klainerman and Machedon, and of Sogge for the
variable coefficient boundaryless case. The estimates are used to prove a local
existence theorem for sem... | math |
258 | Statistical Mechanics of the Periodic Camassa-Holm Equation | math.AP | The paper has been withdrawn | math |
259 | An Analysis on the Shape Equation for Biconcave Axisymmetric Vesicles | math.AP | We study the conditions on the physical parameters in the Helfrich bending
energy of lipid bilayer vesicles. Among embedded surfaces with a biconcave
axisymmetric shape, the variation equation is analyzed in detail. This leads to
simple conditions which guarantee the solution the information about the
geometry. | math |
260 | The resolvent for Laplace-type operators on asymptotically conic spaces | math.AP | Let X be a compact manifold with boundary, and g a scattering metric on X,
which may be either of short range or `gravitational' long range type. Thus, g
gives X the geometric structure of a complete manifold with an asymptotically
conic end. Let H be an operator of the form $H = \Delta + P$, where $\Delta$ is
the Lapl... | math |
261 | On the L^2-stability and L^2 controllability of steady flows of an ideal incompressible fluid | math.AP | The author studies the flows of an ideal incompressible fluid in a
2-dimensional domain, and in particular questions of instability and
controllability. | math |
262 | Navier-Stokes equations and fluid turbulence | math.AP | An Eulerian-Lagrangian approach to incompressible fluids that is convenient
for both analysis and physics is presented. Bounds on burning rates in
combustion and heat transfer in convection are discussed, as well as results
concerning spectra. | math |
263 | Existence and homogenization of the Rayleigh-Bénard problem | math.AP | The Navier-Stokes equation driven by heat conduction is studied. As a
prototype we consider Rayleigh-B\'enard convection, in the Boussinesq
approximation. Under a large aspect ratio assumption, which is the case in
Rayleigh-B\'enard experiments with Prandtl number close to one, we prove the
existence of a global strong... | math |
264 | Multilinear weighted convolution of $L^2$ functions, and applications to non-linear dispersive equations | math.AP | The $X^{s,b}$ spaces, as used by Beals, Bourgain, Kenig-Ponce-Vega,
Klainerman-Machedon and others, are fundamental tools to study the
low-regularity behaviour of non-linear dispersive equations. It is of
particular interest to obtain bilinear or multilinear estimates involving these
spaces. By Plancherel's theorem and... | math |
265 | Global well-posedness below energy space for the 1D Zakharov system | math.AP | The Cauchy problem for the 1-dimensional Zakharov system is shown to be
globally well-posed for large data which not necessarily have finite energy.
The proof combines the local well-posedness result of Ginibre, Tsutsumi, Velo
and a general method introduced by Bourgain to prove a similar result for
nonlinear Schr\"odi... | math |
266 | Local well-posedness of the Yang-Mills equation in the Temporal Gauge below the energy norm | math.AP | We show that the Yang-Mills equation in three dimensions is locally
well-posed in the Temporal gauge for initial data in H^s x H^{s-1} for s > 3/4,
if the norm of the initial data is sufficiently small. The main new ingredients
are a splitting of the connection into curl-free and div-free components, and
some product e... | math |
267 | Nonresonance and global existence of prestressed nonlinear elastic waves | math.AP | The nonlinear hyperbolic system of pde's governing the evolution of the
deformation of isotropic hyperelastic materials is considered. In the absence
of boundaries and with an additional nonresonance or null condition, the system
has global smooth solutions starting close to a one-parameter family of
homogeneous dilati... | math |
268 | An Eulerian-Lagrangian approach to the Navier-Stokes equations | math.AP | This work presents an approach to the Navier-Stokes equations that is phrased
in unbiased Eulerian coordinates, yet describes objects that have Lagrangian
significance: particle paths, their dispersion and diffusion. The commutator
between Lagrangian and Eulerian derivatives plays an important role in the
Navier-Stokes... | math |
269 | Differential operators on equivariant vector bundles over symmetric spaces | math.AP | Generalizing the algebra of motion-invariant differential operators on a
symmetric space we study invariant operators on equivariant vector bundles. We
show that the eigenequation is equivalent to the corresponding eigenequation
with respect to the larger algebra of all invariant operators. We compute the
possible eige... | math |
270 | Invariant measures for Burgers equation with stochastic forcing | math.AP | In this paper we study the following Burgers equation
du/dt + d/dx (u^2/2) = epsilon d^2u/dx^2 + f(x,t)
where f(x,t)=dF/dx(x,t) is a random forcing function, which is periodic in x
and white noise in t. We prove the existence and uniqueness of an invariant
measure by establishing a ``one force, one solution'' princ... | math |
271 | Value Functions for Bolza Problems with Discontinuous Lagrangians and Hamilton-Jacobi Inequalities | math.AP | We investigate the value function of the Bolza problem of the Calculus of
Variations $$
V (t,x)=\inf \{\int_{0}^{t} L (y(s),y'(s))ds + \phi(y(t)) : y \in W^{1,1}
(0,t; R^n) ; y(0)=x \}, $$ with a lower semicontinuous Lagrangian $L$ and a
final cost $\phi$, and show that it is locally Lipschitz for $t>0$ whenever $L$
... | math |
272 | Uniqueness of solutions to Hamilton-Jacobi equations arising in the Calculus of Variations | math.AP | We prove the uniqueness of the viscosity solution to the Hamilton-Jacobi
equation associated with a Bolza problem of the Calculus of Variations,
assuming that the Lagrangian is autonomous, continuous, superlinear, and
satisfies the usual convexity hypothesis. Under the same assumptions we prove
also the uniqueness, in ... | math |
273 | Stability of $L^\infty$ solutions for hyperbolic systems with coinciding shocks and rarefactions | math.AP | We consider a hyperbolic system of conservation laws u_t + f(u)_x = 0 and
u(0,\cdot) = u_0, where each characteristic field is either linearly degenerate
or genuinely nonlinear. Under the assumption of coinciding shock and
rarefaction curves and the existence of a set of Riemann coordinates $w$, we
prove that there exi... | math |
274 | Sharp L1 stability estimates for hyperbolic conservation laws | math.AP | In this paper, we introduce a generalization of Liu-Yang's weighted norm to
linear and to nonlinear hyperbolic equations. Extending a result by Hu and
LeFloch for piecewise constant solutions, we establish sharp L1 continuous
dependence estimates for general solutions of bounded variation. Two different
strategies are ... | math |
275 | Existence of minimal H-bubbles | math.AP | We prove existence of S^2-type parametric surfaces in R^3 having prescribed
noncostant mean curvature. | math |
276 | On the Cauchy- and periodic boundary value problem for a certain class of derivative nonlinear Schroedinger equations | math.AP | The Cauchy- and periodic boundary value problem for the nonlinear
Schroedinger equations in $n$ space dimensions [u_t - i\Delta u = (\nabla
\bar{u})^{\beta}, |\beta|=m \ge 2, u(0)=u_0 \in H^{s+1}_x] is shown to be
locally well posed for $s > s_c := \frac{n}{2} - \frac{1}{m-1}$, $s \ge 0$. In
the special case of space d... | math |
277 | Smooth global Lagrangian flow for the 2D Euler and second-grade fluid equations | math.AP | We present a very simple proof of the global existence of a $C^\infty$
Lagrangian flow map for the 2D Euler and second-grade fluid equations (on a
compact Riemannian manifold with boundary) which has $C^\infty$ dependence on
initial data $u_0$ in the class of $H^s$ divergence-free vector fields for
$s>2$. | math |
278 | On the Stability of the Standard Riemann Semigroup | math.AP | We consider the dependence of the entropic solution of a hyperbolic system of
conservation laws \[ \{\{array}{c} u_t + f(u)_x = 0 u(0,\cdot) = u_0 \{array}
\] on the flux function f. We prove that the solution in Lipschitz continuous
w.r.t.~the $C^0$ norm of the derivative of the perturbation of f. We apply this
result... | math |
279 | Asymptotic solitons of the Johnson equation | math.AP | We prove the existence of non-decaying real solutions of the Johnson
equation, vanishing as $x\to+\infty$. We obtain asymptotic formulas as
$t\to\infty$ for the solutions in the form of an infinite series of asymptotic
solitons with curved lines of constant phase and varying amplitude and width. | math |
280 | Correctors for the homogenization of monotone parabolic operators | math.AP | In the homogenization of monotone parabolic partial differential equations
with oscillations in both the space and time variables the gradients converges
only weakly in $L^p$. In the present paper we construct a family of correctors,
such that, up to a remainder which converges to zero strongly in $L^p$, we
obtain stro... | math |
281 | A note on sigular limits to hyperbolic systems | math.AP | In this note we consider two different singular limits to hyperbolic system
of conservation laws, namely the standard backward schemes for non linear
semigroups and the semidiscrete scheme.
Under the assumption that the rarefaction curve of the corresponding
hyperbolic system are straight lines, we prove the stabilit... | math |
282 | On the support of solutions to the g-KdV equation | math.AP | We discuss the question whether solutions of the initial value problem for
the generalized KdV equation can have compact support at two different times. | math |
283 | Global existence for a quasilinear wave equation outside of star-shaped domains | math.AP | We establish global existence in 3+1 dimensions of small-amplitude solutions
of quasilinear Dirichlet-wave equations satisfying the null condition outside
of star-shapped obstacles. | math |
284 | Counting dimensions of L-harmonic functions | math.AP | In this article, we will consider second order uniformly elliptic operators
of divergence form defined on R^n with measurable coefficients. Mainly, we will
give estimates on the dimension of space of solutions that grow at most
polynomially of degree d. More precisely, in terms of a rectangular coordinate
system {x_1,.... | math |
285 | Regularity of a free boundary with application to the Pompeiu problem | math.AP | In the unit ball B(0,1), let $u$ and $\Omega$ (a domain in $\R$) solve the
following overdetermined problem: $$\Delta u =\chi_\Omega\quad \hbox{in}
B(0,1), \qquad 0 \in \partial \Omega, \qquad u=|\nabla u |=0 \quad \hbox{in}
B(0,1)\setminus \Omega,$$ where $\chi_\Omega$ denotes the characteristic
function, and the equa... | math |
286 | Global Existence for Systems of Nonlinear Wave Equations in 3D with Multiple Speeds | math.AP | Global smooth solutions to the initial value problem for systems of nonlinear
wave equations with multiple propagation speeds will be constructed in the case
of small initial data and nonlinearities satisfying the null condition. | math |
287 | Global regularity of wave maps I. Small critical Sobolev norm in high dimension | math.AP | We show that wave maps from Minkowski space $R^{1+n}$ to a sphere are
globally smooth if the initial data is smooth and has small norm in the
critical Sobolev space $\dot H^{n/2}$ in the high dimensional case $n \geq 5$.
A major difficulty, not present in the earlier results, is that the $\dot
H^{n/2}$ norm barely fail... | math |
288 | On the existence of nontrivial solutions for a nonlinear equation relative to a measure-valued Lagrangian on homogeneous spaces | math.AP | We prove the existence of a non-trivial solution for a nonlinear equation
related to a measure-valued Lagrangian. The result is based on a compact
embedding theorem of the Lagrangian domain and on the application of the
Mountain Pass Theorem joined to a Palais-Smale condition. | math |
289 | Singularities and the wave equation on conic spaces | math.AP | Let $X$ be a manifold with boundary, endowed with a metric with conic
singularities at the boundary components of $X$. Let $u$ be a solution to the
wave equation on $\mathbb{R} \times X$. When a singularity of $u$ strikes a
cone point of $X$, it undergoes a mixture of diffractive spreading and
geometric propagation. | math |
290 | A momotonicity approach to nonlinear Dirichlet problems in perforated domains | math.AP | We study the asymptotic behaviour of solutions to Dirichlet problems in
perforated domains for nonlinear elliptic equations associated with monotone
operators. The main difference with respect to the previous papers on this
subject is that no uniformity is assumed in the monotonicity condition. Under a
very general hyp... | math |
291 | Dromion perturbation for the Davey-Stewartson-1 equations | math.AP | The perturbation of the dromion of the Davey-Stewartson-1 equation is studied
over the large time. | math |
292 | Some Remarks on the Fucik Spectrum of the p-Laplacian and Critical Groups | math.AP | We compute critical groups of variational functionals arising from
quasilinear elliptic boundary value problems with jumping nonlinearities, when
the asymptotic limits of the equation lie in various regions of the plane that
are separated by certain curves of the Fucik spectrum. As an application some
existence and mul... | math |
293 | Some local wellposedness results for nonlinear Schroedinger equations below L^2 | math.AP | We prove some local (in time) wellposedness results for nonlinear
Schroedinger equations with rough data, that is, the initial value belongs to
some Sobolev space of negative index. The proof uses the Fourier restriction
norm method. | math |
294 | Global regularity of wave maps II. Small energy in two dimensions | math.AP | We show that wave maps from Minkowski space $\R^{1+n}$ to a sphere $S^{m-1}$
are globally smooth if the initial data is smooth and has small norm in the
critical Sobolev space $\dot H^{n/2}$, in all dimensions $n \geq 2$. This
generalizes the results in the prequel [math.AP/0010068] of this paper, which
addressed the h... | math |
295 | Multiplicity results for some nonlinear Schroedinger equations with potentials | math.AP | We prove some multiplicity results for a nonlinear equation of Schroedinger
type with potential functions | math |
296 | Fredholm Properties of Elliptic Operators on $\R^n$ | math.AP | We study the Fredholm properties of a general class of elliptic differential
operators on $\R^n$. These results are expressed in terms of a class of
weighted function spaces, which can be locally modeled on a wide variety of
standard function spaces, and a related spectral pencil problem on the sphere,
which is defined... | math |
297 | On Solutions of Three Quasi-Geostrophic Models | math.AP | We consider the 2D quasi-geostrophic model and its two different
regularizations. Global regularity results are established for the regularized
models with subcritical or critical indices. The proof of Onsager's conjecture
concerning weak solutions of the 3D Euler equations and the notion of
dissipative solutions of Du... | math |
298 | Stability of travelling-wave solutions for reaction-diffusion-convection systems | math.AP | We are concerned with the asymptotic behaviour of classical solutions of
systems of the form u_t = Au_xx + f(u, u_x), x in R, t>0, u(x,t) a vector in
RN, with u(x,0)= U(x), where A is a positive-definite diagonal matrix and f is
a 'bistable' nonlinearity satisfying conditions which guarantee the existence
of a comparis... | math |
299 | A model for the quasi-static growth of a brittle fracture: existence and approximation results | math.AP | We give a precise mathematical formulation of a variational model for the
irreversible quasi-static evolution of a brittle fracture proposed by G.A.
Francfort and J.-J. Marigo, and based on Griffith's theory of crack growth. In
the two-dimensional case we prove an existence result for the quasi-static
evolution and sho... | math |
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